Wikiversity talk:Main Page

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{{attention}} To request an edit to the [[Wikiversity:Page protection|protected]] Main Page, add {{tl|editprotected}} to your request. Such requests should either be obvious or uncontroversial, or be discussed to show consensus, so please do not make vague requests here. If possible, describe exactly what changes should be made so that any custodian can quickly satisfy the request.&lt;br&gt;
{{attention}} To raise general topics about [[Wikiversity]], make general suggestions about Wikiversity, to ask questions, or to talk about anything else of a general nature, use the [[Wikiversity:Colloquium|Colloquium]].&lt;br&gt;
{{attention}} To discuss the structure, appearance, etc. of the [[Wikiversity:Main Page|Main Page]], go to the [[Wikiversity:Main page learning project]] and the [[Wikiversity talk:Main page learning project|talk page for the main page learning project]].
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'''''If you wish to post something below, go ahead. It's a talk page. But you are more likely to get a response by going to the [[Wikiversity:Colloquium|Colloquium]], which is where the main talking at Wikiversity goes on! See you there.'''''

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== The Wikiversity:Main page learning project ==

The [[Wikiversity:Main page learning project]] was launched after the redesign of the main page in December 2007. The [[Wikiversity:Main page learning project]] has as its goal &quot;the promotion of responsible involvement of the Wikiversity community in an efficient, productive, open and inclusive maintenance of the Wikiversity main page as a flagship of the activity and values of the Wikiversity community&quot;. If you would like to get involved in the design of the main page, this is where to go. 

If you have general comments about the main page, but you don't especially want to get involved in the main page project, then you can also leave comments on the [[Wikiversity_talk:Main page learning project|talk page for the main page learning project]].

:I've suggested that it might be time to retire the &quot;quote of the day&quot; project and remove the quotes from the Main Page. See: [[Wikiversity talk:Main page learning project/QOTD]]. It might also be appropriate to deprecate the inactive [[Wikiversity:Main page learning project]] and archive it. Thoughts? --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 23:37, 29 November 2019 (UTC)

== add new language university ==
Now that Chinese Wikiversity is created, please add a cross-wiki link to it. --[[User:WQL|WQL]] ([[User talk:WQL|discuss]] • [[Special:Contributions/WQL|contribs]]) 12:52, 12 August 2018 (UTC)

:{{Done}} -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 14:29, 12 August 2018 (UTC)

== Edit request from 204.234.101.112, 14 February 2019 ==

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[[Special:Contributions/204.234.101.112|204.234.101.112]] ([[User talk:204.234.101.112|discuss]]) 21:17, 14 February 2019 (UTC)

:{{Not done}} Empty request -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 01:11, 15 February 2019 (UTC)

== Georgian (ka) wikiversity ==
PLEASE 

Help me to make Georgian (ka) wikiversity--[[User:ჯეო|ჯეო]] ([[User talk:ჯეო|discuss]] • [[Special:Contributions/ჯეო|contribs]]) 17:23, 1 March 2019 (UTC)

:{{at|ჯეო}} See https://beta.wikiversity.org/wiki/Main_Page. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 23:00, 1 March 2019 (UTC)

დიდი მადლობა (Didi Madloba-Thank You)!--[[User:ჯეო|ჯეო]] ([[User talk:ჯეო|discuss]] • [[Special:Contributions/ჯეო|contribs]]) 08:44, 2 March 2019 (UTC)

::Please see [[betawikiversity:Category:KA]]. That is the appropriate place to create learning pages in this language. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 14:11, 10 March 2019 (UTC)

== new langueages ==

we should admit crosing of languajes to have a better understanding--[[Special:Contributions/201.208.239.198|201.208.239.198]] ([[User talk:201.208.239.198|discuss]]) 19:34, 25 July 2019 (UTC)

:This is the English Wikiversity. See [[:es:Portada|Wikiversidad]] for Wikiversity in Spanish. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 22:39, 25 July 2019 (UTC)

== How to change an username? ==

How to change an username? --[[User:Josephina Phoebe White|Josephina Phoebe White]] ([[User talk:Josephina Phoebe White|discuss]] • [[Special:Contributions/Josephina Phoebe White|contribs]]) 07:27, 28 August 2019 (UTC)
*{{ping|Josephina Phoebe White}} You can request at [[Special:GlobalRenameRequest]] --[[User:94rain|94rain]] ([[User talk:94rain|discuss]] • [[Special:Contributions/94rain|contribs]]) 07:29, 28 August 2019 (UTC)
Thanks. --[[User:Josephina Phoebe White|Josephina Phoebe White]] ([[User talk:Josephina Phoebe White|discuss]] • [[Special:Contributions/Josephina Phoebe White|contribs]]) 07:45, 28 August 2019 (UTC)

==Religious user names allowed in Wikiversity?==

https://en.m.wikiversity.org/wiki/Wikiversity:Username

Names of religious figures such as &quot;God&quot;, &quot;Jehovah&quot;,&quot;Buddha&quot;,&quot;Jainism&quot;,&quot;Bonadea&quot;,Hinduism or &quot;Allah&quot;, which user names prohibited

Please answer for my question. This Wikiversity user name policy still alive? Religious user names are prohibited? 

:It isn't a policy, but it's a guideline for people who are wanting to register an account are recommended to follow (as per the page, which could be changed with community consensus). I see no reason for this statement to be &quot;dead&quot;. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 00:15, 2 September 2019 (UTC)
::: Yes: Religious user names are under hedding &quot;Inflammatory usernames&quot;, will be blocked and not allowed.

== LinkedIn ==

I insist that a Wikiversity page should be added on LinkedIn. Wikimedia has its LinkedIn page; Wikipedia, too. But not Wikiversity. I tried to show my Swedish studies but could not choose Wikiversity as the Institution. Why not? Even when it is not a &quot;granting degree&quot; Institution, is is still an Institution, right? When I contacted LinkedIn about this, they sent me the link so that I can create myself the Wikiversity page. But then there is box I must tick: &quot; I confirm I am an approved authority of this Institution to create this page&quot;, which is not the case. But I think there are many Wikiversity experts on here that woud qualify as Wikiversity Linkedin page creators. I can create the page if someone here approves, but I would need some info: # of employees, etc. --[[User:Leonardo T. Cardillo|Leonardo T. Cardillo]] ([[User talk:Leonardo T. Cardillo|discuss]] • [[Special:Contributions/Leonardo T. Cardillo|contribs]]) 23:34, 18 January 2020 (UTC)

:The information would go here [https://www.linkedin.com/company/setup/new/ Wikiversity institution] but it probably should have a bureaucrat or someone from the WMF tick &quot;I verify that I am an authorized representative of this organization and have the right to act on its behalf in the creation and management of this page. The organization and I agree to the additional terms for Pages.&quot; The number of employees (volunteers is not an option but we are unpaid) for our Wikiversity I guess could be the number of active users 201-500. The current logo is File:Wikiversity logo 2017.svg. The website can be https://en.wikiversity.org/wiki/Wikiversity:Main_Page.--[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 00:16, 19 January 2020 (UTC)

{{At|Leonardo T. Cardillo}} Wikiversity is a community. None of us gets to insist that anything happen on behalf of the community unless there is consensus to do so. This requires a discussion in the [[Wikiversity:Colloquium]] and a vote for support or lack thereof. Because this request involves an outside organization, it may also require support from the WMF.

I have some concerns at this point that your passion regarding this issue far exceeds your demonstrated commitment to either Wikiversity or the wider Wikimedia community. It might be better to let this rest for a bit and learn more about how Wikiversity functions before insisting that this be discussed. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 03:29, 19 January 2020 (UTC)

:{{At|Dave Braunschweig}}: I apologize for the use of the word &quot;insist&quot;, I have taken note to not use it anymore here to avoid distractions from the main topic of conversation. Also, I do not like you judge how much my passions should go against my level of contributions. With that being said, and for my personal learning on this environment, can someone please guide me on the very first step I should take to have a Wikiversity page created on LinkedIn? I think you mentioned something like a &quot;poll&quot;, how do I do that? --[[User:Leonardo T. Cardillo|Leonardo T. Cardillo]] ([[User talk:Leonardo T. Cardillo|discuss]] • [[Special:Contributions/Leonardo T. Cardillo|contribs]]) 04:38, 19 January 2020 (UTC)

::{{At|Leonardo T. Cardillo}} I have already guided you on the next step to take. Please read my response carefully. Then slow down and learn more about Wikiversity. We often have people come in with high passions and quick fixes that Wikiversity must make in order to improve. They're typically gone within a month and we're left having to clean up after them. That's not to suggest that this is or isn't a good idea. It is simply to point out that this is a community. You must first learn to work with the community before you try to change it. We look forward to working with you as you figure this out. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 15:31, 19 January 2020 (UTC)

:::{{At|Dave Braunschweig}} Thanks so much for your inputs. I have created this: https://en.wikiversity.org/wiki/Wikiversity:Colloquium#LinkedIn. Please indicate if that is the next step that was intended to be created. Also, please guide on the following ones. Best regards, --[[User:Leonardo T. Cardillo|Leonardo T. Cardillo]] ([[User talk:Leonardo T. Cardillo|discuss]] • [[Special:Contributions/Leonardo T. Cardillo|contribs]]) 16:27, 19 January 2020 (UTC)

== Add New Language ==

Why not bn.wikiversity?  But there is Hindi!  Make it, please.  I am ready to cooperate if needed. [[User:Hirok Raja|Hirok Raja]] ([[User talk:Hirok Raja|discuss]] • [[Special:Contributions/Hirok Raja|contribs]]) 03:07, 1 August 2020 (UTC)

:[[User:Hirok Raja|Hirok Raja]]: please see [[:betawikiversity:|Wikiversity Beta]]. &amp;mdash;Hasley&amp;nbsp;[[user talk:Hasley|&lt;span style=&quot;color: #0645AD; vertical-align: super; font-size: smaller;&quot;&gt;talk&lt;/span&gt;]] 13:04, 1 August 2020 (UTC)

:{{At|Hirok Raja}} Also see [[meta:Wikiversity]]. We are the English Wikiversity. We have no role in setting up new Wikiversity languages. When bn.wikiversity is added, please let us know, and we will add it to our main page. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 13:59, 1 August 2020 (UTC)

== I'm learning Turkish🤩 ==

Hi(to the person reading this)! I'm learning Turkish and I would like someone(native Turkish speaker) to teach how to pronounce Turkish. I do know some words,alphabets and number☺️ and I'm still learning and I hope someone is willing to help me🥺.

@JinahJady! [[User:JanehJody|JanehJody]] ([[User talk:JanehJody|discuss]] • [[Special:Contributions/JanehJody|contribs]]) 18:14, 4 February 2021 (UTC)

:Hi. Welcome to Wikiversity! Please see our [[Turkish|resources relating to the study of the Turkish language]]. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:41, 4 February 2021 (UTC)
::Hi,@[[User:JanehJody|JanehJody]] can i help you ::) [[User:MexmetW|MexmetW]] ([[User talk:MexmetW|discuss]] • [[Special:Contributions/MexmetW|contribs]]) 07:47, 28 September 2022 (UTC)
:Hi,@[[User:JanehJody|JanehJody]] I would love to help you to learning turkish :) [[Special:Contributions/85.105.185.109|85.105.185.109]] ([[User talk:85.105.185.109|discuss]]) 07:31, 28 September 2022 (UTC)

== Is it Wikipedia remodeled or a copy of wikipedia? ==

I am confused--[[User:Noukden|Noukden]] ([[User talk:Noukden|discuss]] • [[Special:Contributions/Noukden|contribs]]) 20:45, 24 May 2021 (UTC)

:{{At|Noukden}} None of the above. See [[What is Wikiversity?]] and [[What Wikiversity is not]]. Wikiversity is learning projects. Link to Wikipedia rather than duplicating it and then add hands-on activities so users can learn by doing. See [[IT Fundamentals]] for one approach. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:15, 25 May 2021 (UTC)

== Action in the earliest? ==

I want to know much more of all action that happend in the earliest centuries. [[User:Dilbkhay|Dilbkhay]] ([[User talk:Dilbkhay|discuss]] • [[Special:Contributions/Dilbkhay|contribs]]) 14:57, 21 August 2021 (UTC)

:Depending upon what you mean by &quot;earliest&quot;, have a look at [[Paleanthropology]] or [[Philosophy/Sciences]]. --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:07, 20 September 2021 (UTC)

== Biology ==

What are the basic principles of ecology [[User:Aludriyo Dominic|Aludriyo Dominic]] ([[User talk:Aludriyo Dominic|discuss]] • [[Special:Contributions/Aludriyo Dominic|contribs]]) 18:25, 25 January 2022 (UTC)

:{{At|Aludriyo Dominic}} Welcome! See [[Wikipedia:Ecology]]. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:17, 26 January 2022 (UTC)
:{{ping|Aludriyo Dominic}} I invite you to read [[User:Atcovi/Science/Ecology]] if you're interested in learning about the basics of Ecology. Also check out the wikipedia link above and [[:Category:Ecology|this category]]. Thanks and weclome! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 03:44, 26 January 2022 (UTC)

I will try to study [[User:Aludriyo Dominic|Aludriyo Dominic]] ([[User talk:Aludriyo Dominic|discuss]] • [[Special:Contributions/Aludriyo Dominic|contribs]]) 05:41, 28 January 2022 (UTC)

== Physics ==

Physics Can Be defined as A Pure Science Subject That deals with the Measurement Of Matter In relation to energy. --{{Unsigned|Oyeyemi Abdul-warith|29 January 2022}}

: Welcome to Wikiversity! Here is a landing page that may be helpful: [[Physics]]. --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 16:42, 29 January 2022 (UTC)

== Popularize ==

Can someone popularize California or the State of Washington on the Main Page? [[Special:Contributions/2604:3D08:6286:7500:B441:2710:77A4:1304|2604:3D08:6286:7500:B441:2710:77A4:1304]] ([[User talk:2604:3D08:6286:7500:B441:2710:77A4:1304|discuss]]) 03:33, 26 June 2022 (UTC)

:No, sorry, promotion isn't part of the [[Wikiversity:Mission]]. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 12:06, 26 June 2022 (UTC)

== [[w:Armistice of WWI|Armistice of WWI]], [[w:Paris Peace Conference|Paris Peace Conference]] and Aftermath ==

The best time to feature this on the main page was last week or yesterday;  the second best time is today.

* [[w:Template:First_World_War_treaties]]  (this template should get transcluded or copied to wikiversity, since this doesn't work: {{w:First_World_War_treaties}} although I wish it would)
* [[Wikiversity:Colloquium#Proclaiming_Armistice_of_WWI_Remembrance_and_Veterans_Day_for_11th_Nov]] our course on WWI is woefully inadequate, but this is a good time to start improving it!
[[User:Jaredscribe|Jaredscribe]] ([[User talk:Jaredscribe|discuss]] • [[Special:Contributions/Jaredscribe|contribs]]) 10:22, 12 November 2023 (UTC)

== Top Staffing Services in Noida for Efficient Hiring Solutions ==

Vision India offers reliable [https://www.vispl.co.in/staffing-services.aspx staffing services in Noida], providing businesses with customized recruitment solutions to meet their specific needs. Whether you're looking for permanent, temporary, or contract staff, Vision India ensures access to a pool of talented professionals across various industries, including IT, healthcare, finance, and engineering. With a deep understanding of the local job market, Vision India leverages a strategic approach to identify the right candidates who not only possess the required skills but also fit well with your company's culture. Their efficient recruitment process helps businesses save time and resources while ensuring high-quality hires that contribute to long-term growth and success. Partnering with Vision India for staffing services ensures a seamless hiring experience, whether you need immediate talent or long-term workforce solutions. [[User:Visionindia11|Visionindia11]] ([[User talk:Visionindia11|discuss]] • [[Special:Contributions/Visionindia11|contribs]]) 08:25, 17 December 2024 (UTC)</text>
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{{attention}} To request an edit to the [[Wikiversity:Page protection|protected]] Main Page, add {{tl|editprotected}} to your request. Such requests should either be obvious or uncontroversial, or be discussed to show consensus, so please do not make vague requests here. If possible, describe exactly what changes should be made so that any custodian can quickly satisfy the request.&lt;br&gt;
{{attention}} To raise general topics about [[Wikiversity]], make general suggestions about Wikiversity, to ask questions, or to talk about anything else of a general nature, use the [[Wikiversity:Colloquium|Colloquium]].&lt;br&gt;
{{attention}} To discuss the structure, appearance, etc. of the [[Wikiversity:Main Page|Main Page]], go to the [[Wikiversity:Main page learning project]] and the [[Wikiversity talk:Main page learning project|talk page for the main page learning project]].
&lt;/div&gt;

----
'''''If you wish to post something below, go ahead. It's a talk page. But you are more likely to get a response by going to the [[Wikiversity:Colloquium|Colloquium]], which is where the main talking at Wikiversity goes on! See you there.'''''

{{archive box|
{{center top}}'''List of talk archives'''{{center bottom}}
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{{Special:Prefixindex/Wikiversity talk:Main Page/Archive |hideredirects=1|stripprefix=1}}
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== The Wikiversity:Main page learning project ==

The [[Wikiversity:Main page learning project]] was launched after the redesign of the main page in December 2007. The [[Wikiversity:Main page learning project]] has as its goal &quot;the promotion of responsible involvement of the Wikiversity community in an efficient, productive, open and inclusive maintenance of the Wikiversity main page as a flagship of the activity and values of the Wikiversity community&quot;. If you would like to get involved in the design of the main page, this is where to go. 

If you have general comments about the main page, but you don't especially want to get involved in the main page project, then you can also leave comments on the [[Wikiversity_talk:Main page learning project|talk page for the main page learning project]].

:I've suggested that it might be time to retire the &quot;quote of the day&quot; project and remove the quotes from the Main Page. See: [[Wikiversity talk:Main page learning project/QOTD]]. It might also be appropriate to deprecate the inactive [[Wikiversity:Main page learning project]] and archive it. Thoughts? --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 23:37, 29 November 2019 (UTC)

== add new language university ==
Now that Chinese Wikiversity is created, please add a cross-wiki link to it. --[[User:WQL|WQL]] ([[User talk:WQL|discuss]] • [[Special:Contributions/WQL|contribs]]) 12:52, 12 August 2018 (UTC)

:{{Done}} -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 14:29, 12 August 2018 (UTC)

== Edit request from 204.234.101.112, 14 February 2019 ==

&lt;nowiki&gt;{{editprotected}}&lt;/nowiki&gt;
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[[Special:Contributions/204.234.101.112|204.234.101.112]] ([[User talk:204.234.101.112|discuss]]) 21:17, 14 February 2019 (UTC)

:{{Not done}} Empty request -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 01:11, 15 February 2019 (UTC)

== Georgian (ka) wikiversity ==
PLEASE 

Help me to make Georgian (ka) wikiversity--[[User:ჯეო|ჯეო]] ([[User talk:ჯეო|discuss]] • [[Special:Contributions/ჯეო|contribs]]) 17:23, 1 March 2019 (UTC)

:{{at|ჯეო}} See https://beta.wikiversity.org/wiki/Main_Page. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 23:00, 1 March 2019 (UTC)

დიდი მადლობა (Didi Madloba-Thank You)!--[[User:ჯეო|ჯეო]] ([[User talk:ჯეო|discuss]] • [[Special:Contributions/ჯეო|contribs]]) 08:44, 2 March 2019 (UTC)

::Please see [[betawikiversity:Category:KA]]. That is the appropriate place to create learning pages in this language. --[[User:Mu301|mikeu]] &lt;sup&gt;[[User talk:Mu301|talk]]&lt;/sup&gt; 14:11, 10 March 2019 (UTC)

== new langueages ==

we should admit crosing of languajes to have a better understanding--[[Special:Contributions/201.208.239.198|201.208.239.198]] ([[User talk:201.208.239.198|discuss]]) 19:34, 25 July 2019 (UTC)

:This is the English Wikiversity. See [[:es:Portada|Wikiversidad]] for Wikiversity in Spanish. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 22:39, 25 July 2019 (UTC)

== How to change an username? ==

How to change an username? --[[User:Josephina Phoebe White|Josephina Phoebe White]] ([[User talk:Josephina Phoebe White|discuss]] • [[Special:Contributions/Josephina Phoebe White|contribs]]) 07:27, 28 August 2019 (UTC)
*{{ping|Josephina Phoebe White}} You can request at [[Special:GlobalRenameRequest]] --[[User:94rain|94rain]] ([[User talk:94rain|discuss]] • [[Special:Contributions/94rain|contribs]]) 07:29, 28 August 2019 (UTC)
Thanks. --[[User:Josephina Phoebe White|Josephina Phoebe White]] ([[User talk:Josephina Phoebe White|discuss]] • [[Special:Contributions/Josephina Phoebe White|contribs]]) 07:45, 28 August 2019 (UTC)

==Religious user names allowed in Wikiversity?==

https://en.m.wikiversity.org/wiki/Wikiversity:Username

Names of religious figures such as &quot;God&quot;, &quot;Jehovah&quot;,&quot;Buddha&quot;,&quot;Jainism&quot;,&quot;Bonadea&quot;,Hinduism or &quot;Allah&quot;, which user names prohibited

Please answer for my question. This Wikiversity user name policy still alive? Religious user names are prohibited? 

:It isn't a policy, but it's a guideline for people who are wanting to register an account are recommended to follow (as per the page, which could be changed with community consensus). I see no reason for this statement to be &quot;dead&quot;. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 00:15, 2 September 2019 (UTC)
::: Yes: Religious user names are under hedding &quot;Inflammatory usernames&quot;, will be blocked and not allowed.

== LinkedIn ==

I insist that a Wikiversity page should be added on LinkedIn. Wikimedia has its LinkedIn page; Wikipedia, too. But not Wikiversity. I tried to show my Swedish studies but could not choose Wikiversity as the Institution. Why not? Even when it is not a &quot;granting degree&quot; Institution, is is still an Institution, right? When I contacted LinkedIn about this, they sent me the link so that I can create myself the Wikiversity page. But then there is box I must tick: &quot; I confirm I am an approved authority of this Institution to create this page&quot;, which is not the case. But I think there are many Wikiversity experts on here that woud qualify as Wikiversity Linkedin page creators. I can create the page if someone here approves, but I would need some info: # of employees, etc. --[[User:Leonardo T. Cardillo|Leonardo T. Cardillo]] ([[User talk:Leonardo T. Cardillo|discuss]] • [[Special:Contributions/Leonardo T. Cardillo|contribs]]) 23:34, 18 January 2020 (UTC)

:The information would go here [https://www.linkedin.com/company/setup/new/ Wikiversity institution] but it probably should have a bureaucrat or someone from the WMF tick &quot;I verify that I am an authorized representative of this organization and have the right to act on its behalf in the creation and management of this page. The organization and I agree to the additional terms for Pages.&quot; The number of employees (volunteers is not an option but we are unpaid) for our Wikiversity I guess could be the number of active users 201-500. The current logo is File:Wikiversity logo 2017.svg. The website can be https://en.wikiversity.org/wiki/Wikiversity:Main_Page.--[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 00:16, 19 January 2020 (UTC)

{{At|Leonardo T. Cardillo}} Wikiversity is a community. None of us gets to insist that anything happen on behalf of the community unless there is consensus to do so. This requires a discussion in the [[Wikiversity:Colloquium]] and a vote for support or lack thereof. Because this request involves an outside organization, it may also require support from the WMF.

I have some concerns at this point that your passion regarding this issue far exceeds your demonstrated commitment to either Wikiversity or the wider Wikimedia community. It might be better to let this rest for a bit and learn more about how Wikiversity functions before insisting that this be discussed. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 03:29, 19 January 2020 (UTC)

:{{At|Dave Braunschweig}}: I apologize for the use of the word &quot;insist&quot;, I have taken note to not use it anymore here to avoid distractions from the main topic of conversation. Also, I do not like you judge how much my passions should go against my level of contributions. With that being said, and for my personal learning on this environment, can someone please guide me on the very first step I should take to have a Wikiversity page created on LinkedIn? I think you mentioned something like a &quot;poll&quot;, how do I do that? --[[User:Leonardo T. Cardillo|Leonardo T. Cardillo]] ([[User talk:Leonardo T. Cardillo|discuss]] • [[Special:Contributions/Leonardo T. Cardillo|contribs]]) 04:38, 19 January 2020 (UTC)

::{{At|Leonardo T. Cardillo}} I have already guided you on the next step to take. Please read my response carefully. Then slow down and learn more about Wikiversity. We often have people come in with high passions and quick fixes that Wikiversity must make in order to improve. They're typically gone within a month and we're left having to clean up after them. That's not to suggest that this is or isn't a good idea. It is simply to point out that this is a community. You must first learn to work with the community before you try to change it. We look forward to working with you as you figure this out. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 15:31, 19 January 2020 (UTC)

:::{{At|Dave Braunschweig}} Thanks so much for your inputs. I have created this: https://en.wikiversity.org/wiki/Wikiversity:Colloquium#LinkedIn. Please indicate if that is the next step that was intended to be created. Also, please guide on the following ones. Best regards, --[[User:Leonardo T. Cardillo|Leonardo T. Cardillo]] ([[User talk:Leonardo T. Cardillo|discuss]] • [[Special:Contributions/Leonardo T. Cardillo|contribs]]) 16:27, 19 January 2020 (UTC)

== Add New Language ==

Why not bn.wikiversity?  But there is Hindi!  Make it, please.  I am ready to cooperate if needed. [[User:Hirok Raja|Hirok Raja]] ([[User talk:Hirok Raja|discuss]] • [[Special:Contributions/Hirok Raja|contribs]]) 03:07, 1 August 2020 (UTC)

:[[User:Hirok Raja|Hirok Raja]]: please see [[:betawikiversity:|Wikiversity Beta]]. &amp;mdash;Hasley&amp;nbsp;[[user talk:Hasley|&lt;span style=&quot;color: #0645AD; vertical-align: super; font-size: smaller;&quot;&gt;talk&lt;/span&gt;]] 13:04, 1 August 2020 (UTC)

:{{At|Hirok Raja}} Also see [[meta:Wikiversity]]. We are the English Wikiversity. We have no role in setting up new Wikiversity languages. When bn.wikiversity is added, please let us know, and we will add it to our main page. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 13:59, 1 August 2020 (UTC)

== I'm learning Turkish🤩 ==

Hi(to the person reading this)! I'm learning Turkish and I would like someone(native Turkish speaker) to teach how to pronounce Turkish. I do know some words,alphabets and number☺️ and I'm still learning and I hope someone is willing to help me🥺.

@JinahJady! [[User:JanehJody|JanehJody]] ([[User talk:JanehJody|discuss]] • [[Special:Contributions/JanehJody|contribs]]) 18:14, 4 February 2021 (UTC)

:Hi. Welcome to Wikiversity! Please see our [[Turkish|resources relating to the study of the Turkish language]]. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:41, 4 February 2021 (UTC)
::Hi,@[[User:JanehJody|JanehJody]] can i help you ::) [[User:MexmetW|MexmetW]] ([[User talk:MexmetW|discuss]] • [[Special:Contributions/MexmetW|contribs]]) 07:47, 28 September 2022 (UTC)
:Hi,@[[User:JanehJody|JanehJody]] I would love to help you to learning turkish :) [[Special:Contributions/85.105.185.109|85.105.185.109]] ([[User talk:85.105.185.109|discuss]]) 07:31, 28 September 2022 (UTC)

== Is it Wikipedia remodeled or a copy of wikipedia? ==

I am confused--[[User:Noukden|Noukden]] ([[User talk:Noukden|discuss]] • [[Special:Contributions/Noukden|contribs]]) 20:45, 24 May 2021 (UTC)

:{{At|Noukden}} None of the above. See [[What is Wikiversity?]] and [[What Wikiversity is not]]. Wikiversity is learning projects. Link to Wikipedia rather than duplicating it and then add hands-on activities so users can learn by doing. See [[IT Fundamentals]] for one approach. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:15, 25 May 2021 (UTC)

== Action in the earliest? ==

I want to know much more of all action that happend in the earliest centuries. [[User:Dilbkhay|Dilbkhay]] ([[User talk:Dilbkhay|discuss]] • [[Special:Contributions/Dilbkhay|contribs]]) 14:57, 21 August 2021 (UTC)

:Depending upon what you mean by &quot;earliest&quot;, have a look at [[Paleanthropology]] or [[Philosophy/Sciences]]. --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 21:07, 20 September 2021 (UTC)

== Biology ==

What are the basic principles of ecology [[User:Aludriyo Dominic|Aludriyo Dominic]] ([[User talk:Aludriyo Dominic|discuss]] • [[Special:Contributions/Aludriyo Dominic|contribs]]) 18:25, 25 January 2022 (UTC)

:{{At|Aludriyo Dominic}} Welcome! See [[Wikipedia:Ecology]]. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 00:17, 26 January 2022 (UTC)
:{{ping|Aludriyo Dominic}} I invite you to read [[User:Atcovi/Science/Ecology]] if you're interested in learning about the basics of Ecology. Also check out the wikipedia link above and [[:Category:Ecology|this category]]. Thanks and weclome! —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 03:44, 26 January 2022 (UTC)

I will try to study [[User:Aludriyo Dominic|Aludriyo Dominic]] ([[User talk:Aludriyo Dominic|discuss]] • [[Special:Contributions/Aludriyo Dominic|contribs]]) 05:41, 28 January 2022 (UTC)

== Physics ==

Physics Can Be defined as A Pure Science Subject That deals with the Measurement Of Matter In relation to energy. --{{Unsigned|Oyeyemi Abdul-warith|29 January 2022}}

: Welcome to Wikiversity! Here is a landing page that may be helpful: [[Physics]]. --[[User:Marshallsumter|Marshallsumter]] ([[User talk:Marshallsumter|discuss]] • [[Special:Contributions/Marshallsumter|contribs]]) 16:42, 29 January 2022 (UTC)

== Popularize ==

Can someone popularize California or the State of Washington on the Main Page? [[Special:Contributions/2604:3D08:6286:7500:B441:2710:77A4:1304|2604:3D08:6286:7500:B441:2710:77A4:1304]] ([[User talk:2604:3D08:6286:7500:B441:2710:77A4:1304|discuss]]) 03:33, 26 June 2022 (UTC)

:No, sorry, promotion isn't part of the [[Wikiversity:Mission]]. -- [[User:Dave Braunschweig|Dave Braunschweig]] ([[User talk:Dave Braunschweig|discuss]] • [[Special:Contributions/Dave Braunschweig|contribs]]) 12:06, 26 June 2022 (UTC)

== [[w:Armistice of WWI|Armistice of WWI]], [[w:Paris Peace Conference|Paris Peace Conference]] and Aftermath ==

The best time to feature this on the main page was last week or yesterday;  the second best time is today.

* [[w:Template:First_World_War_treaties]]  (this template should get transcluded or copied to wikiversity, since this doesn't work: {{w:First_World_War_treaties}} although I wish it would)
* [[Wikiversity:Colloquium#Proclaiming_Armistice_of_WWI_Remembrance_and_Veterans_Day_for_11th_Nov]] our course on WWI is woefully inadequate, but this is a good time to start improving it!
[[User:Jaredscribe|Jaredscribe]] ([[User talk:Jaredscribe|discuss]] • [[Special:Contributions/Jaredscribe|contribs]]) 10:22, 12 November 2023 (UTC)</text>
      <sha1>ntzwji5zesj54h02ses6x2svbulnxxj</sha1>
    </revision>
  </page>
  <page>
    <title>Wikiversity:Colloquium</title>
    <ns>4</ns>
    <id>28</id>
    <revision>
      <id>2692194</id>
      <parentid>2692151</parentid>
      <timestamp>2024-12-16T15:29:02Z</timestamp>
      <contributor>
        <username>Ottawahitech</username>
        <id>2369270</id>
      </contributor>
      <comment>/* Wikiversity - Newsletters */ Reply</comment>
      <origin>2692194</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="76731" sha1="74kxodes2go1x2zbfeu183a5qsug70e" xml:space="preserve">{{Wikiversity:Colloquium/Header}}
&lt;!-- MESSAGES GO BELOW --&gt;
== Reminder! Vote closing soon to fill vacancies of the first U4C ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
:''[[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement – reminder to vote|You can find this message translated into additional languages on Meta-wiki.]] [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement – reminder to vote}}&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]''

Dear all,

The voting period for the Universal Code of Conduct Coordinating Committee (U4C) is closing soon. It is open through 10 August 2024. Read the information on [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2024_Special_Election#Voting|the voting page on Meta-wiki to learn more about voting and voter eligibility]]. If you are eligible to vote and have not voted in this special election, it is important that you vote now.

'''Why should you vote?''' The U4C is a global group dedicated to providing an equitable and consistent implementation of the UCoC. Community input into the committee membership is critical to the success of the UCoC.

Please share this message with members of your community so they can participate as well.

In cooperation with the U4C,&lt;section end=&quot;announcement-content&quot; /&gt;

-- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 15:30, 6 August 2024 (UTC)
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== User group for Wikiversians ==

Was there ever a discussion about the possibility of establishing a user group in the sense of an affiliated organization that would defend the interests of professors and scientists on Wikiversity and possibly actively develop some projects? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:21, 8 August 2024 (UTC)

:Not that I'm aware of. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:20, 8 August 2024 (UTC)
:It's a pleasure to talk to a scientist on Wikiversity. I am a historian of technics and I would like to publish the following biography either on Wikiversity or on Wikipedia: 
:https://en.wikiversity.org/wiki/User:Rbmn/Arthur_Constantin_KREBS_(1850-1935):_Military_engineer,_Automotive_industrialist,_Great_projects_manager
:What would be your advice? [[User:Rbmn|Rbmn]] ([[User talk:Rbmn|discuss]] • [[Special:Contributions/Rbmn|contribs]]) 15:44, 6 October 2024 (UTC)
::The content appears to be largely biographical/encyclopedic, so I think it is likely best suited to Wikipedia. Consider improving/incorporating this content into the existing page: [[w:Arthur Constantin Krebs]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:05, 7 October 2024 (UTC)
::Please do not link to the Wikiversity [[wv:userspace|Userspace]] in Wikipedia articles. You will want to wait until you have a page in the [[wv:mainspace|Wikiversity mainspace]]. You'll also want to use the &lt;code&gt;{{[[:w:Template:Wikiversity|Wikiversity]]}}&lt;/code&gt; template (on Wikipedia) rather than embedding a photo with a link. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:21, 7 October 2024 (UTC)
:I haven't heard anything about this topic. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 21:06, 8 December 2024 (UTC)

== Rich's ''Illustrated Companion'' at Wikiversity: Right place? ==

Hello! I am creating a Wiki-version of a classical glossary (''Illustrated Companion to the Latin Dictionary, and Greek Lexicon'' by Anthony Rich, 1849), which explains the meaning of Latin headwords, primarily those &quot;representing visible objects connected with the arts, manufactures, and every-day life of the Greeks and Romans.&quot; The aim is to help understand what a (classical) Latin text is actually about, instead of merely translating it. I already transcribed the entire text and scanned the images (about 1900) from an original 1849-edition. I am currently working on uploading the images to ''Mediawiki Commons'', which probably will take some time. In the meantime I want to prepare the other aspects of the project (more than 3000 articles, already with many internal links). The important thing: this is ''not'' a ''might exist''-project.  {{Color|red|My question: Is ''Wikiversity'' the proper place for it?}} Although I created an exact rendition of the original text, ''Wikisource'' is not applicable, because the project has a broader scope (adding content to the articles, e. g. links to online editions for quotations, adding images, but also adding entirely new articles). Neither is ''Wikibooks'', because this is not a textbook and may otherwise breach its scope. For more about the project see [[w:User:CalRis25/Temp-RICH-Prospectus|my user-page]] at en.wikipedia. {{Color|Red|So, is Wikiversity the right place for it?}} [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 09:15, 17 August 2024 (UTC)

:Thanks for asking. To be clear, it ''is'' acceptable to make [[:s:en:Category:Wikisource annotations|annotated editions]] of texts at Wikisource and Wikibooks does host at least one [[:b:en:Annotations of The Complete Peanuts|annotated guide to a copyright-protected work]]. So if what you're looking to do is to include inline annotations to a public domain text, you certainly can put that on Wikisource. If you have a textbook or guidebook that is a companion, that would go at Wikibooks. If you have some other kind of learning resources (like maintaining a list of relevant links, organizing a book reading group, etc.), that could go here. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:26, 17 August 2024 (UTC)
::Thank you for your quick answer. Actually, ''Wikibooks'' was my first thought. However, this project is not merely an annotated edition. Although at first it ''will'' be a faithful copy of the original text, I want the project to be &quot;open&quot;, i. e. adding articles should be possible. And the project should enable to do a lot more than mere inline annotation. See section [[w:User:CalRis25/Temp-RICH-Prospectus#Improving_RICH|Improving Rich]] in the project description a my user-page (en.Wikipedia). No ''Mediawiki''-project (Wikisource, Wikibooks, Wikipedia, Wiktionary) seemed to be a sufficiently applicable &quot;fit&quot; for the project, so I thought of Wikiversity as a last resort, because it is supposed to be home to all sorts of &quot;learning resources&quot;. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 09:57, 17 August 2024 (UTC)
:::The scope of Wikiversity ''is'' pretty catch-all and would allow for a pretty flexible place to host most learning resources that don't fit elsewhere.
:::Also, as nitpick, &quot;MediaWiki&quot; is the software that is the basis of these wikis (wikis being collections of interlinked documents that can be edited) and &quot;Wikimedia Foundation&quot; is the non-profit who owns the trademarks and hosts these projects like Wiktionary and Wikivoyage. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 10:06, 17 August 2024 (UTC)
::::Hello Justin, thank you for the reply. '''I think that settles it. I will create this project at ''Wikiversity''.''' Just for additional clarification, why I do so. Let's imagine a full transcription of the original 1849-edition of the ''Illustrated Companion'' by Anthony Rich and call it ''RICH-1849''. We shall call my project, for brevity sake, RICH-2K. And now, let's have a look at the article about the Roman toga (a piece of attire). In ''RICH-1849'' we can can call it ''RICH-1849/Toga'', and it contains ''exactly'' the content of the 1849-book. Now, let's look at the article ''RICH-2K/Toga''. At the beginning its only content would be the article ''RICH-1849/Toga''. Does that make ''RICH-2K/Toga'' and ''RICH-1849/Toga'' the same? Not at all, because in truth ''RICH-2K/Toga'' is a &quot;container&quot; which initially contains only the article ''RICH-1849/Toga'' but later on may include more stuff: images, external links, article text which builds on or extends ''RICH-1849/Toga'' and information from other sources of information (Wikipedia, specialized books). By the way, this added article information would not be a mere copy of the text at en.Wikipedia, because the information needs to looked at through the eyes of someone reading the original text (more citations with direct links to these etc.). [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 11:39, 17 August 2024 (UTC)

== Coming soon: A new sub-referencing feature – try it! ==

&lt;section begin=&quot;Sub-referencing&quot;/&gt;
[[File:Sub-referencing reuse visual.png|{{#ifeq:{{#dir}}|ltr|right|left}}|400px]]
Hello. For many years, community members have requested an easy way to re-use references with different details. Now, a MediaWiki solution is coming: The new sub-referencing feature will work for wikitext and Visual Editor and will enhance the existing reference system. You can continue to use different ways of referencing, but you will probably encounter sub-references in articles written by other users. More information on [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing|the project page]].

'''We want your feedback''' to make sure this feature works well for you:

* [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing#Test|Please try]] the current state of development on beta wiki and [[m:Talk:WMDE Technical Wishes/Sub-referencing|let us know what you think]]. 
* [[m:WMDE Technical Wishes/Sub-referencing/Sign-up|Sign up here]] to get updates and/or invites to participate in user research activities.

[[m:Special:MyLanguage/Wikimedia Deutschland|Wikimedia Deutschland]]’s [[m:Special:MyLanguage/WMDE Technical Wishes|Technical Wishes]] team is planning to bring this feature to Wikimedia wikis later this year. We will reach out to creators/maintainers of tools and templates related to references beforehand.

Please help us spread the message. --[[m:User:Johannes Richter (WMDE)|Johannes Richter (WMDE)]] ([[m:User talk:Johannes Richter (WMDE)|talk]]) 10:36, 19 August 2024 (UTC)
&lt;section end=&quot;Sub-referencing&quot;/&gt;
&lt;!-- Message sent by User:Johannes Richter (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Johannes_Richter_(WMDE)/Sub-referencing/massmessage_list&amp;oldid=27309345 --&gt;

== New [[Template:Form]] ==

Hi! Today I was bold and created [[Template:Form]] (which calls [[Module:WikiForm]] and [[MediaWiki:Gadget-WikiForm.js]]). The template allows to create user-friendly forms that can create pages or add content to existing pages. My motivation and first use case was [[Wikidebate/New|this form]] to create new [[wikidebates]], but I suspect the template can be useful elsewhere on Wikiversity. Let me know if you notice any issues or have any requests or concerns. Kind regards, [[User:Sophivorus|Sophivorus]] ([[User talk:Sophivorus|discuss]] • [[Special:Contributions/Sophivorus|contribs]]) 15:21, 21 August 2024 (UTC)

== Sign up for the language community meeting on August 30th, 15:00 UTC ==

Hi all,

The next language community meeting is scheduled in a few weeks—on August 30th at 15:00 UTC. If you're interested in joining, you can [https://www.mediawiki.org/wiki/Wikimedia_Language_and_Product_Localization/Community_meetings#30_August_2024 sign up on this wiki page].

This participant-driven meeting will focus on sharing language-specific updates related to various projects, discussing technical issues related to language wikis, and working together to find possible solutions. For example, in the last meeting, topics included the Language Converter, the state of language research, updates on the Incubator conversations, and technical challenges around external links not working with special characters on Bengali sites.

Do you have any ideas for topics to share technical updates or discuss challenges? Please add agenda items to the document [https://etherpad.wikimedia.org/p/language-community-meeting-aug-2024 here] and reach out to ssethi(__AT__)wikimedia.org. We look forward to your participation!

[[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 23:20, 22 August 2024 (UTC)
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== Template consolidation: User talk page block notice ==
Wondering if someone who likes templates could have a go at consolidating or helping decide between use of:
* [[Template:Block]]
* [[Template:Blocked]]

Unless I'm missing something, it seems like we don't need both?

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 07:16, 23 August 2024 (UTC)
: I tried to figure out a Wikidata item with most links to projects. I found this: [[Wikidata:Q6379131]], which is Template:Uw-block. There is even a corresponding Wikiversity template, [[Template:Uw-block1]] (not used anywhere).
: My impression is that of the three templates, we only need one. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:43, 13 September 2024 (UTC)

== Announcing the Universal Code of Conduct Coordinating Committee ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
:''[https://lists.wikimedia.org/hyperkitty/list/board-elections@lists.wikimedia.org/thread/OKCCN2CANIH2K7DXJOL2GPVDFWL27R7C/ Original message at wikimedia-l]. [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement - results|You can find this message translated into additional languages on Meta-wiki.]] [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement - results}}&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]''

Hello all,

The scrutineers have finished reviewing the vote and the [[m:Special:MyLanguage/Elections Committee|Elections Committee]] have certified the [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Results|results]] for the [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election|Universal Code of Conduct Coordinating Committee (U4C) special election]].

I am pleased to announce the following individual as regional members of the U4C, who will fulfill a term until 15 June 2026:

* North America (USA and Canada)
** Ajraddatz

The following seats were not filled during this special election:

* Latin America and Caribbean
* Central and East Europe (CEE)
* Sub-Saharan Africa
* South Asia
* The four remaining Community-At-Large seats

Thank you again to everyone who participated in this process and much appreciation to the candidates for your leadership and dedication to the Wikimedia movement and community.

Over the next few weeks, the U4C will begin meeting and planning the 2024-25 year in supporting the implementation and review of the UCoC and Enforcement Guidelines. You can follow their work on [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee|Meta-Wiki]].

On behalf of the U4C and the Elections Committee,&lt;section end=&quot;announcement-content&quot; /&gt;

[[m:User:RamzyM (WMF)|RamzyM (WMF)]] 14:07, 2 September 2024 (UTC)
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== Re: The Vector 2022 skin as the default in two weeks? ==

[[File:Vector 2022 video-en.webm|thumb|A two minute-long video about Vector 2022]]
Hello everyone, I'm reaching out on behalf of the [[mediawikiwiki:Reading/Web|Wikimedia Foundation Web team]] responsible for the MediaWiki skins. I'd like to revisit the topic of making Vector 2022 the default here on English Wikiversity. I [[Wikiversity:Colloquium/archives/September 2022#The Vector 2022 skin as the default in two weeks?|did post a message about this almost two years ago]] (where you can find all the details about the skin), but we didn't finalize it back then.

What happened in the meantime? We built [[mw:Reading/Web/Accessibility for reading|dark mode and different options for font sizes]], and made Vector 2022 the default on most wikis, including all other Wikiversities. With the not-so-new V22 skin being the default, existing and coming features, like dark mode and [[mw:Trust and Safety Product/Temporary Accounts|temporary accounts]] respectively, will become available for logged-out users here.

So, if no large concerns are raised, we will deploy Vector 2022 here in two weeks, in the week of September 16. Do let me know if you have any questions. Thank you! [[User:SGrabarczuk (WMF)|SGrabarczuk (WMF)]] ([[User talk:SGrabarczuk (WMF)|discuss]] • [[Special:Contributions/SGrabarczuk (WMF)|contribs]]) 21:48, 2 September 2024 (UTC)

:Sounds good, Szymon - we look forward to the upcoming change of skin {{smile}} Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 07:35, 13 September 2024 (UTC)
* I for one oppose a switch to Vector 2022. I do not find it preferable. Here is a staggering evidence of user refusal of Vector 2022 once it was deployed: [[W:en:Wikipedia:Requests for comment/Rollback of Vector 2022]], Junuary 2023. 355 voters supported rollback to Vector 2010 whereas 64 opposed, yielding 84.7% support, as clear a supermajority as one may wish. These people opposing Vector 2022 feel the same way as I do. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:48, 13 September 2024 (UTC)
*:Hey @[[User:Dan Polansky|Dan Polansky]]. Thanks for your comment. I'm open to discussion about problems with our software, and I hope we can maintain a respectful tone.
*:I understand that there are users who prefer Vector legacy or other skins, just as there are people who still stick to Monobook. Such people are active across many wikis. They can keep Vector legacy, although non-default skins don't have the support the default ones do. We are rolling out for technical reasons, as I mentioned above, with benefit to not logged-in users.
*:Regarding the rollback RfC on Wikipedia, two neutral users stated that there was no consensus for rollback, RfC is not a vote, and the numbers were different (355:226:24). I believe this all is pretty easy to verify.
*:So to sum up, Vector 2022 needs to become the default, tons and tons of comments were made about the skin and related stuff, and we have taken many ideas into account, and it's totally OK if you stick to Vector legacy.
*:Thanks! [[User:SGrabarczuk (WMF)|SGrabarczuk (WMF)]] ([[User talk:SGrabarczuk (WMF)|discuss]] • [[Special:Contributions/SGrabarczuk (WMF)|contribs]]) 19:30, 16 September 2024 (UTC)
*:: Today, I visited Wikiversity and found it switched to Vector 2022. I changed my preference settings to Vector 2010. From what I understand, non-registered visitors are now defaulted to Vector 2022 despite its unpopularity in [[W:en:Wikipedia:Requests for comment/Rollback of Vector 2022]]. I have not seen any evidence that users prefer Vector 2022, and given the evidence in the linked RfC, I tentatively conclude that the decision to switch has made the site experience worse for the majority of users. The logic of &quot;you can switch&quot; surely applies to Vector 2022 as well: those who prefer it can switch to it. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:08, 17 September 2024 (UTC)

== Have your say: Vote for the 2024 Board of Trustees! ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
Hello all,

The voting period for the [[m:Special:MyLanguage/Wikimedia Foundation elections/2024|2024 Board of Trustees election]] is now open. There are twelve (12) candidates running for four (4) seats on the Board.

Learn more about the candidates by [[m:Special:MyLanguage/Wikimedia Foundation elections/2024/Candidates|reading their statements]] and their [[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Questions_for_candidates|answers to community questions]].

When you are ready, go to the [[Special:SecurePoll/vote/400|SecurePoll]] voting page to vote. '''The vote is open from September 3rd at 00:00 UTC to September 17th at 23:59 UTC'''.

To check your voter eligibility, please visit the [[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Voter_eligibility_guidelines|voter eligibility page]].

Best regards,

The Elections Committee and Board Selection Working Group&lt;section end=&quot;announcement-content&quot; /&gt;

[[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 12:15, 3 September 2024 (UTC)
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== Separate page for hyperbola. ==

Good morning,

I notice that a search for &quot;hyperbola&quot; redirects to &quot;Conic sections&quot;.

At present there is a separate page for &quot;ellipse&quot;. Therefore a separate page for &quot;hyperbola&quot; seems to be justified.

Could this redirection be changed so that search for &quot;hyperbola&quot; goes to a separate page for &quot;hyperbola&quot;?

Many thanks,

[[User:ThaniosAkro|ThaniosAkro]] ([[User talk:ThaniosAkro|discuss]] • [[Special:Contributions/ThaniosAkro|contribs]]) 12:04, 15 September 2024 (UTC)

:It is true that ellipses are covered at [[Conic sections]] (along with hyperbolas, parabolas, etc.) and there is a separate page for [[ellipse]]s that elaborates. We certainly ''could'' have a page about [[hyperbola]]s that is separate, but no one has written sufficient content to spin it off yet. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 12:17, 15 September 2024 (UTC)

== I hereby request for your Unblocking IP address and just reviewed and received a reverted rec ==

Hi there. {{unsigned|Ishmael Raphasha}}
:No one has any clue what you're talking about. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:53, 18 September 2024 (UTC)

== RICH-2K: New project with some initial questions ==

Hello! I'm creating a new learning resource on ''Wikiversity''. The respective project is based on my transcription of a classical dictionary from 1849 by Anthony Rich. For more information about the project see its [[User:CalRis25/RICH: Description|description page]] (see also that page for why not ''Wikisource'' or ''Wikibooks''). The project's scope is fairly big: 3205 article-pages plus 304 REDIRECT-pages. The images (scanned by myself from an original copy) have been uploaded to ''Commons''. I have some initial technical questions (more of these and more detailed ones will follow):

* '''Upload''': Due to the large number of pages it is not realistic to create these manually. Is it possible to bulk-upload these in some way (the Wikitext of the pages is created using a Python-script with one file per article/page)? Is it possible to upload these to a test-environment first where any problems (hopefully none) can be identified and dealt with more easily than on the production-version of ''Wikiversity''?
* '''(Technical) Structure''': I am planning to set up this project at ''&lt;nowiki&gt;https://en.wikiversity.org/wiki/RICH-2K&lt;/nowiki&gt;'' as the main page and anything else as subpages: ''RICH-2K/Subpage_1 ... RICH-2K/Subpage_n''. However, these subpages fall into two categories: 1. Article-pages (content) and 2. Meta/Administrative pages. This project requires search capability restricted to the ''RICH-2K''-namespace. The ''Mediawiki''-software seems to supply a ''Search''-input field with the possibility to restrict the search to some namespace. I would like, however, to restrict the search further to the first group of pages, namely the articles. Is that possible, perhaps by use of (hidden) categories?
* '''External links''': This project will need many external links, and yes, I have read the relevant ''Wikiversity''-pages, but this specific project needs them. The ''Recommended Editions''-page (used for recommended online editions, to which to link when citing texts) alone probably will require several hundred external links. However, only relatively few [[w:Second-level domain|second-level domains]] will be involved, and most of these should be trustworthy (Perseus Digital library, digital collections of universities etc., in some cases, however, also ''Archive.org''). Perhaps there is a list of web-sites, for which external links are generally allowed? And who is allowed to create external links on ''Wikiversity''-pages (I haven't found the relevant policy)?
* '''Categories''': This project requires quite a few of its own categories, which belong to two large groups: 1. Categories (2 levels) of the ''Classed Index'' (about 170 categories), a thematic index of some (but not all) of the articles. 2. Administrative categories. Is there a recommended way to distinguish between different classes of categories within a project (category name or other method)? What about naming conventions for project-specific categories?

I am looking forward to your input. If you think that it's preferable we can move the discussions to the [[User_talk:CalRis25/RICH:_Description|Talk-page]] of the project's description. Thank you in advance. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 05:29, 20 September 2024 (UTC)

:*Admins have access to [[Special:Import]] and can bulk import XML pages. You can create pages in your sandbox if you'd like and make an indefinite amount of them at pages like [[User:CalRis25/sandbox]]. What can and cannot be hosted in user namespace is very loose, but still has to follow in principle Wikiversity's scope.
:*Using subpages is in principle a good way to organize these various resources. Please do not name them after a user name or something obscure. I personally think that &quot;RICH-2K&quot; is a not optimal name. I may recommend something like [[Anthony Rich Dictionary Project]] or [[21st-Century Anthony Rich Dictionary]] or something more obviously intelligible. While we have very few actual policies and guidelines, see [[Wikiversity:Naming conventions]] for a rough consensus of what is probably best practice for naming pages.
:*External linking generally does not use an allowed list (a.k.a. whitelist model), but a disallow (a.k.a. blacklist) model. See [[MediaWiki:Spam-blacklist]] and [[Special:BlockedExternalDomains]] (which is currently empty but is another method of listing blocked domains). It's perfectly fine to aggregate external links in learning resources.
:*I'm not 100% sure what the distinction is that you're drawing, but you can freely arrange categories underneath a main category that has the same name as your larger project. So, following the suggestions I gave, you could have a category like [[:Category:Anthony Rich Dictionary Project]] and then create any number of subcategories that logically help users navigate all these pages. Please make sure the main category you create is itself categorized under some relevant category(ies). If you need help, please ask.
:I think this answers your questions, please let me know if I'm unclear or you have more. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:11, 20 September 2024 (UTC)
::Hello Justin!
::* '''Upload:''' Creating the project in sandbox pages of my User-namespace defeats the purpose, as this is an ''open'' project. Also that would not solve, as such, the problem of having to manually create thousands of pages. I wonder, does ''Wikiversity'' support creation of pages using its API. ''Mediawiki's'' [[mw:API:Main_page|API-description]] seems to imply that it ought to be possible. If that's the case, I should be able to create a Python-script which automatically creates the pages (of course, a few trial pages first).
::* '''(Technical) Structure''': You may be right, here. RICH-2K is, for now, merely a technical name to make a clear but not too verbose distinction between the original text and the current project. I'll give this more thought.
::* '''External links''': I brought this up mainly because when I first edited my ''Wikiversity''-page, I got a message that I was not allowed to create external links. However, I just now tested creating an external link on my user-page and got no error, so this problem seems to be solved.
::* '''Categories''': I think I know what you mean. I'll create a category structure and maybe ask some specific questions once I am ready to do so.
::Thank you for your quick help. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 18:51, 20 September 2024 (UTC)
:::re: upload, I'm just suggesting your sandbox(es) as you asked about &quot;a test-environment&quot;. Anyone can edit someone else's sandboxes, but you typically defer to other users to control what's in their own subpages as a collegial thing. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:39, 23 September 2024 (UTC)

Hello! I have two further questions:
# I created a category-structure for the project. Could you (or someone else) have a look at it ([[User:CalRis25/RICH: Categories]]) and answer the questions in the section [[User:CalRis25/RICH:_Categories#Questions|Questions]]? I gave it some thought and believe that this would work fine for the project.
# ''Project boxes'' (see [[Help:Tour of project boxes]]): It is unclear to me, whether these belong only on the main page of the project (that makes the most sense to me), or on every single subpage.
Thanks in advance for your help. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 17:51, 24 September 2024 (UTC)

:To answer your questions here:
:*No, you are not contravening any policies we have.
:*A leading &quot;The&quot; is acceptable, but if you want it to sort alphabetically, you will have to use &lt;nowiki&gt;{{DEFAULTSORT:}}&lt;/nowiki&gt;. E.g. to get Category:The Best Stuff to sort under &quot;B&quot;, insert &quot;&lt;nowiki&gt;{{DEFAULTSORT:Best Stuff, The}}&lt;/nowiki&gt;.
:*Trailing &quot;etc.&quot; is acceptable.
:*An accent in a category title is acceptable.
:I'll also note that it looks like you have in mind some tracking categories that are redundant. Pages such as [[Special:LonelyPages]] and [[Special:DeadendPages]] already do automatically what you're proposing to do manually.
:As for project boxes, it's typically the case that the subjects are only placed on the main resource, but as you may imagine, [[Help:Tour of project boxes/1|status completion ones]] may vary from subpage to subpage. As with most things at Wikiversity, there are very few actual rules, so it's pretty much the wild west, even tho this project has been around for almost 20&amp;nbsp;years. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:18, 25 September 2024 (UTC)
::Hello Justin, thanks for the DEFAULTSORT-hint for categories beginning with ''The''. I will restrict the project boxes to the main page. As for the the orphaned/dead-end-categories, I prefer these to be project-specific. Once the project is up and running, putting articles &quot;on the map&quot; (making them accessible from other articles and creating links to other articles) is one of the first tasks to be dealt with. I already know which articles are involved and will add these categories to these articles. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 16:51, 25 September 2024 (UTC)

== Your wiki will be in read-only soon ==

&lt;section begin=&quot;server-switch&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;

[[:m:Special:MyLanguage/Tech/Server switch|Read this message in another language]] • [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-Tech%2FServer+switch&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]

The [[foundation:|Wikimedia Foundation]] will switch the traffic between its data centers. This will make sure that Wikipedia and the other Wikimedia wikis can stay online even after a disaster.

All traffic will switch on '''{{#time:j xg|2024-09-25|en}}'''. The switch will start at '''[https://zonestamp.toolforge.org/{{#time:U|2024-09-25T15:00|en}} {{#time:H:i e|2024-09-25T15:00}}]'''.

Unfortunately, because of some limitations in [[mw:Special:MyLanguage/Manual:What is MediaWiki?|MediaWiki]], all editing must stop while the switch is made. We apologize for this disruption, and we are working to minimize it in the future. 

A banner will be displayed on all wikis 30 minutes before this operation happens. This banner will remain visible until the end of the operation. 

'''You will be able to read, but not edit, all wikis for a short period of time.'''

*You will not be able to edit for up to an hour on {{#time:l j xg Y|2024-09-25|en}}.
*If you try to edit or save during these times, you will see an error message. We hope that no edits will be lost during these minutes, but we can't guarantee it. If you see the error message, then please wait until everything is back to normal. Then you should be able to save your edit. But, we recommend that you make a copy of your changes first, just in case.

''Other effects'':

*Background jobs will be slower and some may be dropped. Red links might not be updated as quickly as normal. If you create an article that is already linked somewhere else, the link will stay red longer than usual. Some long-running scripts will have to be stopped.
* We expect the code deployments to happen as any other week. However, some case-by-case code freezes could punctually happen if the operation require them afterwards.
* [[mw:Special:MyLanguage/GitLab|GitLab]] will be unavailable for about 90 minutes.

This project may be postponed if necessary. You can [[wikitech:Switch_Datacenter|read the schedule at wikitech.wikimedia.org]]. Any changes will be announced in the schedule. 

'''Please share this information with your community.'''&lt;/div&gt;&lt;section end=&quot;server-switch&quot;/&gt;

[[User:Trizek_(WMF)|Trizek_(WMF)]], 09:37, 20 September 2024 (UTC)
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== 'Wikidata item' link is moving. Find out where... ==

&lt;div lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot;&gt;&lt;i&gt;Apologies for cross-posting in English. Please consider translating this message.&lt;/i&gt;{{tracked|T66315}}

Hello everyone, a small change will soon be coming to the user-interface of your Wikimedia project.
The [[d:Q16222597|Wikidata item]] [[w:|sitelink]] currently found under the &lt;span style=&quot;color: #54595d;&quot;&gt;&lt;u&gt;''General''&lt;/u&gt;&lt;/span&gt; section of the '''Tools''' sidebar menu will move into the &lt;span style=&quot;color: #54595d;&quot;&gt;&lt;u&gt;''In Other Projects''&lt;/u&gt;&lt;/span&gt; section.

We would like the Wiki communities feedback so please let us know or ask questions on the   [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Discussion page]] before we enable the change which can take place October 4 2024, circa 15:00 UTC+2.
More information can be found on [[m:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|the project page]].&lt;br&gt;&lt;br&gt;We welcome your feedback and questions.&lt;br&gt; [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 18:56, 27 September 2024 (UTC)
&lt;/div&gt;
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==Download as PDF==
[[Phabricator:T376438]]: &quot;Download to PDF&quot; on en.wv is returning error: &quot;{&quot;name&quot;:&quot;HTTPError&quot;,&quot;message&quot;:&quot;500&quot;,&quot;status&quot;:500,&quot;detail&quot;:&quot;Internal Server Error&quot;}&quot;

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:36, 3 October 2024 (UTC)

:I just downloaded this page as a PDF and it worked just fine. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:04, 3 October 2024 (UTC)

== Protected template bug for Pp ==

It seems that templates derivative of {{tlx|Pp}} (compiled in {{tlx|Protection templates}}) are being sorted into protection categories using the name 'Wikipedia' instead of 'Wikiversity' (e.g., [[:Category:Wikipedia pages with incorrect protection templates]]). From what I can tell, it is not in the publicly accessible source code of any of the templates. The only other impacted pages are modules which call {{tlx|pp}}-derivatives (e.g., [[Module:Navbar/styles.css]]).

This does not seem to affect any other pages in [[:Category:Wikiversity protected templates]]. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 18:59, 4 October 2024 (UTC)

:The problem is that &quot;Wikipedia&quot; is [https://en.wikiversity.org/w/index.php?title=Special%3ASearch&amp;limit=500&amp;offset=0&amp;ns828=1&amp;search=Wikipedia&amp;searchToken=9svkpqlxxoquoq7bnkt55ugts mentioned in several modules that were copied over from en.wp]; many of these are legit and many of them need to be replaced with &quot;Wikiversity&quot; ([https://en.wikiversity.org/w/index.php?title=Module%3APp-move-indef&amp;diff=2662815&amp;oldid=1944984 e.g.]) This particular change ''may'' fix all of these issues...? But 1.) it will take time to propagate across the site and 2.) there are still many more &quot;Wikipedia&quot;s that need to be changed, so I'll go thru a few more, but if you want to give me an assist, if you can just check this one week from now and ping me if the problem persists, that would be nice. Sometimes, I make calendar reminders to follow up on these, but I'm not a perfect person. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:55, 5 October 2024 (UTC)

== Invitation to Participate in Wiki Loves Ramadan Community Engagement Survey ==

Dear all,


We are excited to announce the upcoming [[m:Wiki Loves Ramadan|Wiki Loves Ramadan]] event, a global initiative aimed at celebrating Ramadan by enriching Wikipedia and its sister projects with content related to this significant time of year. As we plan to organize this event globally, your insights and experiences are crucial in shaping the best possible participation experience for the community.

To ensure that Wiki Loves Ramadan is engaging, inclusive, and impactful, we kindly invite you to participate in our community engagement survey. Your feedback will help us understand the needs of the community, set the event's focus, and guide our strategies for organizing this global event.

Survey link: https://forms.gle/f66MuzjcPpwzVymu5

Please take a few minutes to share your thoughts. Your input will make a difference!

Thank you for being a part of our journey to make Wiki Loves Ramadan a success.


Warm regards,

User:ZI Jony 03:19, 6 October 2024 (UTC)

Wiki Loves Ramadan Organizing Team
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== 'Edit to my talk page' notification bug? ==

This may belong at the bug tracker, but does anyone else have an issue disabling ''email'' notifications upon  an 'Edit to my talk page' in [[Special:GlobalPreferences]]? Oddly I ''am'' able to disable the global preference on Wikipedia, MediaWiki, etc, but not here. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 09:23, 7 October 2024 (UTC)

:I have not experienced this, but to be clear, do you also have the option to get emails when items on your talk page are edited turned on? —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:39, 7 October 2024 (UTC)
::The only (non-grayed out) options I have enabled for email are 'Failed login attempts' and 'Login from an unfamiliar device'. 'Edit to my talk page' re-checks after every save. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 09:54, 7 October 2024 (UTC)
:::That does sound like a [[phab:]] issue, with the caveat that I don't 100% recall how global preferences work and if they override local ones, etc. If you have parsed that and still have this issue, you'll probably need to file a ticket. Maybe someone else has this issue. Wish I could help. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:57, 7 October 2024 (UTC)
::::[[phab:T376601|Off 'n away]] 🫡 [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 10:35, 7 October 2024 (UTC)

== [[Portal:Computer Science]] ➝ [[Portal:Information sciences]] ==

Seeking consensus to complete the merge into the broader portal. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 06:28, 8 October 2024 (UTC)

:Why should it be merged? Computer Science seems well-enough designed. What is the incentive to collapse it into a broader field of study? —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:18, 8 October 2024 (UTC)
::Portals as top level organizations allowing for content to be best centralized. Also note that I did not start the merge, just offering to finish it. Perhaps a {{tlx|prod}} instead? [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 07:20, 8 October 2024 (UTC)
:::I have no objections, personally. If it gets done, please use a redirect and should someone want to come along to resurrect it later, it will be easier. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:21, 8 October 2024 (UTC)
: Is computer science really a branch of information sciences? I would not think so, but what do I know. Do we have some external resources/links confirm this idea? [[W:Information science]] currently says: &quot;Information science, documentology[1] or informatology[2][3] is an academic field which is primarily concerned with analysis, collection, classification, manipulation, storage, retrieval, movement, dissemination, and protection of information.&quot; --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:49, 11 October 2024 (UTC)
::Looking through [https://stackoverflow.com/q/1047014/22673230] [https://businessdegrees.uab.edu/mis-degree-bachelors/resources/computer-information-systems-vs-computer-science/] [https://www.si.umich.edu/student-experience/what-information-science] a few top (not necessarily RS) searches I'm inclined to agree. I am more familiar with the grafted [[:w:Information and computer science|information ''and'' computer science]] which makes an effort to merge the disciplines, but it does not seem like reaching to say that IS is presented as more applications-concerned (certainly with no lack of theoretical abstraction), whereas CS can be more freely associated with any and all 'science related to computers'. It is easy to reason about the connection between the fields, but I think it is clear academia maintains this taxonomy for a good reason.
::With these considerations, I think I will ''stop'' the process of merging in favor of expanding the existing [[School:Library and Information Science]]. 

::Let me know if there is not consensus to redirect [[Portal:Information sciences]] to [[School:Library and Information Science]] (with enough expansion it can generalize away from just library sciences). [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 16:16, 11 October 2024 (UTC)
::: I do not see that a merge of a ''portal'' to a ''school'' is a good thing. Do you have a clear idea of the concepts of school and portal and how they relate to each other? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:34, 11 October 2024 (UTC)
::::Found [[:Category:Information sciences]]; there are enough existing resources in there to make my other proposed merge excessive. I will simply continue developing the existing [[Portal:Information sciences]] instead. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 17:05, 11 October 2024 (UTC)
::::: Frankly, I would ideally see [[Portal:Information sciences]] deleted: I don't see what it does that a category would not do well enough. There does not seem to be any material specific to &quot;Information sciences&quot; (whatever that is) in that portal at all. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:11, 11 October 2024 (UTC)
::::::Tacked a {{tlx|prod}} for an eventual deletion, but I may still try to develop it as proof of concept at some point. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 17:33, 11 October 2024 (UTC)

== [[:Category:Occupational Epidemiology]] ==

I propose moving the pages in this category (without leaving redirects) to their equivalent under the parent resource [[Occupational Health Risk Surveillance]]. Also due to the number of subpages, it seems &lt;code&gt;|filing=deep&lt;/code&gt; would be a justified. (Also [[Special:PrefixIndex/Occupational_Epidemiology|there are quite a few]] untagged subpages.) [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 05:11, 9 October 2024 (UTC)
: I above all think that the content should be ''moved out of the mainspace'': I do not see readers learning anything from e.g. [[Occupational Epidemiology/Research tools/Reading of scientific articles for learning epidemiology and biostatstics]] or [[Occupational Epidemiology/Research tools/Ongoing projects/Risk Communication in Seafaring/Writing the article guideline IMRAD]]. Wikiversity can be kind enough to host that material in, say, subspace of [[User:Saltrabook]], but more should not be asked, I think. Let us recall that per [[WV:Deletions]], &quot;Resources may be eligible for proposed deletion when education objectives and learning outcomes are scarce, and objections to deletion are unlikely&quot;; I do not see how learning outcomes can be anything but scarce. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 15:04, 11 October 2024 (UTC)
::thank you, agree @ [[User:Saltrabook|Saltrabook]] ([[User talk:Saltrabook|discuss]] • [[Special:Contributions/Saltrabook|contribs]]) 21:03, 13 November 2024 (UTC)

== Active editors ==

It is interesting to observe the stats on [https://stats.wikimedia.org/#/en.wikiversity.org/contributing/active-editors/normal|line|all|(page_type)~content*non-content|monthly active editors] through the project's history. October is our month! [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 20:44, 8 October 2024 (UTC)

:Odd. Maybe related to the school year? —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 02:10, 9 October 2024 (UTC)
::I wonder how many are [[User:Jtneill|Jtneill]]'s crowd... the number is in the hundreds though, so that is one chunky cohort —[[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:16, 9 October 2024 (UTC)
:::Yes, [[Motivation and emotion/Book]] involves ~100-150 students editing most intensely during October each year. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:20, 9 October 2024 (UTC)
::::Neat, that still leaves around ~50-100 other students from other avenues each year since 2021. I also wonder which projects were involved in the COVID enrollment spike. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:26, 9 October 2024 (UTC)
:::::Personally I can admit that my editing is much more active during the school season vs. the summer break, so I'm in the same boat as Jtneill's students. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:24, 13 November 2024 (UTC)

== Intentionally incorrect resource ==

There is a [[Special:Diff/2583464|disclaimer inserted onto a resource]] (by not the original author) that:
&lt;blockquote&gt;I am merely [making this page false] to show you (The viewer) that Wikipedia and this page 'Wikiversity' is bull sh*t and it will not give you the reliability you need when writing an academic piece of writing.&lt;/blockquote&gt;
However, that IP has [[Special:Contributions/86.22.73.151|not made any other edits]], so unless they vandalized via a sock, the intent went un-realized and only that portion need be removed. Bumping here in case there is some obvious jumbo in that essay that someone else can catch. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 16:58, 9 October 2024 (UTC)

:Removed that portion, which was obviously vandalism. No perspective on the rest of the essay. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:38, 9 October 2024 (UTC)

== [[:Category:Proposed guidelines]] ==

Noting for future editors that WV has collapsed all proposals into [[:Category:Proposed policies|proposed policies]]. Seeking consensus to further collapse [[:Category:Wikiversity proposals]] into the former, or to restore [[:Category:Proposed guidelines]]. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 19:19, 9 October 2024 (UTC)

== [[Around Wikiversity in 80 Seconds|Broken 80-second tour]] ==

Bumping a [[Talk:Around_Wikiversity_in_80_Seconds|comment]] on the ''Wikiversity in 80 seconds'' tour. Appears wikisuite is not working with the Vector 2022 appearance. 

Also see [[:w:Wikipedia:Miscellany_for_deletion/Wikiversuite_pages|this thread]] on the Wikiversal package - may not be relevant to Wikiversity, but FYC. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 00:26, 10 October 2024 (UTC)
: I would just delete the material; I do not see value in it. If others agree, I would try to articulate why I think it should be deleted (or move to author user space). --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:57, 13 October 2024 (UTC)
::Just mark as {{tl|historical}}. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:39, 13 October 2024 (UTC)
::: This thing was created by [[User:Planotse]]. His creations are now being discussed in Wikibooks for deletion: [[B:Wikibooks:Requests for deletion#Wikiversal generated pages]]. It seems he used some kind of tool that is no longer available (the above mentioned &quot;Wikiversal&quot; package) to create this kind of slideshow-like material (believing the Wikibooks discussion). I do not see value of this in the mainspace, not even as historical (I am okay with userspace, but maybe even that is not the best option?). A look at the source code of [[Around Wikiversity in 80 Seconds/Introduction]] confirms the words of Omphalographer, namely that &quot;the HTML-heavy markup generated by Wikiversal makes them [the pages] unreasonably difficult to edit.&quot;
::: I went ahead and marked the page for proposed deletion. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:35, 14 October 2024 (UTC)

== Preliminary results of the 2024 Wikimedia Foundation Board of Trustees elections ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
Hello all,

Thank you to everyone who participated in the [[m:Special:MyLanguage/Wikimedia Foundation elections/2024|2024 Wikimedia Foundation Board of Trustees election]]. Close to 6000 community members from more than 180 wiki projects have voted.

The following four candidates were the most voted:

# [[User:Kritzolina|Christel Steigenberger]]
# [[User:Nadzik|Maciej Artur Nadzikiewicz]]
# [[User:Victoria|Victoria Doronina]]
# [[User:Laurentius|Lorenzo Losa]]

While these candidates have been ranked through the vote, they still need to be appointed to the Board of Trustees. They need to pass a successful background check and meet the qualifications outlined in the Bylaws. New trustees will be appointed at the next Board meeting in December 2024.

[[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Results|Learn more about the results on Meta-Wiki.]]

Best regards,

The Elections Committee and Board Selection Working Group

&lt;section end=&quot;announcement-content&quot; /&gt;

[[User:MPossoupe_(WMF)|MPossoupe_(WMF)]] 08:26, 14 October 2024 (UTC)
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== Seeking volunteers to join several of the movement’s committees ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
Each year, typically from October through December, several of the movement’s committees seek new volunteers. 

Read more about the committees on their Meta-wiki pages:
* [[m:Special:MyLanguage/Affiliations_Committee|Affiliations Committee (AffCom)]]
* [[m:Special:MyLanguage/Ombuds_commission|Ombuds commission (OC)]]
* [[m:Special:MyLanguage/Wikimedia Foundation/Legal/Community Resilience and Sustainability/Trust and Safety/Case Review Committee|Case Review Committee (CRC)]]

Applications for the committees open on 16 October 2024. Applications for the Affiliations Committee close on 18 November 2024, and applications for the Ombuds commission and the Case Review Committee close on 2 December 2024. Learn how to apply by [[m:Special:MyLanguage/Wikimedia_Foundation/Legal/Committee_appointments|visiting the appointment page on Meta-wiki]]. Post to the talk page or email [mailto:cst@wikimedia.org cst@wikimedia.org] with any questions you may have.

For the Committee Support team,
&lt;section end=&quot;announcement-content&quot; /&gt;

-- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 23:09, 16 October 2024 (UTC)
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== Interactive elements ==
Can we use interactive elements on Wikiversity? I'd like to add JavaScript to a page. If it's not possible now, where can I suggest this feature? I have a safe integration idea.
[[User:Отец Никифор|Отец Никифор]] ([[User talk:Отец Никифор|discuss]] • [[Special:Contributions/Отец Никифор|contribs]]) 12:10, 17 October 2024 (UTC)
: This is beyond my technical knowledge, but have you checked out:
:* https://www.mediawiki.org/wiki/Manual:Interface/JavaScript?
:* [[Wikipedia:WikiProject JavaScript]]
:* [[MediaWiki:Common.js]]

:What sort of interactive elements are you thinking about?

: Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:39, 18 October 2024 (UTC)
::I was thinking about adding something like a graph with adjustable controls, where users can interact with it and see how different changes affect the outcome. It seems like this could be a useful feature. There might already be discussions about enhancing Wikiversity or similar platforms—perhaps on a relevant talk page or in a Discord group. Do you know where such discussions might be happening? [[User:Отец Никифор|Отец Никифор]] ([[User talk:Отец Никифор|discuss]] • [[Special:Contributions/Отец Никифор|contribs]]) 19:47, 18 October 2024 (UTC)
:::From a quick look, maybe check out:
:::* [[mw:Extension:Graph]]
:::* [[phab:tag/graphs]]
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:40, 18 October 2024 (UTC)
:::: mw:Extension:Graph is currently disabled on Wikipedia etc. wikis, for security reasons, and seems unlikely to be enabled again. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:30, 19 October 2024 (UTC)

== An unexplained spurt of Wikiversity page views ==
The [https://pageviews.wmcloud.org/siteviews/?platform=all-access&amp;source=pageviews&amp;agent=user&amp;start=2024-06-01&amp;end=2024-10-18&amp;sites=en.wikiversity.org|en.wikibooks.org|en.wikiquote.org|en.wikisource.org page view report] shows an unexplained spurt of Wikiversity page views, reaching over 4 times the baseline and then falling back again. Does anyone have any idea what is going on? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:01, 19 October 2024 (UTC)

:Interesting. I wonder why only the English wikiquote and wikiversity and not Wikisource or wikibooks? How reliable do you think those stats are? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 15:44, 8 December 2024 (UTC)

== Center tempate failed on a contributors phone... ==

See the edit comment here - https://en.wikiversity.org/w/index.php?title=Wikiphilosophers&amp;diff=prev&amp;oldid=2673962. I'm puzzled as this is the first failure of this, I've noted recently. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 08:45, 19 October 2024 (UTC)

== Essay-like page in user space that makes little sense and seems incoherent ==
The page [[User:TheoYalur/Illusions]] seems to match the description, at least by my assessment. My understanding is that since the page is only in user space and not in the mainspace, it can stay there even if it has those disqualifying qualities. But if I am wrong and the page belongs deleted, please correct me and let me know. I do not know which policy or guideline, if any, guides the case. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:30, 21 October 2024 (UTC)

== 'Wikidata item' link is moving, finally. ==

Hello everyone, I previously wrote on the 27th September to advise that the ''Wikidata item'' sitelink will change places in the sidebar menu, moving from the '''General''' section into the '''In Other Projects''' section. The scheduled rollout date of 04.10.2024 was delayed due to a necessary request for Mobile/MinervaNeue skin. I am happy to inform that the global rollout can now proceed and will occur later today, 22.10.2024 at 15:00 UTC-2. [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Please let us know]] if you notice any problems or bugs after this change. There should be no need for null-edits or purging cache for the changes to occur. Kind regards, -[[m:User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] 11:28, 22 October 2024 (UTC)
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:Hi @[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]]: I Just noticed your post above, and it is timely.
:I have been participating in the English WikiUniversity for a few years, much less often recently. I seems like something in the way the site displays is different, but I cannot put my finger on it. Your posting gave me a clue. Can you please tell me where  the link to wikidata items has moved to? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:23, 11 December 2024 (UTC)
::Hello @[[User:Ottawahitech|Ottawahitech]], sure, I would be happy to. The button/sitelink name didn't change, just its position. You should find it in the sidebar-menu under the section '''In other projects''' (where the links to all other Wikimedia Projects are displayed). If you do not see it, please reach out to us on the [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Move Wikidata item - Discussion page]]. Thank you, -[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] ([[User talk:Danny Benjafield (WMDE)|discuss]] • [[Special:Contributions/Danny Benjafield (WMDE)|contribs]]) 09:24, 12 December 2024 (UTC)
:::@[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]], thank you for responding. I intend to followup on the ''Move Wikidata item - Discussion page'' as per your post above by putting it on my ever growing todo list.
:::I don't know about others on this wiki, as I said I have not been visiting here frequently, but for me the constant changes are a big distraction. I have been around wikimedia projects since 2007, so why do I have to spend so much time learning and re-learning how to find what I came here for? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:41, 12 December 2024 (UTC)
::::Hi @[[User:Ottawahitech|Ottawahitech]], thanks for you thoughts. Your input whether positive or critical helps us understand the impacts to editors so we welcome your further thoughts when you reach us in your To Do List :)
::::I can't speak about the other changes you've experienced here but I do hope they are made with a spirit of improvement for the community as a whole. -[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] ([[User talk:Danny Benjafield (WMDE)|discuss]] • [[Special:Contributions/Danny Benjafield (WMDE)|contribs]]) 10:43, 16 December 2024 (UTC)

== Final Reminder: Join us in Making Wiki Loves Ramadan Success ==

Dear all,

We’re thrilled to announce the Wiki Loves Ramadan event, a global initiative to celebrate Ramadan by enhancing Wikipedia and its sister projects with valuable content related to this special time of year. As we organize this event globally, we need your valuable input to make it a memorable experience for the community.

Last Call to Participate in Our Survey: To ensure that Wiki Loves Ramadan is inclusive and impactful, we kindly request you to complete our community engagement survey. Your feedback will shape the event’s focus and guide our organizing strategies to better meet community needs.

* Survey Link: [https://docs.google.com/forms/d/e/1FAIpQLSffN4prPtR5DRSq9nH-t1z8hG3jZFBbySrv32YoxV8KbTwxig/viewform?usp=sf_link Complete the Survey]
* Deadline: November 10, 2024
Please take a few minutes to share your thoughts. Your input will truly make a difference!

'''Volunteer Opportunity''': Join the Wiki Loves Ramadan Team! We’re seeking dedicated volunteers for key team roles essential to the success of this initiative. If you’re interested in volunteer roles, we invite you to apply.
* Application Link: [https://docs.google.com/forms/d/e/1FAIpQLSfXiox_eEDH4yJ0gxVBgtL7jPe41TINAWYtpNp1JHSk8zhdgw/viewform?usp=sf_link Apply Here]
* Application Deadline: October 31, 2024
Explore Open Positions: For a detailed list of roles and their responsibilities, please refer to the position descriptions here: [https://docs.google.com/document/d/1oy0_tilC6kow5GGf6cEuFvdFpekcubCqJlaxkxh-jT4/ Position Descriptions]

Thank you for being part of this journey. We look forward to working together to make Wiki Loves Ramadan a success!


Warm regards,&lt;br&gt;
The Wiki Loves Ramadan Organizing Team 05:11, 29 October 2024 (UTC)
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== Android app for Wikiversity ==

Hi, is there an Android app for Wikiversity? How does it work? I have been advised that there is no infrastructure for push notifications for Android apps for sister wikis and I would be interested to know more. Related: [[:phab:T378545]]. Thanks! [[User:Gryllida|Gryllida]] 23:15, 29 October 2024 (UTC)

:Thanks for suggesting this - I agree that it would be useful. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:56, 31 October 2024 (UTC)
:@[[User:Gryllida|Gryllida]]: Would you explain your terminology  for those of us not in the know. What does ''push notifications'' mean? I use [https://www.mediawiki.org/wiki/Help:Notifications notifications] when I am communicating on wikimedia projects, but have never heard this term before. [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:13, 11 December 2024 (UTC)

à

== Import Resource From Wikibooks? ==

Hello! [[wikibooks:Character_List_for_Baxter&amp;Sagart|Character List for Baxter&amp;Sagart]] and related titles [[wikibooks:Wikibooks:Requests_for_deletion#Character_List_for_Baxter&amp;Sagart|are up for deletion at Wikibooks]] because WB policy does not allow dictionaries like them. However, because they are useful as learning tools, I am wondering if they might have a home here at Wikiversity. Pinging @[[User:Tibetologist|Tibetologist]] here to link them in to this discussion, since they are the affected user. Thank you! —[[User:Kittycataclysm|Kittycataclysm]] ([[User talk:Kittycataclysm|discuss]] • [[Special:Contributions/Kittycataclysm|contribs]]) 18:18, 1 November 2024 (UTC)

:Sure, I can do it. That said, as mentioned there, it does seem like something like this is ideally suited for Wiktionary in the Appendix namespace, but I'm not very familiar with CJK characters and languages. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:23, 3 November 2024 (UTC)
::Oh man, these pages are too big to import and while I've already tried a half-dozen times, it will constantly fail. Strictly speaking, we don't have to use the import feature for licensing purposes. We can just copy and paste the contents and list the usernames or on the talk page. I think that's the solution. {{Ping|Tibetologist}}, are you interested in doing that? If you just copied and pasted these pages and then added [[:Category:Chinese]] and maybe include a couple of links to the pages, that would probably be ideal. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:31, 3 November 2024 (UTC)

== Language translation requests? ==

Is there anywhere on Wikiversity to request translation, for example, requesting Latin or French translation? I would be asking from the context as a student, so I would be interested in translation explanation as well. [[User:Indexcard88|Indexcard88]] ([[User talk:Indexcard88|discuss]] • [[Special:Contributions/Indexcard88|contribs]]) 04:56, 20 November 2024 (UTC)

== Sign up for the language community meeting on November 29th, 16:00 UTC ==

Hello everyone,

The next language community meeting is coming up next week, on November 29th, at 16:00 UTC (Zonestamp! For your timezone &lt;https://zonestamp.toolforge.org/1732896000&gt;). If you're interested in joining, you can sign up on this wiki page: &lt;https://www.mediawiki.org/wiki/Wikimedia_Language_and_Product_Localization/Community_meetings#29_November_2024&gt;.

This participant-driven meeting will be organized by the Wikimedia Foundation’s Language Product Localization team and the Language Diversity Hub. There will be presentations on topics like developing language keyboards, the creation of the Moore Wikipedia, and the language support track at Wiki Indaba. We will also have members from the Wayuunaiki community joining us to share their experiences with the Incubator and as a new community within our movement. This meeting will have a Spanish interpretation.

Looking forward to seeing you at the language community meeting! Cheers, [[User:SSethi (WMF)|Srishti]] 19:55, 21 November 2024 (UTC)
&lt;!-- Message sent by User:SSethi (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&amp;oldid=27746256 --&gt;

== Events on Wikiversity ==

Since Wikipedia and Wikivoyage are having their &quot;Asian Month&quot; editathon, I was thinking if we could start up a Wikiversity version of that. This would be an &quot;Asian Month&quot; as well, but it would be about creating resources based on Asia and its culture. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 17:57, 6 December 2024 (UTC)

:Not immediately opposed, but the question is, do we have an active enough community to facilitate this? —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:31, 6 December 2024 (UTC)
::I'm not too sure. As long as we get enough traffic, this could happen. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 08:45, 7 December 2024 (UTC)
:::This is to increase traffic on Wikiversity, which is promoted amongst other communities. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 10:47, 7 December 2024 (UTC)
:Hi @[[User:RockTransport|RockTransport]], This is a good idea, but will it also involve users who are not &quot;professors and scientists&quot;. Just curious. cheers, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:30, 9 December 2024 (UTC)
::Yes, considering the fact that Wikiversity is for everyone, and not just for specific users. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 09:09, 10 December 2024 (UTC)
:::because I'm personally not a &quot;professor&quot; or a &quot;scientist&quot; and because '''anyone''' can create resources on Wikiversity. We want to make Wikiversity open for everyone, and not just for certain users. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 09:10, 10 December 2024 (UTC)
::::I am also not a professor or a scientist, but it seems to me that as result I am viewed here as a visitor rather than someone who can contribute. Just my $.02. [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:05, 12 December 2024 (UTC)

== Wikiversity - Newsletters ==

Hello All,

I wanted to create a newsletter on Wikiversity, which would highlight what is going on in certain months and events on Wikiversity; which would bolster engagement by many people. This would be on the website and would have its dedicated 'Newsletter' tab.

I hope you acknowledge this idea. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 21:05, 8 December 2024 (UTC)

:@[[User:RockTransport|RockTransport]], What sort of things do you plan to include in your newsletter? Will they be different than what is currently in [[Main Page/News]]? Just curious.
:I am also wondering about your motive which I think is: to bolster engagement by many people. I am asking because I wonder if others who are currently active here also think this I is desirable? Have you asked them? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:34, 11 December 2024 (UTC)
::Not yet, which was why I was asking this on the colloquium. I plan to include things that many people have created on Wikiversity over the month, as it is a monthly newsletter. It would be somewhere on the website here. It will be more frequent that the ones seen on [[Main Page/News]]. We will include people's resources to essentially promote them. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 06:50, 12 December 2024 (UTC)
:::@[[User:RockTransport|RockTransport]], I Think what you are saying is that ''Main Page/News'' does not update frequently enough? 
:::If this is the reason, why not start small by simply increasing the frequency of posting news on the main page, instead of trying to start a newsletter?
:::If there is more, can you articulate what else is missing. Thanks in advance, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:51, 12 December 2024 (UTC)
::::I meant going to detail into topics covered in that month, rather than just giving a few points. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 16:53, 12 December 2024 (UTC)
:::::What sort of details did you have in mind? You can pick one of the links provided in [[Main Page/News?]] to illustrate. cheers, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 15:29, 16 December 2024 (UTC)</text>
      <sha1>74kxodes2go1x2zbfeu183a5qsug70e</sha1>
    </revision>
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      <id>2692195</id>
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      <text bytes="76730" sha1="8fw4gjy5i9d6e313phxl0f007apq066" xml:space="preserve">{{Wikiversity:Colloquium/Header}}
&lt;!-- MESSAGES GO BELOW --&gt;
== Reminder! Vote closing soon to fill vacancies of the first U4C ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
:''[[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement – reminder to vote|You can find this message translated into additional languages on Meta-wiki.]] [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement – reminder to vote}}&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]''

Dear all,

The voting period for the Universal Code of Conduct Coordinating Committee (U4C) is closing soon. It is open through 10 August 2024. Read the information on [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2024_Special_Election#Voting|the voting page on Meta-wiki to learn more about voting and voter eligibility]]. If you are eligible to vote and have not voted in this special election, it is important that you vote now.

'''Why should you vote?''' The U4C is a global group dedicated to providing an equitable and consistent implementation of the UCoC. Community input into the committee membership is critical to the success of the UCoC.

Please share this message with members of your community so they can participate as well.

In cooperation with the U4C,&lt;section end=&quot;announcement-content&quot; /&gt;

-- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 15:30, 6 August 2024 (UTC)
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== User group for Wikiversians ==

Was there ever a discussion about the possibility of establishing a user group in the sense of an affiliated organization that would defend the interests of professors and scientists on Wikiversity and possibly actively develop some projects? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:21, 8 August 2024 (UTC)

:Not that I'm aware of. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:20, 8 August 2024 (UTC)
:It's a pleasure to talk to a scientist on Wikiversity. I am a historian of technics and I would like to publish the following biography either on Wikiversity or on Wikipedia: 
:https://en.wikiversity.org/wiki/User:Rbmn/Arthur_Constantin_KREBS_(1850-1935):_Military_engineer,_Automotive_industrialist,_Great_projects_manager
:What would be your advice? [[User:Rbmn|Rbmn]] ([[User talk:Rbmn|discuss]] • [[Special:Contributions/Rbmn|contribs]]) 15:44, 6 October 2024 (UTC)
::The content appears to be largely biographical/encyclopedic, so I think it is likely best suited to Wikipedia. Consider improving/incorporating this content into the existing page: [[w:Arthur Constantin Krebs]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:05, 7 October 2024 (UTC)
::Please do not link to the Wikiversity [[wv:userspace|Userspace]] in Wikipedia articles. You will want to wait until you have a page in the [[wv:mainspace|Wikiversity mainspace]]. You'll also want to use the &lt;code&gt;{{[[:w:Template:Wikiversity|Wikiversity]]}}&lt;/code&gt; template (on Wikipedia) rather than embedding a photo with a link. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:21, 7 October 2024 (UTC)
:I haven't heard anything about this topic. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 21:06, 8 December 2024 (UTC)

== Rich's ''Illustrated Companion'' at Wikiversity: Right place? ==

Hello! I am creating a Wiki-version of a classical glossary (''Illustrated Companion to the Latin Dictionary, and Greek Lexicon'' by Anthony Rich, 1849), which explains the meaning of Latin headwords, primarily those &quot;representing visible objects connected with the arts, manufactures, and every-day life of the Greeks and Romans.&quot; The aim is to help understand what a (classical) Latin text is actually about, instead of merely translating it. I already transcribed the entire text and scanned the images (about 1900) from an original 1849-edition. I am currently working on uploading the images to ''Mediawiki Commons'', which probably will take some time. In the meantime I want to prepare the other aspects of the project (more than 3000 articles, already with many internal links). The important thing: this is ''not'' a ''might exist''-project.  {{Color|red|My question: Is ''Wikiversity'' the proper place for it?}} Although I created an exact rendition of the original text, ''Wikisource'' is not applicable, because the project has a broader scope (adding content to the articles, e. g. links to online editions for quotations, adding images, but also adding entirely new articles). Neither is ''Wikibooks'', because this is not a textbook and may otherwise breach its scope. For more about the project see [[w:User:CalRis25/Temp-RICH-Prospectus|my user-page]] at en.wikipedia. {{Color|Red|So, is Wikiversity the right place for it?}} [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 09:15, 17 August 2024 (UTC)

:Thanks for asking. To be clear, it ''is'' acceptable to make [[:s:en:Category:Wikisource annotations|annotated editions]] of texts at Wikisource and Wikibooks does host at least one [[:b:en:Annotations of The Complete Peanuts|annotated guide to a copyright-protected work]]. So if what you're looking to do is to include inline annotations to a public domain text, you certainly can put that on Wikisource. If you have a textbook or guidebook that is a companion, that would go at Wikibooks. If you have some other kind of learning resources (like maintaining a list of relevant links, organizing a book reading group, etc.), that could go here. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:26, 17 August 2024 (UTC)
::Thank you for your quick answer. Actually, ''Wikibooks'' was my first thought. However, this project is not merely an annotated edition. Although at first it ''will'' be a faithful copy of the original text, I want the project to be &quot;open&quot;, i. e. adding articles should be possible. And the project should enable to do a lot more than mere inline annotation. See section [[w:User:CalRis25/Temp-RICH-Prospectus#Improving_RICH|Improving Rich]] in the project description a my user-page (en.Wikipedia). No ''Mediawiki''-project (Wikisource, Wikibooks, Wikipedia, Wiktionary) seemed to be a sufficiently applicable &quot;fit&quot; for the project, so I thought of Wikiversity as a last resort, because it is supposed to be home to all sorts of &quot;learning resources&quot;. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 09:57, 17 August 2024 (UTC)
:::The scope of Wikiversity ''is'' pretty catch-all and would allow for a pretty flexible place to host most learning resources that don't fit elsewhere.
:::Also, as nitpick, &quot;MediaWiki&quot; is the software that is the basis of these wikis (wikis being collections of interlinked documents that can be edited) and &quot;Wikimedia Foundation&quot; is the non-profit who owns the trademarks and hosts these projects like Wiktionary and Wikivoyage. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 10:06, 17 August 2024 (UTC)
::::Hello Justin, thank you for the reply. '''I think that settles it. I will create this project at ''Wikiversity''.''' Just for additional clarification, why I do so. Let's imagine a full transcription of the original 1849-edition of the ''Illustrated Companion'' by Anthony Rich and call it ''RICH-1849''. We shall call my project, for brevity sake, RICH-2K. And now, let's have a look at the article about the Roman toga (a piece of attire). In ''RICH-1849'' we can can call it ''RICH-1849/Toga'', and it contains ''exactly'' the content of the 1849-book. Now, let's look at the article ''RICH-2K/Toga''. At the beginning its only content would be the article ''RICH-1849/Toga''. Does that make ''RICH-2K/Toga'' and ''RICH-1849/Toga'' the same? Not at all, because in truth ''RICH-2K/Toga'' is a &quot;container&quot; which initially contains only the article ''RICH-1849/Toga'' but later on may include more stuff: images, external links, article text which builds on or extends ''RICH-1849/Toga'' and information from other sources of information (Wikipedia, specialized books). By the way, this added article information would not be a mere copy of the text at en.Wikipedia, because the information needs to looked at through the eyes of someone reading the original text (more citations with direct links to these etc.). [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 11:39, 17 August 2024 (UTC)

== Coming soon: A new sub-referencing feature – try it! ==

&lt;section begin=&quot;Sub-referencing&quot;/&gt;
[[File:Sub-referencing reuse visual.png|{{#ifeq:{{#dir}}|ltr|right|left}}|400px]]
Hello. For many years, community members have requested an easy way to re-use references with different details. Now, a MediaWiki solution is coming: The new sub-referencing feature will work for wikitext and Visual Editor and will enhance the existing reference system. You can continue to use different ways of referencing, but you will probably encounter sub-references in articles written by other users. More information on [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing|the project page]].

'''We want your feedback''' to make sure this feature works well for you:

* [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing#Test|Please try]] the current state of development on beta wiki and [[m:Talk:WMDE Technical Wishes/Sub-referencing|let us know what you think]]. 
* [[m:WMDE Technical Wishes/Sub-referencing/Sign-up|Sign up here]] to get updates and/or invites to participate in user research activities.

[[m:Special:MyLanguage/Wikimedia Deutschland|Wikimedia Deutschland]]’s [[m:Special:MyLanguage/WMDE Technical Wishes|Technical Wishes]] team is planning to bring this feature to Wikimedia wikis later this year. We will reach out to creators/maintainers of tools and templates related to references beforehand.

Please help us spread the message. --[[m:User:Johannes Richter (WMDE)|Johannes Richter (WMDE)]] ([[m:User talk:Johannes Richter (WMDE)|talk]]) 10:36, 19 August 2024 (UTC)
&lt;section end=&quot;Sub-referencing&quot;/&gt;
&lt;!-- Message sent by User:Johannes Richter (WMDE)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=User:Johannes_Richter_(WMDE)/Sub-referencing/massmessage_list&amp;oldid=27309345 --&gt;

== New [[Template:Form]] ==

Hi! Today I was bold and created [[Template:Form]] (which calls [[Module:WikiForm]] and [[MediaWiki:Gadget-WikiForm.js]]). The template allows to create user-friendly forms that can create pages or add content to existing pages. My motivation and first use case was [[Wikidebate/New|this form]] to create new [[wikidebates]], but I suspect the template can be useful elsewhere on Wikiversity. Let me know if you notice any issues or have any requests or concerns. Kind regards, [[User:Sophivorus|Sophivorus]] ([[User talk:Sophivorus|discuss]] • [[Special:Contributions/Sophivorus|contribs]]) 15:21, 21 August 2024 (UTC)

== Sign up for the language community meeting on August 30th, 15:00 UTC ==

Hi all,

The next language community meeting is scheduled in a few weeks—on August 30th at 15:00 UTC. If you're interested in joining, you can [https://www.mediawiki.org/wiki/Wikimedia_Language_and_Product_Localization/Community_meetings#30_August_2024 sign up on this wiki page].

This participant-driven meeting will focus on sharing language-specific updates related to various projects, discussing technical issues related to language wikis, and working together to find possible solutions. For example, in the last meeting, topics included the Language Converter, the state of language research, updates on the Incubator conversations, and technical challenges around external links not working with special characters on Bengali sites.

Do you have any ideas for topics to share technical updates or discuss challenges? Please add agenda items to the document [https://etherpad.wikimedia.org/p/language-community-meeting-aug-2024 here] and reach out to ssethi(__AT__)wikimedia.org. We look forward to your participation!

[[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 23:20, 22 August 2024 (UTC)
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== Template consolidation: User talk page block notice ==
Wondering if someone who likes templates could have a go at consolidating or helping decide between use of:
* [[Template:Block]]
* [[Template:Blocked]]

Unless I'm missing something, it seems like we don't need both?

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 07:16, 23 August 2024 (UTC)
: I tried to figure out a Wikidata item with most links to projects. I found this: [[Wikidata:Q6379131]], which is Template:Uw-block. There is even a corresponding Wikiversity template, [[Template:Uw-block1]] (not used anywhere).
: My impression is that of the three templates, we only need one. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:43, 13 September 2024 (UTC)

== Announcing the Universal Code of Conduct Coordinating Committee ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
:''[https://lists.wikimedia.org/hyperkitty/list/board-elections@lists.wikimedia.org/thread/OKCCN2CANIH2K7DXJOL2GPVDFWL27R7C/ Original message at wikimedia-l]. [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement - results|You can find this message translated into additional languages on Meta-wiki.]] [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement - results}}&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]''

Hello all,

The scrutineers have finished reviewing the vote and the [[m:Special:MyLanguage/Elections Committee|Elections Committee]] have certified the [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Results|results]] for the [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election|Universal Code of Conduct Coordinating Committee (U4C) special election]].

I am pleased to announce the following individual as regional members of the U4C, who will fulfill a term until 15 June 2026:

* North America (USA and Canada)
** Ajraddatz

The following seats were not filled during this special election:

* Latin America and Caribbean
* Central and East Europe (CEE)
* Sub-Saharan Africa
* South Asia
* The four remaining Community-At-Large seats

Thank you again to everyone who participated in this process and much appreciation to the candidates for your leadership and dedication to the Wikimedia movement and community.

Over the next few weeks, the U4C will begin meeting and planning the 2024-25 year in supporting the implementation and review of the UCoC and Enforcement Guidelines. You can follow their work on [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee|Meta-Wiki]].

On behalf of the U4C and the Elections Committee,&lt;section end=&quot;announcement-content&quot; /&gt;

[[m:User:RamzyM (WMF)|RamzyM (WMF)]] 14:07, 2 September 2024 (UTC)
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== Re: The Vector 2022 skin as the default in two weeks? ==

[[File:Vector 2022 video-en.webm|thumb|A two minute-long video about Vector 2022]]
Hello everyone, I'm reaching out on behalf of the [[mediawikiwiki:Reading/Web|Wikimedia Foundation Web team]] responsible for the MediaWiki skins. I'd like to revisit the topic of making Vector 2022 the default here on English Wikiversity. I [[Wikiversity:Colloquium/archives/September 2022#The Vector 2022 skin as the default in two weeks?|did post a message about this almost two years ago]] (where you can find all the details about the skin), but we didn't finalize it back then.

What happened in the meantime? We built [[mw:Reading/Web/Accessibility for reading|dark mode and different options for font sizes]], and made Vector 2022 the default on most wikis, including all other Wikiversities. With the not-so-new V22 skin being the default, existing and coming features, like dark mode and [[mw:Trust and Safety Product/Temporary Accounts|temporary accounts]] respectively, will become available for logged-out users here.

So, if no large concerns are raised, we will deploy Vector 2022 here in two weeks, in the week of September 16. Do let me know if you have any questions. Thank you! [[User:SGrabarczuk (WMF)|SGrabarczuk (WMF)]] ([[User talk:SGrabarczuk (WMF)|discuss]] • [[Special:Contributions/SGrabarczuk (WMF)|contribs]]) 21:48, 2 September 2024 (UTC)

:Sounds good, Szymon - we look forward to the upcoming change of skin {{smile}} Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 07:35, 13 September 2024 (UTC)
* I for one oppose a switch to Vector 2022. I do not find it preferable. Here is a staggering evidence of user refusal of Vector 2022 once it was deployed: [[W:en:Wikipedia:Requests for comment/Rollback of Vector 2022]], Junuary 2023. 355 voters supported rollback to Vector 2010 whereas 64 opposed, yielding 84.7% support, as clear a supermajority as one may wish. These people opposing Vector 2022 feel the same way as I do. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:48, 13 September 2024 (UTC)
*:Hey @[[User:Dan Polansky|Dan Polansky]]. Thanks for your comment. I'm open to discussion about problems with our software, and I hope we can maintain a respectful tone.
*:I understand that there are users who prefer Vector legacy or other skins, just as there are people who still stick to Monobook. Such people are active across many wikis. They can keep Vector legacy, although non-default skins don't have the support the default ones do. We are rolling out for technical reasons, as I mentioned above, with benefit to not logged-in users.
*:Regarding the rollback RfC on Wikipedia, two neutral users stated that there was no consensus for rollback, RfC is not a vote, and the numbers were different (355:226:24). I believe this all is pretty easy to verify.
*:So to sum up, Vector 2022 needs to become the default, tons and tons of comments were made about the skin and related stuff, and we have taken many ideas into account, and it's totally OK if you stick to Vector legacy.
*:Thanks! [[User:SGrabarczuk (WMF)|SGrabarczuk (WMF)]] ([[User talk:SGrabarczuk (WMF)|discuss]] • [[Special:Contributions/SGrabarczuk (WMF)|contribs]]) 19:30, 16 September 2024 (UTC)
*:: Today, I visited Wikiversity and found it switched to Vector 2022. I changed my preference settings to Vector 2010. From what I understand, non-registered visitors are now defaulted to Vector 2022 despite its unpopularity in [[W:en:Wikipedia:Requests for comment/Rollback of Vector 2022]]. I have not seen any evidence that users prefer Vector 2022, and given the evidence in the linked RfC, I tentatively conclude that the decision to switch has made the site experience worse for the majority of users. The logic of &quot;you can switch&quot; surely applies to Vector 2022 as well: those who prefer it can switch to it. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:08, 17 September 2024 (UTC)

== Have your say: Vote for the 2024 Board of Trustees! ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
Hello all,

The voting period for the [[m:Special:MyLanguage/Wikimedia Foundation elections/2024|2024 Board of Trustees election]] is now open. There are twelve (12) candidates running for four (4) seats on the Board.

Learn more about the candidates by [[m:Special:MyLanguage/Wikimedia Foundation elections/2024/Candidates|reading their statements]] and their [[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Questions_for_candidates|answers to community questions]].

When you are ready, go to the [[Special:SecurePoll/vote/400|SecurePoll]] voting page to vote. '''The vote is open from September 3rd at 00:00 UTC to September 17th at 23:59 UTC'''.

To check your voter eligibility, please visit the [[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Voter_eligibility_guidelines|voter eligibility page]].

Best regards,

The Elections Committee and Board Selection Working Group&lt;section end=&quot;announcement-content&quot; /&gt;

[[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 12:15, 3 September 2024 (UTC)
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== Separate page for hyperbola. ==

Good morning,

I notice that a search for &quot;hyperbola&quot; redirects to &quot;Conic sections&quot;.

At present there is a separate page for &quot;ellipse&quot;. Therefore a separate page for &quot;hyperbola&quot; seems to be justified.

Could this redirection be changed so that search for &quot;hyperbola&quot; goes to a separate page for &quot;hyperbola&quot;?

Many thanks,

[[User:ThaniosAkro|ThaniosAkro]] ([[User talk:ThaniosAkro|discuss]] • [[Special:Contributions/ThaniosAkro|contribs]]) 12:04, 15 September 2024 (UTC)

:It is true that ellipses are covered at [[Conic sections]] (along with hyperbolas, parabolas, etc.) and there is a separate page for [[ellipse]]s that elaborates. We certainly ''could'' have a page about [[hyperbola]]s that is separate, but no one has written sufficient content to spin it off yet. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 12:17, 15 September 2024 (UTC)

== I hereby request for your Unblocking IP address and just reviewed and received a reverted rec ==

Hi there. {{unsigned|Ishmael Raphasha}}
:No one has any clue what you're talking about. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:53, 18 September 2024 (UTC)

== RICH-2K: New project with some initial questions ==

Hello! I'm creating a new learning resource on ''Wikiversity''. The respective project is based on my transcription of a classical dictionary from 1849 by Anthony Rich. For more information about the project see its [[User:CalRis25/RICH: Description|description page]] (see also that page for why not ''Wikisource'' or ''Wikibooks''). The project's scope is fairly big: 3205 article-pages plus 304 REDIRECT-pages. The images (scanned by myself from an original copy) have been uploaded to ''Commons''. I have some initial technical questions (more of these and more detailed ones will follow):

* '''Upload''': Due to the large number of pages it is not realistic to create these manually. Is it possible to bulk-upload these in some way (the Wikitext of the pages is created using a Python-script with one file per article/page)? Is it possible to upload these to a test-environment first where any problems (hopefully none) can be identified and dealt with more easily than on the production-version of ''Wikiversity''?
* '''(Technical) Structure''': I am planning to set up this project at ''&lt;nowiki&gt;https://en.wikiversity.org/wiki/RICH-2K&lt;/nowiki&gt;'' as the main page and anything else as subpages: ''RICH-2K/Subpage_1 ... RICH-2K/Subpage_n''. However, these subpages fall into two categories: 1. Article-pages (content) and 2. Meta/Administrative pages. This project requires search capability restricted to the ''RICH-2K''-namespace. The ''Mediawiki''-software seems to supply a ''Search''-input field with the possibility to restrict the search to some namespace. I would like, however, to restrict the search further to the first group of pages, namely the articles. Is that possible, perhaps by use of (hidden) categories?
* '''External links''': This project will need many external links, and yes, I have read the relevant ''Wikiversity''-pages, but this specific project needs them. The ''Recommended Editions''-page (used for recommended online editions, to which to link when citing texts) alone probably will require several hundred external links. However, only relatively few [[w:Second-level domain|second-level domains]] will be involved, and most of these should be trustworthy (Perseus Digital library, digital collections of universities etc., in some cases, however, also ''Archive.org''). Perhaps there is a list of web-sites, for which external links are generally allowed? And who is allowed to create external links on ''Wikiversity''-pages (I haven't found the relevant policy)?
* '''Categories''': This project requires quite a few of its own categories, which belong to two large groups: 1. Categories (2 levels) of the ''Classed Index'' (about 170 categories), a thematic index of some (but not all) of the articles. 2. Administrative categories. Is there a recommended way to distinguish between different classes of categories within a project (category name or other method)? What about naming conventions for project-specific categories?

I am looking forward to your input. If you think that it's preferable we can move the discussions to the [[User_talk:CalRis25/RICH:_Description|Talk-page]] of the project's description. Thank you in advance. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 05:29, 20 September 2024 (UTC)

:*Admins have access to [[Special:Import]] and can bulk import XML pages. You can create pages in your sandbox if you'd like and make an indefinite amount of them at pages like [[User:CalRis25/sandbox]]. What can and cannot be hosted in user namespace is very loose, but still has to follow in principle Wikiversity's scope.
:*Using subpages is in principle a good way to organize these various resources. Please do not name them after a user name or something obscure. I personally think that &quot;RICH-2K&quot; is a not optimal name. I may recommend something like [[Anthony Rich Dictionary Project]] or [[21st-Century Anthony Rich Dictionary]] or something more obviously intelligible. While we have very few actual policies and guidelines, see [[Wikiversity:Naming conventions]] for a rough consensus of what is probably best practice for naming pages.
:*External linking generally does not use an allowed list (a.k.a. whitelist model), but a disallow (a.k.a. blacklist) model. See [[MediaWiki:Spam-blacklist]] and [[Special:BlockedExternalDomains]] (which is currently empty but is another method of listing blocked domains). It's perfectly fine to aggregate external links in learning resources.
:*I'm not 100% sure what the distinction is that you're drawing, but you can freely arrange categories underneath a main category that has the same name as your larger project. So, following the suggestions I gave, you could have a category like [[:Category:Anthony Rich Dictionary Project]] and then create any number of subcategories that logically help users navigate all these pages. Please make sure the main category you create is itself categorized under some relevant category(ies). If you need help, please ask.
:I think this answers your questions, please let me know if I'm unclear or you have more. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:11, 20 September 2024 (UTC)
::Hello Justin!
::* '''Upload:''' Creating the project in sandbox pages of my User-namespace defeats the purpose, as this is an ''open'' project. Also that would not solve, as such, the problem of having to manually create thousands of pages. I wonder, does ''Wikiversity'' support creation of pages using its API. ''Mediawiki's'' [[mw:API:Main_page|API-description]] seems to imply that it ought to be possible. If that's the case, I should be able to create a Python-script which automatically creates the pages (of course, a few trial pages first).
::* '''(Technical) Structure''': You may be right, here. RICH-2K is, for now, merely a technical name to make a clear but not too verbose distinction between the original text and the current project. I'll give this more thought.
::* '''External links''': I brought this up mainly because when I first edited my ''Wikiversity''-page, I got a message that I was not allowed to create external links. However, I just now tested creating an external link on my user-page and got no error, so this problem seems to be solved.
::* '''Categories''': I think I know what you mean. I'll create a category structure and maybe ask some specific questions once I am ready to do so.
::Thank you for your quick help. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 18:51, 20 September 2024 (UTC)
:::re: upload, I'm just suggesting your sandbox(es) as you asked about &quot;a test-environment&quot;. Anyone can edit someone else's sandboxes, but you typically defer to other users to control what's in their own subpages as a collegial thing. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:39, 23 September 2024 (UTC)

Hello! I have two further questions:
# I created a category-structure for the project. Could you (or someone else) have a look at it ([[User:CalRis25/RICH: Categories]]) and answer the questions in the section [[User:CalRis25/RICH:_Categories#Questions|Questions]]? I gave it some thought and believe that this would work fine for the project.
# ''Project boxes'' (see [[Help:Tour of project boxes]]): It is unclear to me, whether these belong only on the main page of the project (that makes the most sense to me), or on every single subpage.
Thanks in advance for your help. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 17:51, 24 September 2024 (UTC)

:To answer your questions here:
:*No, you are not contravening any policies we have.
:*A leading &quot;The&quot; is acceptable, but if you want it to sort alphabetically, you will have to use &lt;nowiki&gt;{{DEFAULTSORT:}}&lt;/nowiki&gt;. E.g. to get Category:The Best Stuff to sort under &quot;B&quot;, insert &quot;&lt;nowiki&gt;{{DEFAULTSORT:Best Stuff, The}}&lt;/nowiki&gt;.
:*Trailing &quot;etc.&quot; is acceptable.
:*An accent in a category title is acceptable.
:I'll also note that it looks like you have in mind some tracking categories that are redundant. Pages such as [[Special:LonelyPages]] and [[Special:DeadendPages]] already do automatically what you're proposing to do manually.
:As for project boxes, it's typically the case that the subjects are only placed on the main resource, but as you may imagine, [[Help:Tour of project boxes/1|status completion ones]] may vary from subpage to subpage. As with most things at Wikiversity, there are very few actual rules, so it's pretty much the wild west, even tho this project has been around for almost 20&amp;nbsp;years. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:18, 25 September 2024 (UTC)
::Hello Justin, thanks for the DEFAULTSORT-hint for categories beginning with ''The''. I will restrict the project boxes to the main page. As for the the orphaned/dead-end-categories, I prefer these to be project-specific. Once the project is up and running, putting articles &quot;on the map&quot; (making them accessible from other articles and creating links to other articles) is one of the first tasks to be dealt with. I already know which articles are involved and will add these categories to these articles. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 16:51, 25 September 2024 (UTC)

== Your wiki will be in read-only soon ==

&lt;section begin=&quot;server-switch&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;

[[:m:Special:MyLanguage/Tech/Server switch|Read this message in another language]] • [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-Tech%2FServer+switch&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]

The [[foundation:|Wikimedia Foundation]] will switch the traffic between its data centers. This will make sure that Wikipedia and the other Wikimedia wikis can stay online even after a disaster.

All traffic will switch on '''{{#time:j xg|2024-09-25|en}}'''. The switch will start at '''[https://zonestamp.toolforge.org/{{#time:U|2024-09-25T15:00|en}} {{#time:H:i e|2024-09-25T15:00}}]'''.

Unfortunately, because of some limitations in [[mw:Special:MyLanguage/Manual:What is MediaWiki?|MediaWiki]], all editing must stop while the switch is made. We apologize for this disruption, and we are working to minimize it in the future. 

A banner will be displayed on all wikis 30 minutes before this operation happens. This banner will remain visible until the end of the operation. 

'''You will be able to read, but not edit, all wikis for a short period of time.'''

*You will not be able to edit for up to an hour on {{#time:l j xg Y|2024-09-25|en}}.
*If you try to edit or save during these times, you will see an error message. We hope that no edits will be lost during these minutes, but we can't guarantee it. If you see the error message, then please wait until everything is back to normal. Then you should be able to save your edit. But, we recommend that you make a copy of your changes first, just in case.

''Other effects'':

*Background jobs will be slower and some may be dropped. Red links might not be updated as quickly as normal. If you create an article that is already linked somewhere else, the link will stay red longer than usual. Some long-running scripts will have to be stopped.
* We expect the code deployments to happen as any other week. However, some case-by-case code freezes could punctually happen if the operation require them afterwards.
* [[mw:Special:MyLanguage/GitLab|GitLab]] will be unavailable for about 90 minutes.

This project may be postponed if necessary. You can [[wikitech:Switch_Datacenter|read the schedule at wikitech.wikimedia.org]]. Any changes will be announced in the schedule. 

'''Please share this information with your community.'''&lt;/div&gt;&lt;section end=&quot;server-switch&quot;/&gt;

[[User:Trizek_(WMF)|Trizek_(WMF)]], 09:37, 20 September 2024 (UTC)
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== 'Wikidata item' link is moving. Find out where... ==

&lt;div lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot;&gt;&lt;i&gt;Apologies for cross-posting in English. Please consider translating this message.&lt;/i&gt;{{tracked|T66315}}

Hello everyone, a small change will soon be coming to the user-interface of your Wikimedia project.
The [[d:Q16222597|Wikidata item]] [[w:|sitelink]] currently found under the &lt;span style=&quot;color: #54595d;&quot;&gt;&lt;u&gt;''General''&lt;/u&gt;&lt;/span&gt; section of the '''Tools''' sidebar menu will move into the &lt;span style=&quot;color: #54595d;&quot;&gt;&lt;u&gt;''In Other Projects''&lt;/u&gt;&lt;/span&gt; section.

We would like the Wiki communities feedback so please let us know or ask questions on the   [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Discussion page]] before we enable the change which can take place October 4 2024, circa 15:00 UTC+2.
More information can be found on [[m:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|the project page]].&lt;br&gt;&lt;br&gt;We welcome your feedback and questions.&lt;br&gt; [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 18:56, 27 September 2024 (UTC)
&lt;/div&gt;
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==Download as PDF==
[[Phabricator:T376438]]: &quot;Download to PDF&quot; on en.wv is returning error: &quot;{&quot;name&quot;:&quot;HTTPError&quot;,&quot;message&quot;:&quot;500&quot;,&quot;status&quot;:500,&quot;detail&quot;:&quot;Internal Server Error&quot;}&quot;

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:36, 3 October 2024 (UTC)

:I just downloaded this page as a PDF and it worked just fine. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:04, 3 October 2024 (UTC)

== Protected template bug for Pp ==

It seems that templates derivative of {{tlx|Pp}} (compiled in {{tlx|Protection templates}}) are being sorted into protection categories using the name 'Wikipedia' instead of 'Wikiversity' (e.g., [[:Category:Wikipedia pages with incorrect protection templates]]). From what I can tell, it is not in the publicly accessible source code of any of the templates. The only other impacted pages are modules which call {{tlx|pp}}-derivatives (e.g., [[Module:Navbar/styles.css]]).

This does not seem to affect any other pages in [[:Category:Wikiversity protected templates]]. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 18:59, 4 October 2024 (UTC)

:The problem is that &quot;Wikipedia&quot; is [https://en.wikiversity.org/w/index.php?title=Special%3ASearch&amp;limit=500&amp;offset=0&amp;ns828=1&amp;search=Wikipedia&amp;searchToken=9svkpqlxxoquoq7bnkt55ugts mentioned in several modules that were copied over from en.wp]; many of these are legit and many of them need to be replaced with &quot;Wikiversity&quot; ([https://en.wikiversity.org/w/index.php?title=Module%3APp-move-indef&amp;diff=2662815&amp;oldid=1944984 e.g.]) This particular change ''may'' fix all of these issues...? But 1.) it will take time to propagate across the site and 2.) there are still many more &quot;Wikipedia&quot;s that need to be changed, so I'll go thru a few more, but if you want to give me an assist, if you can just check this one week from now and ping me if the problem persists, that would be nice. Sometimes, I make calendar reminders to follow up on these, but I'm not a perfect person. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:55, 5 October 2024 (UTC)

== Invitation to Participate in Wiki Loves Ramadan Community Engagement Survey ==

Dear all,


We are excited to announce the upcoming [[m:Wiki Loves Ramadan|Wiki Loves Ramadan]] event, a global initiative aimed at celebrating Ramadan by enriching Wikipedia and its sister projects with content related to this significant time of year. As we plan to organize this event globally, your insights and experiences are crucial in shaping the best possible participation experience for the community.

To ensure that Wiki Loves Ramadan is engaging, inclusive, and impactful, we kindly invite you to participate in our community engagement survey. Your feedback will help us understand the needs of the community, set the event's focus, and guide our strategies for organizing this global event.

Survey link: https://forms.gle/f66MuzjcPpwzVymu5

Please take a few minutes to share your thoughts. Your input will make a difference!

Thank you for being a part of our journey to make Wiki Loves Ramadan a success.


Warm regards,

User:ZI Jony 03:19, 6 October 2024 (UTC)

Wiki Loves Ramadan Organizing Team
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== 'Edit to my talk page' notification bug? ==

This may belong at the bug tracker, but does anyone else have an issue disabling ''email'' notifications upon  an 'Edit to my talk page' in [[Special:GlobalPreferences]]? Oddly I ''am'' able to disable the global preference on Wikipedia, MediaWiki, etc, but not here. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 09:23, 7 October 2024 (UTC)

:I have not experienced this, but to be clear, do you also have the option to get emails when items on your talk page are edited turned on? —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:39, 7 October 2024 (UTC)
::The only (non-grayed out) options I have enabled for email are 'Failed login attempts' and 'Login from an unfamiliar device'. 'Edit to my talk page' re-checks after every save. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 09:54, 7 October 2024 (UTC)
:::That does sound like a [[phab:]] issue, with the caveat that I don't 100% recall how global preferences work and if they override local ones, etc. If you have parsed that and still have this issue, you'll probably need to file a ticket. Maybe someone else has this issue. Wish I could help. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:57, 7 October 2024 (UTC)
::::[[phab:T376601|Off 'n away]] 🫡 [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 10:35, 7 October 2024 (UTC)

== [[Portal:Computer Science]] ➝ [[Portal:Information sciences]] ==

Seeking consensus to complete the merge into the broader portal. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 06:28, 8 October 2024 (UTC)

:Why should it be merged? Computer Science seems well-enough designed. What is the incentive to collapse it into a broader field of study? —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:18, 8 October 2024 (UTC)
::Portals as top level organizations allowing for content to be best centralized. Also note that I did not start the merge, just offering to finish it. Perhaps a {{tlx|prod}} instead? [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 07:20, 8 October 2024 (UTC)
:::I have no objections, personally. If it gets done, please use a redirect and should someone want to come along to resurrect it later, it will be easier. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:21, 8 October 2024 (UTC)
: Is computer science really a branch of information sciences? I would not think so, but what do I know. Do we have some external resources/links confirm this idea? [[W:Information science]] currently says: &quot;Information science, documentology[1] or informatology[2][3] is an academic field which is primarily concerned with analysis, collection, classification, manipulation, storage, retrieval, movement, dissemination, and protection of information.&quot; --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:49, 11 October 2024 (UTC)
::Looking through [https://stackoverflow.com/q/1047014/22673230] [https://businessdegrees.uab.edu/mis-degree-bachelors/resources/computer-information-systems-vs-computer-science/] [https://www.si.umich.edu/student-experience/what-information-science] a few top (not necessarily RS) searches I'm inclined to agree. I am more familiar with the grafted [[:w:Information and computer science|information ''and'' computer science]] which makes an effort to merge the disciplines, but it does not seem like reaching to say that IS is presented as more applications-concerned (certainly with no lack of theoretical abstraction), whereas CS can be more freely associated with any and all 'science related to computers'. It is easy to reason about the connection between the fields, but I think it is clear academia maintains this taxonomy for a good reason.
::With these considerations, I think I will ''stop'' the process of merging in favor of expanding the existing [[School:Library and Information Science]]. 

::Let me know if there is not consensus to redirect [[Portal:Information sciences]] to [[School:Library and Information Science]] (with enough expansion it can generalize away from just library sciences). [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 16:16, 11 October 2024 (UTC)
::: I do not see that a merge of a ''portal'' to a ''school'' is a good thing. Do you have a clear idea of the concepts of school and portal and how they relate to each other? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:34, 11 October 2024 (UTC)
::::Found [[:Category:Information sciences]]; there are enough existing resources in there to make my other proposed merge excessive. I will simply continue developing the existing [[Portal:Information sciences]] instead. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 17:05, 11 October 2024 (UTC)
::::: Frankly, I would ideally see [[Portal:Information sciences]] deleted: I don't see what it does that a category would not do well enough. There does not seem to be any material specific to &quot;Information sciences&quot; (whatever that is) in that portal at all. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:11, 11 October 2024 (UTC)
::::::Tacked a {{tlx|prod}} for an eventual deletion, but I may still try to develop it as proof of concept at some point. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 17:33, 11 October 2024 (UTC)

== [[:Category:Occupational Epidemiology]] ==

I propose moving the pages in this category (without leaving redirects) to their equivalent under the parent resource [[Occupational Health Risk Surveillance]]. Also due to the number of subpages, it seems &lt;code&gt;|filing=deep&lt;/code&gt; would be a justified. (Also [[Special:PrefixIndex/Occupational_Epidemiology|there are quite a few]] untagged subpages.) [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 05:11, 9 October 2024 (UTC)
: I above all think that the content should be ''moved out of the mainspace'': I do not see readers learning anything from e.g. [[Occupational Epidemiology/Research tools/Reading of scientific articles for learning epidemiology and biostatstics]] or [[Occupational Epidemiology/Research tools/Ongoing projects/Risk Communication in Seafaring/Writing the article guideline IMRAD]]. Wikiversity can be kind enough to host that material in, say, subspace of [[User:Saltrabook]], but more should not be asked, I think. Let us recall that per [[WV:Deletions]], &quot;Resources may be eligible for proposed deletion when education objectives and learning outcomes are scarce, and objections to deletion are unlikely&quot;; I do not see how learning outcomes can be anything but scarce. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 15:04, 11 October 2024 (UTC)
::thank you, agree @ [[User:Saltrabook|Saltrabook]] ([[User talk:Saltrabook|discuss]] • [[Special:Contributions/Saltrabook|contribs]]) 21:03, 13 November 2024 (UTC)

== Active editors ==

It is interesting to observe the stats on [https://stats.wikimedia.org/#/en.wikiversity.org/contributing/active-editors/normal|line|all|(page_type)~content*non-content|monthly active editors] through the project's history. October is our month! [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 20:44, 8 October 2024 (UTC)

:Odd. Maybe related to the school year? —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 02:10, 9 October 2024 (UTC)
::I wonder how many are [[User:Jtneill|Jtneill]]'s crowd... the number is in the hundreds though, so that is one chunky cohort —[[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:16, 9 October 2024 (UTC)
:::Yes, [[Motivation and emotion/Book]] involves ~100-150 students editing most intensely during October each year. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:20, 9 October 2024 (UTC)
::::Neat, that still leaves around ~50-100 other students from other avenues each year since 2021. I also wonder which projects were involved in the COVID enrollment spike. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:26, 9 October 2024 (UTC)
:::::Personally I can admit that my editing is much more active during the school season vs. the summer break, so I'm in the same boat as Jtneill's students. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:24, 13 November 2024 (UTC)

== Intentionally incorrect resource ==

There is a [[Special:Diff/2583464|disclaimer inserted onto a resource]] (by not the original author) that:
&lt;blockquote&gt;I am merely [making this page false] to show you (The viewer) that Wikipedia and this page 'Wikiversity' is bull sh*t and it will not give you the reliability you need when writing an academic piece of writing.&lt;/blockquote&gt;
However, that IP has [[Special:Contributions/86.22.73.151|not made any other edits]], so unless they vandalized via a sock, the intent went un-realized and only that portion need be removed. Bumping here in case there is some obvious jumbo in that essay that someone else can catch. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 16:58, 9 October 2024 (UTC)

:Removed that portion, which was obviously vandalism. No perspective on the rest of the essay. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:38, 9 October 2024 (UTC)

== [[:Category:Proposed guidelines]] ==

Noting for future editors that WV has collapsed all proposals into [[:Category:Proposed policies|proposed policies]]. Seeking consensus to further collapse [[:Category:Wikiversity proposals]] into the former, or to restore [[:Category:Proposed guidelines]]. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 19:19, 9 October 2024 (UTC)

== [[Around Wikiversity in 80 Seconds|Broken 80-second tour]] ==

Bumping a [[Talk:Around_Wikiversity_in_80_Seconds|comment]] on the ''Wikiversity in 80 seconds'' tour. Appears wikisuite is not working with the Vector 2022 appearance. 

Also see [[:w:Wikipedia:Miscellany_for_deletion/Wikiversuite_pages|this thread]] on the Wikiversal package - may not be relevant to Wikiversity, but FYC. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 00:26, 10 October 2024 (UTC)
: I would just delete the material; I do not see value in it. If others agree, I would try to articulate why I think it should be deleted (or move to author user space). --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:57, 13 October 2024 (UTC)
::Just mark as {{tl|historical}}. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:39, 13 October 2024 (UTC)
::: This thing was created by [[User:Planotse]]. His creations are now being discussed in Wikibooks for deletion: [[B:Wikibooks:Requests for deletion#Wikiversal generated pages]]. It seems he used some kind of tool that is no longer available (the above mentioned &quot;Wikiversal&quot; package) to create this kind of slideshow-like material (believing the Wikibooks discussion). I do not see value of this in the mainspace, not even as historical (I am okay with userspace, but maybe even that is not the best option?). A look at the source code of [[Around Wikiversity in 80 Seconds/Introduction]] confirms the words of Omphalographer, namely that &quot;the HTML-heavy markup generated by Wikiversal makes them [the pages] unreasonably difficult to edit.&quot;
::: I went ahead and marked the page for proposed deletion. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:35, 14 October 2024 (UTC)

== Preliminary results of the 2024 Wikimedia Foundation Board of Trustees elections ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
Hello all,

Thank you to everyone who participated in the [[m:Special:MyLanguage/Wikimedia Foundation elections/2024|2024 Wikimedia Foundation Board of Trustees election]]. Close to 6000 community members from more than 180 wiki projects have voted.

The following four candidates were the most voted:

# [[User:Kritzolina|Christel Steigenberger]]
# [[User:Nadzik|Maciej Artur Nadzikiewicz]]
# [[User:Victoria|Victoria Doronina]]
# [[User:Laurentius|Lorenzo Losa]]

While these candidates have been ranked through the vote, they still need to be appointed to the Board of Trustees. They need to pass a successful background check and meet the qualifications outlined in the Bylaws. New trustees will be appointed at the next Board meeting in December 2024.

[[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Results|Learn more about the results on Meta-Wiki.]]

Best regards,

The Elections Committee and Board Selection Working Group

&lt;section end=&quot;announcement-content&quot; /&gt;

[[User:MPossoupe_(WMF)|MPossoupe_(WMF)]] 08:26, 14 October 2024 (UTC)
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== Seeking volunteers to join several of the movement’s committees ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
Each year, typically from October through December, several of the movement’s committees seek new volunteers. 

Read more about the committees on their Meta-wiki pages:
* [[m:Special:MyLanguage/Affiliations_Committee|Affiliations Committee (AffCom)]]
* [[m:Special:MyLanguage/Ombuds_commission|Ombuds commission (OC)]]
* [[m:Special:MyLanguage/Wikimedia Foundation/Legal/Community Resilience and Sustainability/Trust and Safety/Case Review Committee|Case Review Committee (CRC)]]

Applications for the committees open on 16 October 2024. Applications for the Affiliations Committee close on 18 November 2024, and applications for the Ombuds commission and the Case Review Committee close on 2 December 2024. Learn how to apply by [[m:Special:MyLanguage/Wikimedia_Foundation/Legal/Committee_appointments|visiting the appointment page on Meta-wiki]]. Post to the talk page or email [mailto:cst@wikimedia.org cst@wikimedia.org] with any questions you may have.

For the Committee Support team,
&lt;section end=&quot;announcement-content&quot; /&gt;

-- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 23:09, 16 October 2024 (UTC)
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== Interactive elements ==
Can we use interactive elements on Wikiversity? I'd like to add JavaScript to a page. If it's not possible now, where can I suggest this feature? I have a safe integration idea.
[[User:Отец Никифор|Отец Никифор]] ([[User talk:Отец Никифор|discuss]] • [[Special:Contributions/Отец Никифор|contribs]]) 12:10, 17 October 2024 (UTC)
: This is beyond my technical knowledge, but have you checked out:
:* https://www.mediawiki.org/wiki/Manual:Interface/JavaScript?
:* [[Wikipedia:WikiProject JavaScript]]
:* [[MediaWiki:Common.js]]

:What sort of interactive elements are you thinking about?

: Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:39, 18 October 2024 (UTC)
::I was thinking about adding something like a graph with adjustable controls, where users can interact with it and see how different changes affect the outcome. It seems like this could be a useful feature. There might already be discussions about enhancing Wikiversity or similar platforms—perhaps on a relevant talk page or in a Discord group. Do you know where such discussions might be happening? [[User:Отец Никифор|Отец Никифор]] ([[User talk:Отец Никифор|discuss]] • [[Special:Contributions/Отец Никифор|contribs]]) 19:47, 18 October 2024 (UTC)
:::From a quick look, maybe check out:
:::* [[mw:Extension:Graph]]
:::* [[phab:tag/graphs]]
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:40, 18 October 2024 (UTC)
:::: mw:Extension:Graph is currently disabled on Wikipedia etc. wikis, for security reasons, and seems unlikely to be enabled again. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:30, 19 October 2024 (UTC)

== An unexplained spurt of Wikiversity page views ==
The [https://pageviews.wmcloud.org/siteviews/?platform=all-access&amp;source=pageviews&amp;agent=user&amp;start=2024-06-01&amp;end=2024-10-18&amp;sites=en.wikiversity.org|en.wikibooks.org|en.wikiquote.org|en.wikisource.org page view report] shows an unexplained spurt of Wikiversity page views, reaching over 4 times the baseline and then falling back again. Does anyone have any idea what is going on? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:01, 19 October 2024 (UTC)

:Interesting. I wonder why only the English wikiquote and wikiversity and not Wikisource or wikibooks? How reliable do you think those stats are? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 15:44, 8 December 2024 (UTC)

== Center tempate failed on a contributors phone... ==

See the edit comment here - https://en.wikiversity.org/w/index.php?title=Wikiphilosophers&amp;diff=prev&amp;oldid=2673962. I'm puzzled as this is the first failure of this, I've noted recently. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 08:45, 19 October 2024 (UTC)

== Essay-like page in user space that makes little sense and seems incoherent ==
The page [[User:TheoYalur/Illusions]] seems to match the description, at least by my assessment. My understanding is that since the page is only in user space and not in the mainspace, it can stay there even if it has those disqualifying qualities. But if I am wrong and the page belongs deleted, please correct me and let me know. I do not know which policy or guideline, if any, guides the case. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:30, 21 October 2024 (UTC)

== 'Wikidata item' link is moving, finally. ==

Hello everyone, I previously wrote on the 27th September to advise that the ''Wikidata item'' sitelink will change places in the sidebar menu, moving from the '''General''' section into the '''In Other Projects''' section. The scheduled rollout date of 04.10.2024 was delayed due to a necessary request for Mobile/MinervaNeue skin. I am happy to inform that the global rollout can now proceed and will occur later today, 22.10.2024 at 15:00 UTC-2. [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Please let us know]] if you notice any problems or bugs after this change. There should be no need for null-edits or purging cache for the changes to occur. Kind regards, -[[m:User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] 11:28, 22 October 2024 (UTC)
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:Hi @[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]]: I Just noticed your post above, and it is timely.
:I have been participating in the English WikiUniversity for a few years, much less often recently. I seems like something in the way the site displays is different, but I cannot put my finger on it. Your posting gave me a clue. Can you please tell me where  the link to wikidata items has moved to? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:23, 11 December 2024 (UTC)
::Hello @[[User:Ottawahitech|Ottawahitech]], sure, I would be happy to. The button/sitelink name didn't change, just its position. You should find it in the sidebar-menu under the section '''In other projects''' (where the links to all other Wikimedia Projects are displayed). If you do not see it, please reach out to us on the [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Move Wikidata item - Discussion page]]. Thank you, -[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] ([[User talk:Danny Benjafield (WMDE)|discuss]] • [[Special:Contributions/Danny Benjafield (WMDE)|contribs]]) 09:24, 12 December 2024 (UTC)
:::@[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]], thank you for responding. I intend to followup on the ''Move Wikidata item - Discussion page'' as per your post above by putting it on my ever growing todo list.
:::I don't know about others on this wiki, as I said I have not been visiting here frequently, but for me the constant changes are a big distraction. I have been around wikimedia projects since 2007, so why do I have to spend so much time learning and re-learning how to find what I came here for? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:41, 12 December 2024 (UTC)
::::Hi @[[User:Ottawahitech|Ottawahitech]], thanks for you thoughts. Your input whether positive or critical helps us understand the impacts to editors so we welcome your further thoughts when you reach us in your To Do List :)
::::I can't speak about the other changes you've experienced here but I do hope they are made with a spirit of improvement for the community as a whole. -[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] ([[User talk:Danny Benjafield (WMDE)|discuss]] • [[Special:Contributions/Danny Benjafield (WMDE)|contribs]]) 10:43, 16 December 2024 (UTC)

== Final Reminder: Join us in Making Wiki Loves Ramadan Success ==

Dear all,

We’re thrilled to announce the Wiki Loves Ramadan event, a global initiative to celebrate Ramadan by enhancing Wikipedia and its sister projects with valuable content related to this special time of year. As we organize this event globally, we need your valuable input to make it a memorable experience for the community.

Last Call to Participate in Our Survey: To ensure that Wiki Loves Ramadan is inclusive and impactful, we kindly request you to complete our community engagement survey. Your feedback will shape the event’s focus and guide our organizing strategies to better meet community needs.

* Survey Link: [https://docs.google.com/forms/d/e/1FAIpQLSffN4prPtR5DRSq9nH-t1z8hG3jZFBbySrv32YoxV8KbTwxig/viewform?usp=sf_link Complete the Survey]
* Deadline: November 10, 2024
Please take a few minutes to share your thoughts. Your input will truly make a difference!

'''Volunteer Opportunity''': Join the Wiki Loves Ramadan Team! We’re seeking dedicated volunteers for key team roles essential to the success of this initiative. If you’re interested in volunteer roles, we invite you to apply.
* Application Link: [https://docs.google.com/forms/d/e/1FAIpQLSfXiox_eEDH4yJ0gxVBgtL7jPe41TINAWYtpNp1JHSk8zhdgw/viewform?usp=sf_link Apply Here]
* Application Deadline: October 31, 2024
Explore Open Positions: For a detailed list of roles and their responsibilities, please refer to the position descriptions here: [https://docs.google.com/document/d/1oy0_tilC6kow5GGf6cEuFvdFpekcubCqJlaxkxh-jT4/ Position Descriptions]

Thank you for being part of this journey. We look forward to working together to make Wiki Loves Ramadan a success!


Warm regards,&lt;br&gt;
The Wiki Loves Ramadan Organizing Team 05:11, 29 October 2024 (UTC)
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== Android app for Wikiversity ==

Hi, is there an Android app for Wikiversity? How does it work? I have been advised that there is no infrastructure for push notifications for Android apps for sister wikis and I would be interested to know more. Related: [[:phab:T378545]]. Thanks! [[User:Gryllida|Gryllida]] 23:15, 29 October 2024 (UTC)

:Thanks for suggesting this - I agree that it would be useful. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:56, 31 October 2024 (UTC)
:@[[User:Gryllida|Gryllida]]: Would you explain your terminology  for those of us not in the know. What does ''push notifications'' mean? I use [https://www.mediawiki.org/wiki/Help:Notifications notifications] when I am communicating on wikimedia projects, but have never heard this term before. [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:13, 11 December 2024 (UTC)

à

== Import Resource From Wikibooks? ==

Hello! [[wikibooks:Character_List_for_Baxter&amp;Sagart|Character List for Baxter&amp;Sagart]] and related titles [[wikibooks:Wikibooks:Requests_for_deletion#Character_List_for_Baxter&amp;Sagart|are up for deletion at Wikibooks]] because WB policy does not allow dictionaries like them. However, because they are useful as learning tools, I am wondering if they might have a home here at Wikiversity. Pinging @[[User:Tibetologist|Tibetologist]] here to link them in to this discussion, since they are the affected user. Thank you! —[[User:Kittycataclysm|Kittycataclysm]] ([[User talk:Kittycataclysm|discuss]] • [[Special:Contributions/Kittycataclysm|contribs]]) 18:18, 1 November 2024 (UTC)

:Sure, I can do it. That said, as mentioned there, it does seem like something like this is ideally suited for Wiktionary in the Appendix namespace, but I'm not very familiar with CJK characters and languages. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:23, 3 November 2024 (UTC)
::Oh man, these pages are too big to import and while I've already tried a half-dozen times, it will constantly fail. Strictly speaking, we don't have to use the import feature for licensing purposes. We can just copy and paste the contents and list the usernames or on the talk page. I think that's the solution. {{Ping|Tibetologist}}, are you interested in doing that? If you just copied and pasted these pages and then added [[:Category:Chinese]] and maybe include a couple of links to the pages, that would probably be ideal. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:31, 3 November 2024 (UTC)

== Language translation requests? ==

Is there anywhere on Wikiversity to request translation, for example, requesting Latin or French translation? I would be asking from the context as a student, so I would be interested in translation explanation as well. [[User:Indexcard88|Indexcard88]] ([[User talk:Indexcard88|discuss]] • [[Special:Contributions/Indexcard88|contribs]]) 04:56, 20 November 2024 (UTC)

== Sign up for the language community meeting on November 29th, 16:00 UTC ==

Hello everyone,

The next language community meeting is coming up next week, on November 29th, at 16:00 UTC (Zonestamp! For your timezone &lt;https://zonestamp.toolforge.org/1732896000&gt;). If you're interested in joining, you can sign up on this wiki page: &lt;https://www.mediawiki.org/wiki/Wikimedia_Language_and_Product_Localization/Community_meetings#29_November_2024&gt;.

This participant-driven meeting will be organized by the Wikimedia Foundation’s Language Product Localization team and the Language Diversity Hub. There will be presentations on topics like developing language keyboards, the creation of the Moore Wikipedia, and the language support track at Wiki Indaba. We will also have members from the Wayuunaiki community joining us to share their experiences with the Incubator and as a new community within our movement. This meeting will have a Spanish interpretation.

Looking forward to seeing you at the language community meeting! Cheers, [[User:SSethi (WMF)|Srishti]] 19:55, 21 November 2024 (UTC)
&lt;!-- Message sent by User:SSethi (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Distribution_list/Global_message_delivery&amp;oldid=27746256 --&gt;

== Events on Wikiversity ==

Since Wikipedia and Wikivoyage are having their &quot;Asian Month&quot; editathon, I was thinking if we could start up a Wikiversity version of that. This would be an &quot;Asian Month&quot; as well, but it would be about creating resources based on Asia and its culture. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 17:57, 6 December 2024 (UTC)

:Not immediately opposed, but the question is, do we have an active enough community to facilitate this? —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:31, 6 December 2024 (UTC)
::I'm not too sure. As long as we get enough traffic, this could happen. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 08:45, 7 December 2024 (UTC)
:::This is to increase traffic on Wikiversity, which is promoted amongst other communities. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 10:47, 7 December 2024 (UTC)
:Hi @[[User:RockTransport|RockTransport]], This is a good idea, but will it also involve users who are not &quot;professors and scientists&quot;. Just curious. cheers, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:30, 9 December 2024 (UTC)
::Yes, considering the fact that Wikiversity is for everyone, and not just for specific users. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 09:09, 10 December 2024 (UTC)
:::because I'm personally not a &quot;professor&quot; or a &quot;scientist&quot; and because '''anyone''' can create resources on Wikiversity. We want to make Wikiversity open for everyone, and not just for certain users. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 09:10, 10 December 2024 (UTC)
::::I am also not a professor or a scientist, but it seems to me that as result I am viewed here as a visitor rather than someone who can contribute. Just my $.02. [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:05, 12 December 2024 (UTC)

== Wikiversity - Newsletters ==

Hello All,

I wanted to create a newsletter on Wikiversity, which would highlight what is going on in certain months and events on Wikiversity; which would bolster engagement by many people. This would be on the website and would have its dedicated 'Newsletter' tab.

I hope you acknowledge this idea. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 21:05, 8 December 2024 (UTC)

:@[[User:RockTransport|RockTransport]], What sort of things do you plan to include in your newsletter? Will they be different than what is currently in [[Main Page/News]]? Just curious.
:I am also wondering about your motive which I think is: to bolster engagement by many people. I am asking because I wonder if others who are currently active here also think this I is desirable? Have you asked them? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:34, 11 December 2024 (UTC)
::Not yet, which was why I was asking this on the colloquium. I plan to include things that many people have created on Wikiversity over the month, as it is a monthly newsletter. It would be somewhere on the website here. It will be more frequent that the ones seen on [[Main Page/News]]. We will include people's resources to essentially promote them. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 06:50, 12 December 2024 (UTC)
:::@[[User:RockTransport|RockTransport]], I Think what you are saying is that ''Main Page/News'' does not update frequently enough? 
:::If this is the reason, why not start small by simply increasing the frequency of posting news on the main page, instead of trying to start a newsletter?
:::If there is more, can you articulate what else is missing. Thanks in advance, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:51, 12 December 2024 (UTC)
::::I meant going to detail into topics covered in that month, rather than just giving a few points. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 16:53, 12 December 2024 (UTC)
:::::What sort of details did you have in mind? You can pick one of the links provided in [[Main Page/News]] to illustrate. cheers, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 15:29, 16 December 2024 (UTC)</text>
      <sha1>8fw4gjy5i9d6e313phxl0f007apq066</sha1>
    </revision>
    <revision>
      <id>2692201</id>
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      <text bytes="77304" sha1="isv49nakfh8xvbhgosu5xp0rvaoy2k1" xml:space="preserve">{{Wikiversity:Colloquium/Header}}
&lt;!-- MESSAGES GO BELOW --&gt;
== Reminder! Vote closing soon to fill vacancies of the first U4C ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
:''[[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement – reminder to vote|You can find this message translated into additional languages on Meta-wiki.]] [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement – reminder to vote}}&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]''

Dear all,

The voting period for the Universal Code of Conduct Coordinating Committee (U4C) is closing soon. It is open through 10 August 2024. Read the information on [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2024_Special_Election#Voting|the voting page on Meta-wiki to learn more about voting and voter eligibility]]. If you are eligible to vote and have not voted in this special election, it is important that you vote now.

'''Why should you vote?''' The U4C is a global group dedicated to providing an equitable and consistent implementation of the UCoC. Community input into the committee membership is critical to the success of the UCoC.

Please share this message with members of your community so they can participate as well.

In cooperation with the U4C,&lt;section end=&quot;announcement-content&quot; /&gt;

-- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 15:30, 6 August 2024 (UTC)
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== User group for Wikiversians ==

Was there ever a discussion about the possibility of establishing a user group in the sense of an affiliated organization that would defend the interests of professors and scientists on Wikiversity and possibly actively develop some projects? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:21, 8 August 2024 (UTC)

:Not that I'm aware of. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:20, 8 August 2024 (UTC)
:It's a pleasure to talk to a scientist on Wikiversity. I am a historian of technics and I would like to publish the following biography either on Wikiversity or on Wikipedia: 
:https://en.wikiversity.org/wiki/User:Rbmn/Arthur_Constantin_KREBS_(1850-1935):_Military_engineer,_Automotive_industrialist,_Great_projects_manager
:What would be your advice? [[User:Rbmn|Rbmn]] ([[User talk:Rbmn|discuss]] • [[Special:Contributions/Rbmn|contribs]]) 15:44, 6 October 2024 (UTC)
::The content appears to be largely biographical/encyclopedic, so I think it is likely best suited to Wikipedia. Consider improving/incorporating this content into the existing page: [[w:Arthur Constantin Krebs]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:05, 7 October 2024 (UTC)
::Please do not link to the Wikiversity [[wv:userspace|Userspace]] in Wikipedia articles. You will want to wait until you have a page in the [[wv:mainspace|Wikiversity mainspace]]. You'll also want to use the &lt;code&gt;{{[[:w:Template:Wikiversity|Wikiversity]]}}&lt;/code&gt; template (on Wikipedia) rather than embedding a photo with a link. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:21, 7 October 2024 (UTC)
:I haven't heard anything about this topic. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 21:06, 8 December 2024 (UTC)

== Rich's ''Illustrated Companion'' at Wikiversity: Right place? ==

Hello! I am creating a Wiki-version of a classical glossary (''Illustrated Companion to the Latin Dictionary, and Greek Lexicon'' by Anthony Rich, 1849), which explains the meaning of Latin headwords, primarily those &quot;representing visible objects connected with the arts, manufactures, and every-day life of the Greeks and Romans.&quot; The aim is to help understand what a (classical) Latin text is actually about, instead of merely translating it. I already transcribed the entire text and scanned the images (about 1900) from an original 1849-edition. I am currently working on uploading the images to ''Mediawiki Commons'', which probably will take some time. In the meantime I want to prepare the other aspects of the project (more than 3000 articles, already with many internal links). The important thing: this is ''not'' a ''might exist''-project.  {{Color|red|My question: Is ''Wikiversity'' the proper place for it?}} Although I created an exact rendition of the original text, ''Wikisource'' is not applicable, because the project has a broader scope (adding content to the articles, e. g. links to online editions for quotations, adding images, but also adding entirely new articles). Neither is ''Wikibooks'', because this is not a textbook and may otherwise breach its scope. For more about the project see [[w:User:CalRis25/Temp-RICH-Prospectus|my user-page]] at en.wikipedia. {{Color|Red|So, is Wikiversity the right place for it?}} [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 09:15, 17 August 2024 (UTC)

:Thanks for asking. To be clear, it ''is'' acceptable to make [[:s:en:Category:Wikisource annotations|annotated editions]] of texts at Wikisource and Wikibooks does host at least one [[:b:en:Annotations of The Complete Peanuts|annotated guide to a copyright-protected work]]. So if what you're looking to do is to include inline annotations to a public domain text, you certainly can put that on Wikisource. If you have a textbook or guidebook that is a companion, that would go at Wikibooks. If you have some other kind of learning resources (like maintaining a list of relevant links, organizing a book reading group, etc.), that could go here. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:26, 17 August 2024 (UTC)
::Thank you for your quick answer. Actually, ''Wikibooks'' was my first thought. However, this project is not merely an annotated edition. Although at first it ''will'' be a faithful copy of the original text, I want the project to be &quot;open&quot;, i. e. adding articles should be possible. And the project should enable to do a lot more than mere inline annotation. See section [[w:User:CalRis25/Temp-RICH-Prospectus#Improving_RICH|Improving Rich]] in the project description a my user-page (en.Wikipedia). No ''Mediawiki''-project (Wikisource, Wikibooks, Wikipedia, Wiktionary) seemed to be a sufficiently applicable &quot;fit&quot; for the project, so I thought of Wikiversity as a last resort, because it is supposed to be home to all sorts of &quot;learning resources&quot;. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 09:57, 17 August 2024 (UTC)
:::The scope of Wikiversity ''is'' pretty catch-all and would allow for a pretty flexible place to host most learning resources that don't fit elsewhere.
:::Also, as nitpick, &quot;MediaWiki&quot; is the software that is the basis of these wikis (wikis being collections of interlinked documents that can be edited) and &quot;Wikimedia Foundation&quot; is the non-profit who owns the trademarks and hosts these projects like Wiktionary and Wikivoyage. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 10:06, 17 August 2024 (UTC)
::::Hello Justin, thank you for the reply. '''I think that settles it. I will create this project at ''Wikiversity''.''' Just for additional clarification, why I do so. Let's imagine a full transcription of the original 1849-edition of the ''Illustrated Companion'' by Anthony Rich and call it ''RICH-1849''. We shall call my project, for brevity sake, RICH-2K. And now, let's have a look at the article about the Roman toga (a piece of attire). In ''RICH-1849'' we can can call it ''RICH-1849/Toga'', and it contains ''exactly'' the content of the 1849-book. Now, let's look at the article ''RICH-2K/Toga''. At the beginning its only content would be the article ''RICH-1849/Toga''. Does that make ''RICH-2K/Toga'' and ''RICH-1849/Toga'' the same? Not at all, because in truth ''RICH-2K/Toga'' is a &quot;container&quot; which initially contains only the article ''RICH-1849/Toga'' but later on may include more stuff: images, external links, article text which builds on or extends ''RICH-1849/Toga'' and information from other sources of information (Wikipedia, specialized books). By the way, this added article information would not be a mere copy of the text at en.Wikipedia, because the information needs to looked at through the eyes of someone reading the original text (more citations with direct links to these etc.). [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 11:39, 17 August 2024 (UTC)

== Coming soon: A new sub-referencing feature – try it! ==

&lt;section begin=&quot;Sub-referencing&quot;/&gt;
[[File:Sub-referencing reuse visual.png|{{#ifeq:{{#dir}}|ltr|right|left}}|400px]]
Hello. For many years, community members have requested an easy way to re-use references with different details. Now, a MediaWiki solution is coming: The new sub-referencing feature will work for wikitext and Visual Editor and will enhance the existing reference system. You can continue to use different ways of referencing, but you will probably encounter sub-references in articles written by other users. More information on [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing|the project page]].

'''We want your feedback''' to make sure this feature works well for you:

* [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing#Test|Please try]] the current state of development on beta wiki and [[m:Talk:WMDE Technical Wishes/Sub-referencing|let us know what you think]]. 
* [[m:WMDE Technical Wishes/Sub-referencing/Sign-up|Sign up here]] to get updates and/or invites to participate in user research activities.

[[m:Special:MyLanguage/Wikimedia Deutschland|Wikimedia Deutschland]]’s [[m:Special:MyLanguage/WMDE Technical Wishes|Technical Wishes]] team is planning to bring this feature to Wikimedia wikis later this year. We will reach out to creators/maintainers of tools and templates related to references beforehand.

Please help us spread the message. --[[m:User:Johannes Richter (WMDE)|Johannes Richter (WMDE)]] ([[m:User talk:Johannes Richter (WMDE)|talk]]) 10:36, 19 August 2024 (UTC)
&lt;section end=&quot;Sub-referencing&quot;/&gt;
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== New [[Template:Form]] ==

Hi! Today I was bold and created [[Template:Form]] (which calls [[Module:WikiForm]] and [[MediaWiki:Gadget-WikiForm.js]]). The template allows to create user-friendly forms that can create pages or add content to existing pages. My motivation and first use case was [[Wikidebate/New|this form]] to create new [[wikidebates]], but I suspect the template can be useful elsewhere on Wikiversity. Let me know if you notice any issues or have any requests or concerns. Kind regards, [[User:Sophivorus|Sophivorus]] ([[User talk:Sophivorus|discuss]] • [[Special:Contributions/Sophivorus|contribs]]) 15:21, 21 August 2024 (UTC)

== Sign up for the language community meeting on August 30th, 15:00 UTC ==

Hi all,

The next language community meeting is scheduled in a few weeks—on August 30th at 15:00 UTC. If you're interested in joining, you can [https://www.mediawiki.org/wiki/Wikimedia_Language_and_Product_Localization/Community_meetings#30_August_2024 sign up on this wiki page].

This participant-driven meeting will focus on sharing language-specific updates related to various projects, discussing technical issues related to language wikis, and working together to find possible solutions. For example, in the last meeting, topics included the Language Converter, the state of language research, updates on the Incubator conversations, and technical challenges around external links not working with special characters on Bengali sites.

Do you have any ideas for topics to share technical updates or discuss challenges? Please add agenda items to the document [https://etherpad.wikimedia.org/p/language-community-meeting-aug-2024 here] and reach out to ssethi(__AT__)wikimedia.org. We look forward to your participation!

[[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 23:20, 22 August 2024 (UTC)
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== Template consolidation: User talk page block notice ==
Wondering if someone who likes templates could have a go at consolidating or helping decide between use of:
* [[Template:Block]]
* [[Template:Blocked]]

Unless I'm missing something, it seems like we don't need both?

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 07:16, 23 August 2024 (UTC)
: I tried to figure out a Wikidata item with most links to projects. I found this: [[Wikidata:Q6379131]], which is Template:Uw-block. There is even a corresponding Wikiversity template, [[Template:Uw-block1]] (not used anywhere).
: My impression is that of the three templates, we only need one. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:43, 13 September 2024 (UTC)

== Announcing the Universal Code of Conduct Coordinating Committee ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
:''[https://lists.wikimedia.org/hyperkitty/list/board-elections@lists.wikimedia.org/thread/OKCCN2CANIH2K7DXJOL2GPVDFWL27R7C/ Original message at wikimedia-l]. [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement - results|You can find this message translated into additional languages on Meta-wiki.]] [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement - results}}&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]''

Hello all,

The scrutineers have finished reviewing the vote and the [[m:Special:MyLanguage/Elections Committee|Elections Committee]] have certified the [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Results|results]] for the [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election|Universal Code of Conduct Coordinating Committee (U4C) special election]].

I am pleased to announce the following individual as regional members of the U4C, who will fulfill a term until 15 June 2026:

* North America (USA and Canada)
** Ajraddatz

The following seats were not filled during this special election:

* Latin America and Caribbean
* Central and East Europe (CEE)
* Sub-Saharan Africa
* South Asia
* The four remaining Community-At-Large seats

Thank you again to everyone who participated in this process and much appreciation to the candidates for your leadership and dedication to the Wikimedia movement and community.

Over the next few weeks, the U4C will begin meeting and planning the 2024-25 year in supporting the implementation and review of the UCoC and Enforcement Guidelines. You can follow their work on [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee|Meta-Wiki]].

On behalf of the U4C and the Elections Committee,&lt;section end=&quot;announcement-content&quot; /&gt;

[[m:User:RamzyM (WMF)|RamzyM (WMF)]] 14:07, 2 September 2024 (UTC)
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== Re: The Vector 2022 skin as the default in two weeks? ==

[[File:Vector 2022 video-en.webm|thumb|A two minute-long video about Vector 2022]]
Hello everyone, I'm reaching out on behalf of the [[mediawikiwiki:Reading/Web|Wikimedia Foundation Web team]] responsible for the MediaWiki skins. I'd like to revisit the topic of making Vector 2022 the default here on English Wikiversity. I [[Wikiversity:Colloquium/archives/September 2022#The Vector 2022 skin as the default in two weeks?|did post a message about this almost two years ago]] (where you can find all the details about the skin), but we didn't finalize it back then.

What happened in the meantime? We built [[mw:Reading/Web/Accessibility for reading|dark mode and different options for font sizes]], and made Vector 2022 the default on most wikis, including all other Wikiversities. With the not-so-new V22 skin being the default, existing and coming features, like dark mode and [[mw:Trust and Safety Product/Temporary Accounts|temporary accounts]] respectively, will become available for logged-out users here.

So, if no large concerns are raised, we will deploy Vector 2022 here in two weeks, in the week of September 16. Do let me know if you have any questions. Thank you! [[User:SGrabarczuk (WMF)|SGrabarczuk (WMF)]] ([[User talk:SGrabarczuk (WMF)|discuss]] • [[Special:Contributions/SGrabarczuk (WMF)|contribs]]) 21:48, 2 September 2024 (UTC)

:Sounds good, Szymon - we look forward to the upcoming change of skin {{smile}} Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 07:35, 13 September 2024 (UTC)
* I for one oppose a switch to Vector 2022. I do not find it preferable. Here is a staggering evidence of user refusal of Vector 2022 once it was deployed: [[W:en:Wikipedia:Requests for comment/Rollback of Vector 2022]], Junuary 2023. 355 voters supported rollback to Vector 2010 whereas 64 opposed, yielding 84.7% support, as clear a supermajority as one may wish. These people opposing Vector 2022 feel the same way as I do. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:48, 13 September 2024 (UTC)
*:Hey @[[User:Dan Polansky|Dan Polansky]]. Thanks for your comment. I'm open to discussion about problems with our software, and I hope we can maintain a respectful tone.
*:I understand that there are users who prefer Vector legacy or other skins, just as there are people who still stick to Monobook. Such people are active across many wikis. They can keep Vector legacy, although non-default skins don't have the support the default ones do. We are rolling out for technical reasons, as I mentioned above, with benefit to not logged-in users.
*:Regarding the rollback RfC on Wikipedia, two neutral users stated that there was no consensus for rollback, RfC is not a vote, and the numbers were different (355:226:24). I believe this all is pretty easy to verify.
*:So to sum up, Vector 2022 needs to become the default, tons and tons of comments were made about the skin and related stuff, and we have taken many ideas into account, and it's totally OK if you stick to Vector legacy.
*:Thanks! [[User:SGrabarczuk (WMF)|SGrabarczuk (WMF)]] ([[User talk:SGrabarczuk (WMF)|discuss]] • [[Special:Contributions/SGrabarczuk (WMF)|contribs]]) 19:30, 16 September 2024 (UTC)
*:: Today, I visited Wikiversity and found it switched to Vector 2022. I changed my preference settings to Vector 2010. From what I understand, non-registered visitors are now defaulted to Vector 2022 despite its unpopularity in [[W:en:Wikipedia:Requests for comment/Rollback of Vector 2022]]. I have not seen any evidence that users prefer Vector 2022, and given the evidence in the linked RfC, I tentatively conclude that the decision to switch has made the site experience worse for the majority of users. The logic of &quot;you can switch&quot; surely applies to Vector 2022 as well: those who prefer it can switch to it. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:08, 17 September 2024 (UTC)

== Have your say: Vote for the 2024 Board of Trustees! ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
Hello all,

The voting period for the [[m:Special:MyLanguage/Wikimedia Foundation elections/2024|2024 Board of Trustees election]] is now open. There are twelve (12) candidates running for four (4) seats on the Board.

Learn more about the candidates by [[m:Special:MyLanguage/Wikimedia Foundation elections/2024/Candidates|reading their statements]] and their [[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Questions_for_candidates|answers to community questions]].

When you are ready, go to the [[Special:SecurePoll/vote/400|SecurePoll]] voting page to vote. '''The vote is open from September 3rd at 00:00 UTC to September 17th at 23:59 UTC'''.

To check your voter eligibility, please visit the [[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Voter_eligibility_guidelines|voter eligibility page]].

Best regards,

The Elections Committee and Board Selection Working Group&lt;section end=&quot;announcement-content&quot; /&gt;

[[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 12:15, 3 September 2024 (UTC)
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== Separate page for hyperbola. ==

Good morning,

I notice that a search for &quot;hyperbola&quot; redirects to &quot;Conic sections&quot;.

At present there is a separate page for &quot;ellipse&quot;. Therefore a separate page for &quot;hyperbola&quot; seems to be justified.

Could this redirection be changed so that search for &quot;hyperbola&quot; goes to a separate page for &quot;hyperbola&quot;?

Many thanks,

[[User:ThaniosAkro|ThaniosAkro]] ([[User talk:ThaniosAkro|discuss]] • [[Special:Contributions/ThaniosAkro|contribs]]) 12:04, 15 September 2024 (UTC)

:It is true that ellipses are covered at [[Conic sections]] (along with hyperbolas, parabolas, etc.) and there is a separate page for [[ellipse]]s that elaborates. We certainly ''could'' have a page about [[hyperbola]]s that is separate, but no one has written sufficient content to spin it off yet. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 12:17, 15 September 2024 (UTC)

== I hereby request for your Unblocking IP address and just reviewed and received a reverted rec ==

Hi there. {{unsigned|Ishmael Raphasha}}
:No one has any clue what you're talking about. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:53, 18 September 2024 (UTC)

== RICH-2K: New project with some initial questions ==

Hello! I'm creating a new learning resource on ''Wikiversity''. The respective project is based on my transcription of a classical dictionary from 1849 by Anthony Rich. For more information about the project see its [[User:CalRis25/RICH: Description|description page]] (see also that page for why not ''Wikisource'' or ''Wikibooks''). The project's scope is fairly big: 3205 article-pages plus 304 REDIRECT-pages. The images (scanned by myself from an original copy) have been uploaded to ''Commons''. I have some initial technical questions (more of these and more detailed ones will follow):

* '''Upload''': Due to the large number of pages it is not realistic to create these manually. Is it possible to bulk-upload these in some way (the Wikitext of the pages is created using a Python-script with one file per article/page)? Is it possible to upload these to a test-environment first where any problems (hopefully none) can be identified and dealt with more easily than on the production-version of ''Wikiversity''?
* '''(Technical) Structure''': I am planning to set up this project at ''&lt;nowiki&gt;https://en.wikiversity.org/wiki/RICH-2K&lt;/nowiki&gt;'' as the main page and anything else as subpages: ''RICH-2K/Subpage_1 ... RICH-2K/Subpage_n''. However, these subpages fall into two categories: 1. Article-pages (content) and 2. Meta/Administrative pages. This project requires search capability restricted to the ''RICH-2K''-namespace. The ''Mediawiki''-software seems to supply a ''Search''-input field with the possibility to restrict the search to some namespace. I would like, however, to restrict the search further to the first group of pages, namely the articles. Is that possible, perhaps by use of (hidden) categories?
* '''External links''': This project will need many external links, and yes, I have read the relevant ''Wikiversity''-pages, but this specific project needs them. The ''Recommended Editions''-page (used for recommended online editions, to which to link when citing texts) alone probably will require several hundred external links. However, only relatively few [[w:Second-level domain|second-level domains]] will be involved, and most of these should be trustworthy (Perseus Digital library, digital collections of universities etc., in some cases, however, also ''Archive.org''). Perhaps there is a list of web-sites, for which external links are generally allowed? And who is allowed to create external links on ''Wikiversity''-pages (I haven't found the relevant policy)?
* '''Categories''': This project requires quite a few of its own categories, which belong to two large groups: 1. Categories (2 levels) of the ''Classed Index'' (about 170 categories), a thematic index of some (but not all) of the articles. 2. Administrative categories. Is there a recommended way to distinguish between different classes of categories within a project (category name or other method)? What about naming conventions for project-specific categories?

I am looking forward to your input. If you think that it's preferable we can move the discussions to the [[User_talk:CalRis25/RICH:_Description|Talk-page]] of the project's description. Thank you in advance. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 05:29, 20 September 2024 (UTC)

:*Admins have access to [[Special:Import]] and can bulk import XML pages. You can create pages in your sandbox if you'd like and make an indefinite amount of them at pages like [[User:CalRis25/sandbox]]. What can and cannot be hosted in user namespace is very loose, but still has to follow in principle Wikiversity's scope.
:*Using subpages is in principle a good way to organize these various resources. Please do not name them after a user name or something obscure. I personally think that &quot;RICH-2K&quot; is a not optimal name. I may recommend something like [[Anthony Rich Dictionary Project]] or [[21st-Century Anthony Rich Dictionary]] or something more obviously intelligible. While we have very few actual policies and guidelines, see [[Wikiversity:Naming conventions]] for a rough consensus of what is probably best practice for naming pages.
:*External linking generally does not use an allowed list (a.k.a. whitelist model), but a disallow (a.k.a. blacklist) model. See [[MediaWiki:Spam-blacklist]] and [[Special:BlockedExternalDomains]] (which is currently empty but is another method of listing blocked domains). It's perfectly fine to aggregate external links in learning resources.
:*I'm not 100% sure what the distinction is that you're drawing, but you can freely arrange categories underneath a main category that has the same name as your larger project. So, following the suggestions I gave, you could have a category like [[:Category:Anthony Rich Dictionary Project]] and then create any number of subcategories that logically help users navigate all these pages. Please make sure the main category you create is itself categorized under some relevant category(ies). If you need help, please ask.
:I think this answers your questions, please let me know if I'm unclear or you have more. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:11, 20 September 2024 (UTC)
::Hello Justin!
::* '''Upload:''' Creating the project in sandbox pages of my User-namespace defeats the purpose, as this is an ''open'' project. Also that would not solve, as such, the problem of having to manually create thousands of pages. I wonder, does ''Wikiversity'' support creation of pages using its API. ''Mediawiki's'' [[mw:API:Main_page|API-description]] seems to imply that it ought to be possible. If that's the case, I should be able to create a Python-script which automatically creates the pages (of course, a few trial pages first).
::* '''(Technical) Structure''': You may be right, here. RICH-2K is, for now, merely a technical name to make a clear but not too verbose distinction between the original text and the current project. I'll give this more thought.
::* '''External links''': I brought this up mainly because when I first edited my ''Wikiversity''-page, I got a message that I was not allowed to create external links. However, I just now tested creating an external link on my user-page and got no error, so this problem seems to be solved.
::* '''Categories''': I think I know what you mean. I'll create a category structure and maybe ask some specific questions once I am ready to do so.
::Thank you for your quick help. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 18:51, 20 September 2024 (UTC)
:::re: upload, I'm just suggesting your sandbox(es) as you asked about &quot;a test-environment&quot;. Anyone can edit someone else's sandboxes, but you typically defer to other users to control what's in their own subpages as a collegial thing. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:39, 23 September 2024 (UTC)

Hello! I have two further questions:
# I created a category-structure for the project. Could you (or someone else) have a look at it ([[User:CalRis25/RICH: Categories]]) and answer the questions in the section [[User:CalRis25/RICH:_Categories#Questions|Questions]]? I gave it some thought and believe that this would work fine for the project.
# ''Project boxes'' (see [[Help:Tour of project boxes]]): It is unclear to me, whether these belong only on the main page of the project (that makes the most sense to me), or on every single subpage.
Thanks in advance for your help. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 17:51, 24 September 2024 (UTC)

:To answer your questions here:
:*No, you are not contravening any policies we have.
:*A leading &quot;The&quot; is acceptable, but if you want it to sort alphabetically, you will have to use &lt;nowiki&gt;{{DEFAULTSORT:}}&lt;/nowiki&gt;. E.g. to get Category:The Best Stuff to sort under &quot;B&quot;, insert &quot;&lt;nowiki&gt;{{DEFAULTSORT:Best Stuff, The}}&lt;/nowiki&gt;.
:*Trailing &quot;etc.&quot; is acceptable.
:*An accent in a category title is acceptable.
:I'll also note that it looks like you have in mind some tracking categories that are redundant. Pages such as [[Special:LonelyPages]] and [[Special:DeadendPages]] already do automatically what you're proposing to do manually.
:As for project boxes, it's typically the case that the subjects are only placed on the main resource, but as you may imagine, [[Help:Tour of project boxes/1|status completion ones]] may vary from subpage to subpage. As with most things at Wikiversity, there are very few actual rules, so it's pretty much the wild west, even tho this project has been around for almost 20&amp;nbsp;years. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:18, 25 September 2024 (UTC)
::Hello Justin, thanks for the DEFAULTSORT-hint for categories beginning with ''The''. I will restrict the project boxes to the main page. As for the the orphaned/dead-end-categories, I prefer these to be project-specific. Once the project is up and running, putting articles &quot;on the map&quot; (making them accessible from other articles and creating links to other articles) is one of the first tasks to be dealt with. I already know which articles are involved and will add these categories to these articles. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 16:51, 25 September 2024 (UTC)

== Your wiki will be in read-only soon ==

&lt;section begin=&quot;server-switch&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;

[[:m:Special:MyLanguage/Tech/Server switch|Read this message in another language]] • [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-Tech%2FServer+switch&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]

The [[foundation:|Wikimedia Foundation]] will switch the traffic between its data centers. This will make sure that Wikipedia and the other Wikimedia wikis can stay online even after a disaster.

All traffic will switch on '''{{#time:j xg|2024-09-25|en}}'''. The switch will start at '''[https://zonestamp.toolforge.org/{{#time:U|2024-09-25T15:00|en}} {{#time:H:i e|2024-09-25T15:00}}]'''.

Unfortunately, because of some limitations in [[mw:Special:MyLanguage/Manual:What is MediaWiki?|MediaWiki]], all editing must stop while the switch is made. We apologize for this disruption, and we are working to minimize it in the future. 

A banner will be displayed on all wikis 30 minutes before this operation happens. This banner will remain visible until the end of the operation. 

'''You will be able to read, but not edit, all wikis for a short period of time.'''

*You will not be able to edit for up to an hour on {{#time:l j xg Y|2024-09-25|en}}.
*If you try to edit or save during these times, you will see an error message. We hope that no edits will be lost during these minutes, but we can't guarantee it. If you see the error message, then please wait until everything is back to normal. Then you should be able to save your edit. But, we recommend that you make a copy of your changes first, just in case.

''Other effects'':

*Background jobs will be slower and some may be dropped. Red links might not be updated as quickly as normal. If you create an article that is already linked somewhere else, the link will stay red longer than usual. Some long-running scripts will have to be stopped.
* We expect the code deployments to happen as any other week. However, some case-by-case code freezes could punctually happen if the operation require them afterwards.
* [[mw:Special:MyLanguage/GitLab|GitLab]] will be unavailable for about 90 minutes.

This project may be postponed if necessary. You can [[wikitech:Switch_Datacenter|read the schedule at wikitech.wikimedia.org]]. Any changes will be announced in the schedule. 

'''Please share this information with your community.'''&lt;/div&gt;&lt;section end=&quot;server-switch&quot;/&gt;

[[User:Trizek_(WMF)|Trizek_(WMF)]], 09:37, 20 September 2024 (UTC)
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== 'Wikidata item' link is moving. Find out where... ==

&lt;div lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot;&gt;&lt;i&gt;Apologies for cross-posting in English. Please consider translating this message.&lt;/i&gt;{{tracked|T66315}}

Hello everyone, a small change will soon be coming to the user-interface of your Wikimedia project.
The [[d:Q16222597|Wikidata item]] [[w:|sitelink]] currently found under the &lt;span style=&quot;color: #54595d;&quot;&gt;&lt;u&gt;''General''&lt;/u&gt;&lt;/span&gt; section of the '''Tools''' sidebar menu will move into the &lt;span style=&quot;color: #54595d;&quot;&gt;&lt;u&gt;''In Other Projects''&lt;/u&gt;&lt;/span&gt; section.

We would like the Wiki communities feedback so please let us know or ask questions on the   [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Discussion page]] before we enable the change which can take place October 4 2024, circa 15:00 UTC+2.
More information can be found on [[m:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|the project page]].&lt;br&gt;&lt;br&gt;We welcome your feedback and questions.&lt;br&gt; [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 18:56, 27 September 2024 (UTC)
&lt;/div&gt;
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==Download as PDF==
[[Phabricator:T376438]]: &quot;Download to PDF&quot; on en.wv is returning error: &quot;{&quot;name&quot;:&quot;HTTPError&quot;,&quot;message&quot;:&quot;500&quot;,&quot;status&quot;:500,&quot;detail&quot;:&quot;Internal Server Error&quot;}&quot;

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:36, 3 October 2024 (UTC)

:I just downloaded this page as a PDF and it worked just fine. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:04, 3 October 2024 (UTC)

== Protected template bug for Pp ==

It seems that templates derivative of {{tlx|Pp}} (compiled in {{tlx|Protection templates}}) are being sorted into protection categories using the name 'Wikipedia' instead of 'Wikiversity' (e.g., [[:Category:Wikipedia pages with incorrect protection templates]]). From what I can tell, it is not in the publicly accessible source code of any of the templates. The only other impacted pages are modules which call {{tlx|pp}}-derivatives (e.g., [[Module:Navbar/styles.css]]).

This does not seem to affect any other pages in [[:Category:Wikiversity protected templates]]. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 18:59, 4 October 2024 (UTC)

:The problem is that &quot;Wikipedia&quot; is [https://en.wikiversity.org/w/index.php?title=Special%3ASearch&amp;limit=500&amp;offset=0&amp;ns828=1&amp;search=Wikipedia&amp;searchToken=9svkpqlxxoquoq7bnkt55ugts mentioned in several modules that were copied over from en.wp]; many of these are legit and many of them need to be replaced with &quot;Wikiversity&quot; ([https://en.wikiversity.org/w/index.php?title=Module%3APp-move-indef&amp;diff=2662815&amp;oldid=1944984 e.g.]) This particular change ''may'' fix all of these issues...? But 1.) it will take time to propagate across the site and 2.) there are still many more &quot;Wikipedia&quot;s that need to be changed, so I'll go thru a few more, but if you want to give me an assist, if you can just check this one week from now and ping me if the problem persists, that would be nice. Sometimes, I make calendar reminders to follow up on these, but I'm not a perfect person. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:55, 5 October 2024 (UTC)

== Invitation to Participate in Wiki Loves Ramadan Community Engagement Survey ==

Dear all,


We are excited to announce the upcoming [[m:Wiki Loves Ramadan|Wiki Loves Ramadan]] event, a global initiative aimed at celebrating Ramadan by enriching Wikipedia and its sister projects with content related to this significant time of year. As we plan to organize this event globally, your insights and experiences are crucial in shaping the best possible participation experience for the community.

To ensure that Wiki Loves Ramadan is engaging, inclusive, and impactful, we kindly invite you to participate in our community engagement survey. Your feedback will help us understand the needs of the community, set the event's focus, and guide our strategies for organizing this global event.

Survey link: https://forms.gle/f66MuzjcPpwzVymu5

Please take a few minutes to share your thoughts. Your input will make a difference!

Thank you for being a part of our journey to make Wiki Loves Ramadan a success.


Warm regards,

User:ZI Jony 03:19, 6 October 2024 (UTC)

Wiki Loves Ramadan Organizing Team
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== 'Edit to my talk page' notification bug? ==

This may belong at the bug tracker, but does anyone else have an issue disabling ''email'' notifications upon  an 'Edit to my talk page' in [[Special:GlobalPreferences]]? Oddly I ''am'' able to disable the global preference on Wikipedia, MediaWiki, etc, but not here. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 09:23, 7 October 2024 (UTC)

:I have not experienced this, but to be clear, do you also have the option to get emails when items on your talk page are edited turned on? —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:39, 7 October 2024 (UTC)
::The only (non-grayed out) options I have enabled for email are 'Failed login attempts' and 'Login from an unfamiliar device'. 'Edit to my talk page' re-checks after every save. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 09:54, 7 October 2024 (UTC)
:::That does sound like a [[phab:]] issue, with the caveat that I don't 100% recall how global preferences work and if they override local ones, etc. If you have parsed that and still have this issue, you'll probably need to file a ticket. Maybe someone else has this issue. Wish I could help. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:57, 7 October 2024 (UTC)
::::[[phab:T376601|Off 'n away]] 🫡 [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 10:35, 7 October 2024 (UTC)

== [[Portal:Computer Science]] ➝ [[Portal:Information sciences]] ==

Seeking consensus to complete the merge into the broader portal. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 06:28, 8 October 2024 (UTC)

:Why should it be merged? Computer Science seems well-enough designed. What is the incentive to collapse it into a broader field of study? —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:18, 8 October 2024 (UTC)
::Portals as top level organizations allowing for content to be best centralized. Also note that I did not start the merge, just offering to finish it. Perhaps a {{tlx|prod}} instead? [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 07:20, 8 October 2024 (UTC)
:::I have no objections, personally. If it gets done, please use a redirect and should someone want to come along to resurrect it later, it will be easier. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:21, 8 October 2024 (UTC)
: Is computer science really a branch of information sciences? I would not think so, but what do I know. Do we have some external resources/links confirm this idea? [[W:Information science]] currently says: &quot;Information science, documentology[1] or informatology[2][3] is an academic field which is primarily concerned with analysis, collection, classification, manipulation, storage, retrieval, movement, dissemination, and protection of information.&quot; --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:49, 11 October 2024 (UTC)
::Looking through [https://stackoverflow.com/q/1047014/22673230] [https://businessdegrees.uab.edu/mis-degree-bachelors/resources/computer-information-systems-vs-computer-science/] [https://www.si.umich.edu/student-experience/what-information-science] a few top (not necessarily RS) searches I'm inclined to agree. I am more familiar with the grafted [[:w:Information and computer science|information ''and'' computer science]] which makes an effort to merge the disciplines, but it does not seem like reaching to say that IS is presented as more applications-concerned (certainly with no lack of theoretical abstraction), whereas CS can be more freely associated with any and all 'science related to computers'. It is easy to reason about the connection between the fields, but I think it is clear academia maintains this taxonomy for a good reason.
::With these considerations, I think I will ''stop'' the process of merging in favor of expanding the existing [[School:Library and Information Science]]. 

::Let me know if there is not consensus to redirect [[Portal:Information sciences]] to [[School:Library and Information Science]] (with enough expansion it can generalize away from just library sciences). [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 16:16, 11 October 2024 (UTC)
::: I do not see that a merge of a ''portal'' to a ''school'' is a good thing. Do you have a clear idea of the concepts of school and portal and how they relate to each other? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:34, 11 October 2024 (UTC)
::::Found [[:Category:Information sciences]]; there are enough existing resources in there to make my other proposed merge excessive. I will simply continue developing the existing [[Portal:Information sciences]] instead. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 17:05, 11 October 2024 (UTC)
::::: Frankly, I would ideally see [[Portal:Information sciences]] deleted: I don't see what it does that a category would not do well enough. There does not seem to be any material specific to &quot;Information sciences&quot; (whatever that is) in that portal at all. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:11, 11 October 2024 (UTC)
::::::Tacked a {{tlx|prod}} for an eventual deletion, but I may still try to develop it as proof of concept at some point. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 17:33, 11 October 2024 (UTC)

== [[:Category:Occupational Epidemiology]] ==

I propose moving the pages in this category (without leaving redirects) to their equivalent under the parent resource [[Occupational Health Risk Surveillance]]. Also due to the number of subpages, it seems &lt;code&gt;|filing=deep&lt;/code&gt; would be a justified. (Also [[Special:PrefixIndex/Occupational_Epidemiology|there are quite a few]] untagged subpages.) [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 05:11, 9 October 2024 (UTC)
: I above all think that the content should be ''moved out of the mainspace'': I do not see readers learning anything from e.g. [[Occupational Epidemiology/Research tools/Reading of scientific articles for learning epidemiology and biostatstics]] or [[Occupational Epidemiology/Research tools/Ongoing projects/Risk Communication in Seafaring/Writing the article guideline IMRAD]]. Wikiversity can be kind enough to host that material in, say, subspace of [[User:Saltrabook]], but more should not be asked, I think. Let us recall that per [[WV:Deletions]], &quot;Resources may be eligible for proposed deletion when education objectives and learning outcomes are scarce, and objections to deletion are unlikely&quot;; I do not see how learning outcomes can be anything but scarce. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 15:04, 11 October 2024 (UTC)
::thank you, agree @ [[User:Saltrabook|Saltrabook]] ([[User talk:Saltrabook|discuss]] • [[Special:Contributions/Saltrabook|contribs]]) 21:03, 13 November 2024 (UTC)

== Active editors ==

It is interesting to observe the stats on [https://stats.wikimedia.org/#/en.wikiversity.org/contributing/active-editors/normal|line|all|(page_type)~content*non-content|monthly active editors] through the project's history. October is our month! [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 20:44, 8 October 2024 (UTC)

:Odd. Maybe related to the school year? —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 02:10, 9 October 2024 (UTC)
::I wonder how many are [[User:Jtneill|Jtneill]]'s crowd... the number is in the hundreds though, so that is one chunky cohort —[[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:16, 9 October 2024 (UTC)
:::Yes, [[Motivation and emotion/Book]] involves ~100-150 students editing most intensely during October each year. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:20, 9 October 2024 (UTC)
::::Neat, that still leaves around ~50-100 other students from other avenues each year since 2021. I also wonder which projects were involved in the COVID enrollment spike. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:26, 9 October 2024 (UTC)
:::::Personally I can admit that my editing is much more active during the school season vs. the summer break, so I'm in the same boat as Jtneill's students. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:24, 13 November 2024 (UTC)
:@[[User:Tule-hog|Tule-hog]] This is an interesting topic, but it is not clear to me as an outsider what you and other participants in this discussion find interesting. I  find this graph not very meaningful because it does not tell me if the number of  Active editors has gone up or down during the period covered, which I think was 2000-now. 
:I can see a big jump between 2000 and 2007, but what happened since then? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 15:45, 16 December 2024 (UTC)

== Intentionally incorrect resource ==

There is a [[Special:Diff/2583464|disclaimer inserted onto a resource]] (by not the original author) that:
&lt;blockquote&gt;I am merely [making this page false] to show you (The viewer) that Wikipedia and this page 'Wikiversity' is bull sh*t and it will not give you the reliability you need when writing an academic piece of writing.&lt;/blockquote&gt;
However, that IP has [[Special:Contributions/86.22.73.151|not made any other edits]], so unless they vandalized via a sock, the intent went un-realized and only that portion need be removed. Bumping here in case there is some obvious jumbo in that essay that someone else can catch. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 16:58, 9 October 2024 (UTC)

:Removed that portion, which was obviously vandalism. No perspective on the rest of the essay. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:38, 9 October 2024 (UTC)

== [[:Category:Proposed guidelines]] ==

Noting for future editors that WV has collapsed all proposals into [[:Category:Proposed policies|proposed policies]]. Seeking consensus to further collapse [[:Category:Wikiversity proposals]] into the former, or to restore [[:Category:Proposed guidelines]]. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 19:19, 9 October 2024 (UTC)

== [[Around Wikiversity in 80 Seconds|Broken 80-second tour]] ==

Bumping a [[Talk:Around_Wikiversity_in_80_Seconds|comment]] on the ''Wikiversity in 80 seconds'' tour. Appears wikisuite is not working with the Vector 2022 appearance. 

Also see [[:w:Wikipedia:Miscellany_for_deletion/Wikiversuite_pages|this thread]] on the Wikiversal package - may not be relevant to Wikiversity, but FYC. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 00:26, 10 October 2024 (UTC)
: I would just delete the material; I do not see value in it. If others agree, I would try to articulate why I think it should be deleted (or move to author user space). --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:57, 13 October 2024 (UTC)
::Just mark as {{tl|historical}}. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:39, 13 October 2024 (UTC)
::: This thing was created by [[User:Planotse]]. His creations are now being discussed in Wikibooks for deletion: [[B:Wikibooks:Requests for deletion#Wikiversal generated pages]]. It seems he used some kind of tool that is no longer available (the above mentioned &quot;Wikiversal&quot; package) to create this kind of slideshow-like material (believing the Wikibooks discussion). I do not see value of this in the mainspace, not even as historical (I am okay with userspace, but maybe even that is not the best option?). A look at the source code of [[Around Wikiversity in 80 Seconds/Introduction]] confirms the words of Omphalographer, namely that &quot;the HTML-heavy markup generated by Wikiversal makes them [the pages] unreasonably difficult to edit.&quot;
::: I went ahead and marked the page for proposed deletion. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:35, 14 October 2024 (UTC)

== Preliminary results of the 2024 Wikimedia Foundation Board of Trustees elections ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
Hello all,

Thank you to everyone who participated in the [[m:Special:MyLanguage/Wikimedia Foundation elections/2024|2024 Wikimedia Foundation Board of Trustees election]]. Close to 6000 community members from more than 180 wiki projects have voted.

The following four candidates were the most voted:

# [[User:Kritzolina|Christel Steigenberger]]
# [[User:Nadzik|Maciej Artur Nadzikiewicz]]
# [[User:Victoria|Victoria Doronina]]
# [[User:Laurentius|Lorenzo Losa]]

While these candidates have been ranked through the vote, they still need to be appointed to the Board of Trustees. They need to pass a successful background check and meet the qualifications outlined in the Bylaws. New trustees will be appointed at the next Board meeting in December 2024.

[[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Results|Learn more about the results on Meta-Wiki.]]

Best regards,

The Elections Committee and Board Selection Working Group

&lt;section end=&quot;announcement-content&quot; /&gt;

[[User:MPossoupe_(WMF)|MPossoupe_(WMF)]] 08:26, 14 October 2024 (UTC)
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== Seeking volunteers to join several of the movement’s committees ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
Each year, typically from October through December, several of the movement’s committees seek new volunteers. 

Read more about the committees on their Meta-wiki pages:
* [[m:Special:MyLanguage/Affiliations_Committee|Affiliations Committee (AffCom)]]
* [[m:Special:MyLanguage/Ombuds_commission|Ombuds commission (OC)]]
* [[m:Special:MyLanguage/Wikimedia Foundation/Legal/Community Resilience and Sustainability/Trust and Safety/Case Review Committee|Case Review Committee (CRC)]]

Applications for the committees open on 16 October 2024. Applications for the Affiliations Committee close on 18 November 2024, and applications for the Ombuds commission and the Case Review Committee close on 2 December 2024. Learn how to apply by [[m:Special:MyLanguage/Wikimedia_Foundation/Legal/Committee_appointments|visiting the appointment page on Meta-wiki]]. Post to the talk page or email [mailto:cst@wikimedia.org cst@wikimedia.org] with any questions you may have.

For the Committee Support team,
&lt;section end=&quot;announcement-content&quot; /&gt;

-- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 23:09, 16 October 2024 (UTC)
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== Interactive elements ==
Can we use interactive elements on Wikiversity? I'd like to add JavaScript to a page. If it's not possible now, where can I suggest this feature? I have a safe integration idea.
[[User:Отец Никифор|Отец Никифор]] ([[User talk:Отец Никифор|discuss]] • [[Special:Contributions/Отец Никифор|contribs]]) 12:10, 17 October 2024 (UTC)
: This is beyond my technical knowledge, but have you checked out:
:* https://www.mediawiki.org/wiki/Manual:Interface/JavaScript?
:* [[Wikipedia:WikiProject JavaScript]]
:* [[MediaWiki:Common.js]]

:What sort of interactive elements are you thinking about?

: Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:39, 18 October 2024 (UTC)
::I was thinking about adding something like a graph with adjustable controls, where users can interact with it and see how different changes affect the outcome. It seems like this could be a useful feature. There might already be discussions about enhancing Wikiversity or similar platforms—perhaps on a relevant talk page or in a Discord group. Do you know where such discussions might be happening? [[User:Отец Никифор|Отец Никифор]] ([[User talk:Отец Никифор|discuss]] • [[Special:Contributions/Отец Никифор|contribs]]) 19:47, 18 October 2024 (UTC)
:::From a quick look, maybe check out:
:::* [[mw:Extension:Graph]]
:::* [[phab:tag/graphs]]
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:40, 18 October 2024 (UTC)
:::: mw:Extension:Graph is currently disabled on Wikipedia etc. wikis, for security reasons, and seems unlikely to be enabled again. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:30, 19 October 2024 (UTC)

== An unexplained spurt of Wikiversity page views ==
The [https://pageviews.wmcloud.org/siteviews/?platform=all-access&amp;source=pageviews&amp;agent=user&amp;start=2024-06-01&amp;end=2024-10-18&amp;sites=en.wikiversity.org|en.wikibooks.org|en.wikiquote.org|en.wikisource.org page view report] shows an unexplained spurt of Wikiversity page views, reaching over 4 times the baseline and then falling back again. Does anyone have any idea what is going on? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:01, 19 October 2024 (UTC)

:Interesting. I wonder why only the English wikiquote and wikiversity and not Wikisource or wikibooks? How reliable do you think those stats are? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 15:44, 8 December 2024 (UTC)

== Center tempate failed on a contributors phone... ==

See the edit comment here - https://en.wikiversity.org/w/index.php?title=Wikiphilosophers&amp;diff=prev&amp;oldid=2673962. I'm puzzled as this is the first failure of this, I've noted recently. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 08:45, 19 October 2024 (UTC)

== Essay-like page in user space that makes little sense and seems incoherent ==
The page [[User:TheoYalur/Illusions]] seems to match the description, at least by my assessment. My understanding is that since the page is only in user space and not in the mainspace, it can stay there even if it has those disqualifying qualities. But if I am wrong and the page belongs deleted, please correct me and let me know. I do not know which policy or guideline, if any, guides the case. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:30, 21 October 2024 (UTC)

== 'Wikidata item' link is moving, finally. ==

Hello everyone, I previously wrote on the 27th September to advise that the ''Wikidata item'' sitelink will change places in the sidebar menu, moving from the '''General''' section into the '''In Other Projects''' section. The scheduled rollout date of 04.10.2024 was delayed due to a necessary request for Mobile/MinervaNeue skin. I am happy to inform that the global rollout can now proceed and will occur later today, 22.10.2024 at 15:00 UTC-2. [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Please let us know]] if you notice any problems or bugs after this change. There should be no need for null-edits or purging cache for the changes to occur. Kind regards, -[[m:User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] 11:28, 22 October 2024 (UTC)
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:Hi @[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]]: I Just noticed your post above, and it is timely.
:I have been participating in the English WikiUniversity for a few years, much less often recently. I seems like something in the way the site displays is different, but I cannot put my finger on it. Your posting gave me a clue. Can you please tell me where  the link to wikidata items has moved to? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:23, 11 December 2024 (UTC)
::Hello @[[User:Ottawahitech|Ottawahitech]], sure, I would be happy to. The button/sitelink name didn't change, just its position. You should find it in the sidebar-menu under the section '''In other projects''' (where the links to all other Wikimedia Projects are displayed). If you do not see it, please reach out to us on the [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Move Wikidata item - Discussion page]]. Thank you, -[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] ([[User talk:Danny Benjafield (WMDE)|discuss]] • [[Special:Contributions/Danny Benjafield (WMDE)|contribs]]) 09:24, 12 December 2024 (UTC)
:::@[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]], thank you for responding. I intend to followup on the ''Move Wikidata item - Discussion page'' as per your post above by putting it on my ever growing todo list.
:::I don't know about others on this wiki, as I said I have not been visiting here frequently, but for me the constant changes are a big distraction. I have been around wikimedia projects since 2007, so why do I have to spend so much time learning and re-learning how to find what I came here for? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:41, 12 December 2024 (UTC)
::::Hi @[[User:Ottawahitech|Ottawahitech]], thanks for you thoughts. Your input whether positive or critical helps us understand the impacts to editors so we welcome your further thoughts when you reach us in your To Do List :)
::::I can't speak about the other changes you've experienced here but I do hope they are made with a spirit of improvement for the community as a whole. -[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] ([[User talk:Danny Benjafield (WMDE)|discuss]] • [[Special:Contributions/Danny Benjafield (WMDE)|contribs]]) 10:43, 16 December 2024 (UTC)

== Final Reminder: Join us in Making Wiki Loves Ramadan Success ==

Dear all,

We’re thrilled to announce the Wiki Loves Ramadan event, a global initiative to celebrate Ramadan by enhancing Wikipedia and its sister projects with valuable content related to this special time of year. As we organize this event globally, we need your valuable input to make it a memorable experience for the community.

Last Call to Participate in Our Survey: To ensure that Wiki Loves Ramadan is inclusive and impactful, we kindly request you to complete our community engagement survey. Your feedback will shape the event’s focus and guide our organizing strategies to better meet community needs.

* Survey Link: [https://docs.google.com/forms/d/e/1FAIpQLSffN4prPtR5DRSq9nH-t1z8hG3jZFBbySrv32YoxV8KbTwxig/viewform?usp=sf_link Complete the Survey]
* Deadline: November 10, 2024
Please take a few minutes to share your thoughts. Your input will truly make a difference!

'''Volunteer Opportunity''': Join the Wiki Loves Ramadan Team! We’re seeking dedicated volunteers for key team roles essential to the success of this initiative. If you’re interested in volunteer roles, we invite you to apply.
* Application Link: [https://docs.google.com/forms/d/e/1FAIpQLSfXiox_eEDH4yJ0gxVBgtL7jPe41TINAWYtpNp1JHSk8zhdgw/viewform?usp=sf_link Apply Here]
* Application Deadline: October 31, 2024
Explore Open Positions: For a detailed list of roles and their responsibilities, please refer to the position descriptions here: [https://docs.google.com/document/d/1oy0_tilC6kow5GGf6cEuFvdFpekcubCqJlaxkxh-jT4/ Position Descriptions]

Thank you for being part of this journey. We look forward to working together to make Wiki Loves Ramadan a success!


Warm regards,&lt;br&gt;
The Wiki Loves Ramadan Organizing Team 05:11, 29 October 2024 (UTC)
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== Android app for Wikiversity ==

Hi, is there an Android app for Wikiversity? How does it work? I have been advised that there is no infrastructure for push notifications for Android apps for sister wikis and I would be interested to know more. Related: [[:phab:T378545]]. Thanks! [[User:Gryllida|Gryllida]] 23:15, 29 October 2024 (UTC)

:Thanks for suggesting this - I agree that it would be useful. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:56, 31 October 2024 (UTC)
:@[[User:Gryllida|Gryllida]]: Would you explain your terminology  for those of us not in the know. What does ''push notifications'' mean? I use [https://www.mediawiki.org/wiki/Help:Notifications notifications] when I am communicating on wikimedia projects, but have never heard this term before. [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:13, 11 December 2024 (UTC)

à

== Import Resource From Wikibooks? ==

Hello! [[wikibooks:Character_List_for_Baxter&amp;Sagart|Character List for Baxter&amp;Sagart]] and related titles [[wikibooks:Wikibooks:Requests_for_deletion#Character_List_for_Baxter&amp;Sagart|are up for deletion at Wikibooks]] because WB policy does not allow dictionaries like them. However, because they are useful as learning tools, I am wondering if they might have a home here at Wikiversity. Pinging @[[User:Tibetologist|Tibetologist]] here to link them in to this discussion, since they are the affected user. Thank you! —[[User:Kittycataclysm|Kittycataclysm]] ([[User talk:Kittycataclysm|discuss]] • [[Special:Contributions/Kittycataclysm|contribs]]) 18:18, 1 November 2024 (UTC)

:Sure, I can do it. That said, as mentioned there, it does seem like something like this is ideally suited for Wiktionary in the Appendix namespace, but I'm not very familiar with CJK characters and languages. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:23, 3 November 2024 (UTC)
::Oh man, these pages are too big to import and while I've already tried a half-dozen times, it will constantly fail. Strictly speaking, we don't have to use the import feature for licensing purposes. We can just copy and paste the contents and list the usernames or on the talk page. I think that's the solution. {{Ping|Tibetologist}}, are you interested in doing that? If you just copied and pasted these pages and then added [[:Category:Chinese]] and maybe include a couple of links to the pages, that would probably be ideal. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:31, 3 November 2024 (UTC)

== Language translation requests? ==

Is there anywhere on Wikiversity to request translation, for example, requesting Latin or French translation? I would be asking from the context as a student, so I would be interested in translation explanation as well. [[User:Indexcard88|Indexcard88]] ([[User talk:Indexcard88|discuss]] • [[Special:Contributions/Indexcard88|contribs]]) 04:56, 20 November 2024 (UTC)

== Sign up for the language community meeting on November 29th, 16:00 UTC ==

Hello everyone,

The next language community meeting is coming up next week, on November 29th, at 16:00 UTC (Zonestamp! For your timezone &lt;https://zonestamp.toolforge.org/1732896000&gt;). If you're interested in joining, you can sign up on this wiki page: &lt;https://www.mediawiki.org/wiki/Wikimedia_Language_and_Product_Localization/Community_meetings#29_November_2024&gt;.

This participant-driven meeting will be organized by the Wikimedia Foundation’s Language Product Localization team and the Language Diversity Hub. There will be presentations on topics like developing language keyboards, the creation of the Moore Wikipedia, and the language support track at Wiki Indaba. We will also have members from the Wayuunaiki community joining us to share their experiences with the Incubator and as a new community within our movement. This meeting will have a Spanish interpretation.

Looking forward to seeing you at the language community meeting! Cheers, [[User:SSethi (WMF)|Srishti]] 19:55, 21 November 2024 (UTC)
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== Events on Wikiversity ==

Since Wikipedia and Wikivoyage are having their &quot;Asian Month&quot; editathon, I was thinking if we could start up a Wikiversity version of that. This would be an &quot;Asian Month&quot; as well, but it would be about creating resources based on Asia and its culture. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 17:57, 6 December 2024 (UTC)

:Not immediately opposed, but the question is, do we have an active enough community to facilitate this? —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:31, 6 December 2024 (UTC)
::I'm not too sure. As long as we get enough traffic, this could happen. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 08:45, 7 December 2024 (UTC)
:::This is to increase traffic on Wikiversity, which is promoted amongst other communities. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 10:47, 7 December 2024 (UTC)
:Hi @[[User:RockTransport|RockTransport]], This is a good idea, but will it also involve users who are not &quot;professors and scientists&quot;. Just curious. cheers, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:30, 9 December 2024 (UTC)
::Yes, considering the fact that Wikiversity is for everyone, and not just for specific users. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 09:09, 10 December 2024 (UTC)
:::because I'm personally not a &quot;professor&quot; or a &quot;scientist&quot; and because '''anyone''' can create resources on Wikiversity. We want to make Wikiversity open for everyone, and not just for certain users. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 09:10, 10 December 2024 (UTC)
::::I am also not a professor or a scientist, but it seems to me that as result I am viewed here as a visitor rather than someone who can contribute. Just my $.02. [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:05, 12 December 2024 (UTC)

== Wikiversity - Newsletters ==

Hello All,

I wanted to create a newsletter on Wikiversity, which would highlight what is going on in certain months and events on Wikiversity; which would bolster engagement by many people. This would be on the website and would have its dedicated 'Newsletter' tab.

I hope you acknowledge this idea. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 21:05, 8 December 2024 (UTC)

:@[[User:RockTransport|RockTransport]], What sort of things do you plan to include in your newsletter? Will they be different than what is currently in [[Main Page/News]]? Just curious.
:I am also wondering about your motive which I think is: to bolster engagement by many people. I am asking because I wonder if others who are currently active here also think this I is desirable? Have you asked them? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:34, 11 December 2024 (UTC)
::Not yet, which was why I was asking this on the colloquium. I plan to include things that many people have created on Wikiversity over the month, as it is a monthly newsletter. It would be somewhere on the website here. It will be more frequent that the ones seen on [[Main Page/News]]. We will include people's resources to essentially promote them. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 06:50, 12 December 2024 (UTC)
:::@[[User:RockTransport|RockTransport]], I Think what you are saying is that ''Main Page/News'' does not update frequently enough? 
:::If this is the reason, why not start small by simply increasing the frequency of posting news on the main page, instead of trying to start a newsletter?
:::If there is more, can you articulate what else is missing. Thanks in advance, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:51, 12 December 2024 (UTC)
::::I meant going to detail into topics covered in that month, rather than just giving a few points. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 16:53, 12 December 2024 (UTC)
:::::What sort of details did you have in mind? You can pick one of the links provided in [[Main Page/News]] to illustrate. cheers, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 15:29, 16 December 2024 (UTC)</text>
      <sha1>isv49nakfh8xvbhgosu5xp0rvaoy2k1</sha1>
    </revision>
    <revision>
      <id>2692204</id>
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        <id>2992610</id>
      </contributor>
      <comment>/* Wikiversity - Newsletters */ Reply</comment>
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      <text bytes="77681" sha1="mjuisqvu2yvsdtmi6jwpuwm856tq212" xml:space="preserve">{{Wikiversity:Colloquium/Header}}
&lt;!-- MESSAGES GO BELOW --&gt;
== Reminder! Vote closing soon to fill vacancies of the first U4C ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
:''[[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement – reminder to vote|You can find this message translated into additional languages on Meta-wiki.]] [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement – reminder to vote}}&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]''

Dear all,

The voting period for the Universal Code of Conduct Coordinating Committee (U4C) is closing soon. It is open through 10 August 2024. Read the information on [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2024_Special_Election#Voting|the voting page on Meta-wiki to learn more about voting and voter eligibility]]. If you are eligible to vote and have not voted in this special election, it is important that you vote now.

'''Why should you vote?''' The U4C is a global group dedicated to providing an equitable and consistent implementation of the UCoC. Community input into the committee membership is critical to the success of the UCoC.

Please share this message with members of your community so they can participate as well.

In cooperation with the U4C,&lt;section end=&quot;announcement-content&quot; /&gt;

-- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 15:30, 6 August 2024 (UTC)
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== User group for Wikiversians ==

Was there ever a discussion about the possibility of establishing a user group in the sense of an affiliated organization that would defend the interests of professors and scientists on Wikiversity and possibly actively develop some projects? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:21, 8 August 2024 (UTC)

:Not that I'm aware of. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:20, 8 August 2024 (UTC)
:It's a pleasure to talk to a scientist on Wikiversity. I am a historian of technics and I would like to publish the following biography either on Wikiversity or on Wikipedia: 
:https://en.wikiversity.org/wiki/User:Rbmn/Arthur_Constantin_KREBS_(1850-1935):_Military_engineer,_Automotive_industrialist,_Great_projects_manager
:What would be your advice? [[User:Rbmn|Rbmn]] ([[User talk:Rbmn|discuss]] • [[Special:Contributions/Rbmn|contribs]]) 15:44, 6 October 2024 (UTC)
::The content appears to be largely biographical/encyclopedic, so I think it is likely best suited to Wikipedia. Consider improving/incorporating this content into the existing page: [[w:Arthur Constantin Krebs]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:05, 7 October 2024 (UTC)
::Please do not link to the Wikiversity [[wv:userspace|Userspace]] in Wikipedia articles. You will want to wait until you have a page in the [[wv:mainspace|Wikiversity mainspace]]. You'll also want to use the &lt;code&gt;{{[[:w:Template:Wikiversity|Wikiversity]]}}&lt;/code&gt; template (on Wikipedia) rather than embedding a photo with a link. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:21, 7 October 2024 (UTC)
:I haven't heard anything about this topic. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 21:06, 8 December 2024 (UTC)

== Rich's ''Illustrated Companion'' at Wikiversity: Right place? ==

Hello! I am creating a Wiki-version of a classical glossary (''Illustrated Companion to the Latin Dictionary, and Greek Lexicon'' by Anthony Rich, 1849), which explains the meaning of Latin headwords, primarily those &quot;representing visible objects connected with the arts, manufactures, and every-day life of the Greeks and Romans.&quot; The aim is to help understand what a (classical) Latin text is actually about, instead of merely translating it. I already transcribed the entire text and scanned the images (about 1900) from an original 1849-edition. I am currently working on uploading the images to ''Mediawiki Commons'', which probably will take some time. In the meantime I want to prepare the other aspects of the project (more than 3000 articles, already with many internal links). The important thing: this is ''not'' a ''might exist''-project.  {{Color|red|My question: Is ''Wikiversity'' the proper place for it?}} Although I created an exact rendition of the original text, ''Wikisource'' is not applicable, because the project has a broader scope (adding content to the articles, e. g. links to online editions for quotations, adding images, but also adding entirely new articles). Neither is ''Wikibooks'', because this is not a textbook and may otherwise breach its scope. For more about the project see [[w:User:CalRis25/Temp-RICH-Prospectus|my user-page]] at en.wikipedia. {{Color|Red|So, is Wikiversity the right place for it?}} [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 09:15, 17 August 2024 (UTC)

:Thanks for asking. To be clear, it ''is'' acceptable to make [[:s:en:Category:Wikisource annotations|annotated editions]] of texts at Wikisource and Wikibooks does host at least one [[:b:en:Annotations of The Complete Peanuts|annotated guide to a copyright-protected work]]. So if what you're looking to do is to include inline annotations to a public domain text, you certainly can put that on Wikisource. If you have a textbook or guidebook that is a companion, that would go at Wikibooks. If you have some other kind of learning resources (like maintaining a list of relevant links, organizing a book reading group, etc.), that could go here. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:26, 17 August 2024 (UTC)
::Thank you for your quick answer. Actually, ''Wikibooks'' was my first thought. However, this project is not merely an annotated edition. Although at first it ''will'' be a faithful copy of the original text, I want the project to be &quot;open&quot;, i. e. adding articles should be possible. And the project should enable to do a lot more than mere inline annotation. See section [[w:User:CalRis25/Temp-RICH-Prospectus#Improving_RICH|Improving Rich]] in the project description a my user-page (en.Wikipedia). No ''Mediawiki''-project (Wikisource, Wikibooks, Wikipedia, Wiktionary) seemed to be a sufficiently applicable &quot;fit&quot; for the project, so I thought of Wikiversity as a last resort, because it is supposed to be home to all sorts of &quot;learning resources&quot;. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 09:57, 17 August 2024 (UTC)
:::The scope of Wikiversity ''is'' pretty catch-all and would allow for a pretty flexible place to host most learning resources that don't fit elsewhere.
:::Also, as nitpick, &quot;MediaWiki&quot; is the software that is the basis of these wikis (wikis being collections of interlinked documents that can be edited) and &quot;Wikimedia Foundation&quot; is the non-profit who owns the trademarks and hosts these projects like Wiktionary and Wikivoyage. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 10:06, 17 August 2024 (UTC)
::::Hello Justin, thank you for the reply. '''I think that settles it. I will create this project at ''Wikiversity''.''' Just for additional clarification, why I do so. Let's imagine a full transcription of the original 1849-edition of the ''Illustrated Companion'' by Anthony Rich and call it ''RICH-1849''. We shall call my project, for brevity sake, RICH-2K. And now, let's have a look at the article about the Roman toga (a piece of attire). In ''RICH-1849'' we can can call it ''RICH-1849/Toga'', and it contains ''exactly'' the content of the 1849-book. Now, let's look at the article ''RICH-2K/Toga''. At the beginning its only content would be the article ''RICH-1849/Toga''. Does that make ''RICH-2K/Toga'' and ''RICH-1849/Toga'' the same? Not at all, because in truth ''RICH-2K/Toga'' is a &quot;container&quot; which initially contains only the article ''RICH-1849/Toga'' but later on may include more stuff: images, external links, article text which builds on or extends ''RICH-1849/Toga'' and information from other sources of information (Wikipedia, specialized books). By the way, this added article information would not be a mere copy of the text at en.Wikipedia, because the information needs to looked at through the eyes of someone reading the original text (more citations with direct links to these etc.). [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 11:39, 17 August 2024 (UTC)

== Coming soon: A new sub-referencing feature – try it! ==

&lt;section begin=&quot;Sub-referencing&quot;/&gt;
[[File:Sub-referencing reuse visual.png|{{#ifeq:{{#dir}}|ltr|right|left}}|400px]]
Hello. For many years, community members have requested an easy way to re-use references with different details. Now, a MediaWiki solution is coming: The new sub-referencing feature will work for wikitext and Visual Editor and will enhance the existing reference system. You can continue to use different ways of referencing, but you will probably encounter sub-references in articles written by other users. More information on [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing|the project page]].

'''We want your feedback''' to make sure this feature works well for you:

* [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing#Test|Please try]] the current state of development on beta wiki and [[m:Talk:WMDE Technical Wishes/Sub-referencing|let us know what you think]]. 
* [[m:WMDE Technical Wishes/Sub-referencing/Sign-up|Sign up here]] to get updates and/or invites to participate in user research activities.

[[m:Special:MyLanguage/Wikimedia Deutschland|Wikimedia Deutschland]]’s [[m:Special:MyLanguage/WMDE Technical Wishes|Technical Wishes]] team is planning to bring this feature to Wikimedia wikis later this year. We will reach out to creators/maintainers of tools and templates related to references beforehand.

Please help us spread the message. --[[m:User:Johannes Richter (WMDE)|Johannes Richter (WMDE)]] ([[m:User talk:Johannes Richter (WMDE)|talk]]) 10:36, 19 August 2024 (UTC)
&lt;section end=&quot;Sub-referencing&quot;/&gt;
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== New [[Template:Form]] ==

Hi! Today I was bold and created [[Template:Form]] (which calls [[Module:WikiForm]] and [[MediaWiki:Gadget-WikiForm.js]]). The template allows to create user-friendly forms that can create pages or add content to existing pages. My motivation and first use case was [[Wikidebate/New|this form]] to create new [[wikidebates]], but I suspect the template can be useful elsewhere on Wikiversity. Let me know if you notice any issues or have any requests or concerns. Kind regards, [[User:Sophivorus|Sophivorus]] ([[User talk:Sophivorus|discuss]] • [[Special:Contributions/Sophivorus|contribs]]) 15:21, 21 August 2024 (UTC)

== Sign up for the language community meeting on August 30th, 15:00 UTC ==

Hi all,

The next language community meeting is scheduled in a few weeks—on August 30th at 15:00 UTC. If you're interested in joining, you can [https://www.mediawiki.org/wiki/Wikimedia_Language_and_Product_Localization/Community_meetings#30_August_2024 sign up on this wiki page].

This participant-driven meeting will focus on sharing language-specific updates related to various projects, discussing technical issues related to language wikis, and working together to find possible solutions. For example, in the last meeting, topics included the Language Converter, the state of language research, updates on the Incubator conversations, and technical challenges around external links not working with special characters on Bengali sites.

Do you have any ideas for topics to share technical updates or discuss challenges? Please add agenda items to the document [https://etherpad.wikimedia.org/p/language-community-meeting-aug-2024 here] and reach out to ssethi(__AT__)wikimedia.org. We look forward to your participation!

[[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 23:20, 22 August 2024 (UTC)
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== Template consolidation: User talk page block notice ==
Wondering if someone who likes templates could have a go at consolidating or helping decide between use of:
* [[Template:Block]]
* [[Template:Blocked]]

Unless I'm missing something, it seems like we don't need both?

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 07:16, 23 August 2024 (UTC)
: I tried to figure out a Wikidata item with most links to projects. I found this: [[Wikidata:Q6379131]], which is Template:Uw-block. There is even a corresponding Wikiversity template, [[Template:Uw-block1]] (not used anywhere).
: My impression is that of the three templates, we only need one. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:43, 13 September 2024 (UTC)

== Announcing the Universal Code of Conduct Coordinating Committee ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
:''[https://lists.wikimedia.org/hyperkitty/list/board-elections@lists.wikimedia.org/thread/OKCCN2CANIH2K7DXJOL2GPVDFWL27R7C/ Original message at wikimedia-l]. [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement - results|You can find this message translated into additional languages on Meta-wiki.]] [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement - results}}&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]''

Hello all,

The scrutineers have finished reviewing the vote and the [[m:Special:MyLanguage/Elections Committee|Elections Committee]] have certified the [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Results|results]] for the [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election|Universal Code of Conduct Coordinating Committee (U4C) special election]].

I am pleased to announce the following individual as regional members of the U4C, who will fulfill a term until 15 June 2026:

* North America (USA and Canada)
** Ajraddatz

The following seats were not filled during this special election:

* Latin America and Caribbean
* Central and East Europe (CEE)
* Sub-Saharan Africa
* South Asia
* The four remaining Community-At-Large seats

Thank you again to everyone who participated in this process and much appreciation to the candidates for your leadership and dedication to the Wikimedia movement and community.

Over the next few weeks, the U4C will begin meeting and planning the 2024-25 year in supporting the implementation and review of the UCoC and Enforcement Guidelines. You can follow their work on [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee|Meta-Wiki]].

On behalf of the U4C and the Elections Committee,&lt;section end=&quot;announcement-content&quot; /&gt;

[[m:User:RamzyM (WMF)|RamzyM (WMF)]] 14:07, 2 September 2024 (UTC)
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== Re: The Vector 2022 skin as the default in two weeks? ==

[[File:Vector 2022 video-en.webm|thumb|A two minute-long video about Vector 2022]]
Hello everyone, I'm reaching out on behalf of the [[mediawikiwiki:Reading/Web|Wikimedia Foundation Web team]] responsible for the MediaWiki skins. I'd like to revisit the topic of making Vector 2022 the default here on English Wikiversity. I [[Wikiversity:Colloquium/archives/September 2022#The Vector 2022 skin as the default in two weeks?|did post a message about this almost two years ago]] (where you can find all the details about the skin), but we didn't finalize it back then.

What happened in the meantime? We built [[mw:Reading/Web/Accessibility for reading|dark mode and different options for font sizes]], and made Vector 2022 the default on most wikis, including all other Wikiversities. With the not-so-new V22 skin being the default, existing and coming features, like dark mode and [[mw:Trust and Safety Product/Temporary Accounts|temporary accounts]] respectively, will become available for logged-out users here.

So, if no large concerns are raised, we will deploy Vector 2022 here in two weeks, in the week of September 16. Do let me know if you have any questions. Thank you! [[User:SGrabarczuk (WMF)|SGrabarczuk (WMF)]] ([[User talk:SGrabarczuk (WMF)|discuss]] • [[Special:Contributions/SGrabarczuk (WMF)|contribs]]) 21:48, 2 September 2024 (UTC)

:Sounds good, Szymon - we look forward to the upcoming change of skin {{smile}} Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 07:35, 13 September 2024 (UTC)
* I for one oppose a switch to Vector 2022. I do not find it preferable. Here is a staggering evidence of user refusal of Vector 2022 once it was deployed: [[W:en:Wikipedia:Requests for comment/Rollback of Vector 2022]], Junuary 2023. 355 voters supported rollback to Vector 2010 whereas 64 opposed, yielding 84.7% support, as clear a supermajority as one may wish. These people opposing Vector 2022 feel the same way as I do. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:48, 13 September 2024 (UTC)
*:Hey @[[User:Dan Polansky|Dan Polansky]]. Thanks for your comment. I'm open to discussion about problems with our software, and I hope we can maintain a respectful tone.
*:I understand that there are users who prefer Vector legacy or other skins, just as there are people who still stick to Monobook. Such people are active across many wikis. They can keep Vector legacy, although non-default skins don't have the support the default ones do. We are rolling out for technical reasons, as I mentioned above, with benefit to not logged-in users.
*:Regarding the rollback RfC on Wikipedia, two neutral users stated that there was no consensus for rollback, RfC is not a vote, and the numbers were different (355:226:24). I believe this all is pretty easy to verify.
*:So to sum up, Vector 2022 needs to become the default, tons and tons of comments were made about the skin and related stuff, and we have taken many ideas into account, and it's totally OK if you stick to Vector legacy.
*:Thanks! [[User:SGrabarczuk (WMF)|SGrabarczuk (WMF)]] ([[User talk:SGrabarczuk (WMF)|discuss]] • [[Special:Contributions/SGrabarczuk (WMF)|contribs]]) 19:30, 16 September 2024 (UTC)
*:: Today, I visited Wikiversity and found it switched to Vector 2022. I changed my preference settings to Vector 2010. From what I understand, non-registered visitors are now defaulted to Vector 2022 despite its unpopularity in [[W:en:Wikipedia:Requests for comment/Rollback of Vector 2022]]. I have not seen any evidence that users prefer Vector 2022, and given the evidence in the linked RfC, I tentatively conclude that the decision to switch has made the site experience worse for the majority of users. The logic of &quot;you can switch&quot; surely applies to Vector 2022 as well: those who prefer it can switch to it. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:08, 17 September 2024 (UTC)

== Have your say: Vote for the 2024 Board of Trustees! ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
Hello all,

The voting period for the [[m:Special:MyLanguage/Wikimedia Foundation elections/2024|2024 Board of Trustees election]] is now open. There are twelve (12) candidates running for four (4) seats on the Board.

Learn more about the candidates by [[m:Special:MyLanguage/Wikimedia Foundation elections/2024/Candidates|reading their statements]] and their [[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Questions_for_candidates|answers to community questions]].

When you are ready, go to the [[Special:SecurePoll/vote/400|SecurePoll]] voting page to vote. '''The vote is open from September 3rd at 00:00 UTC to September 17th at 23:59 UTC'''.

To check your voter eligibility, please visit the [[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Voter_eligibility_guidelines|voter eligibility page]].

Best regards,

The Elections Committee and Board Selection Working Group&lt;section end=&quot;announcement-content&quot; /&gt;

[[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 12:15, 3 September 2024 (UTC)
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== Separate page for hyperbola. ==

Good morning,

I notice that a search for &quot;hyperbola&quot; redirects to &quot;Conic sections&quot;.

At present there is a separate page for &quot;ellipse&quot;. Therefore a separate page for &quot;hyperbola&quot; seems to be justified.

Could this redirection be changed so that search for &quot;hyperbola&quot; goes to a separate page for &quot;hyperbola&quot;?

Many thanks,

[[User:ThaniosAkro|ThaniosAkro]] ([[User talk:ThaniosAkro|discuss]] • [[Special:Contributions/ThaniosAkro|contribs]]) 12:04, 15 September 2024 (UTC)

:It is true that ellipses are covered at [[Conic sections]] (along with hyperbolas, parabolas, etc.) and there is a separate page for [[ellipse]]s that elaborates. We certainly ''could'' have a page about [[hyperbola]]s that is separate, but no one has written sufficient content to spin it off yet. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 12:17, 15 September 2024 (UTC)

== I hereby request for your Unblocking IP address and just reviewed and received a reverted rec ==

Hi there. {{unsigned|Ishmael Raphasha}}
:No one has any clue what you're talking about. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:53, 18 September 2024 (UTC)

== RICH-2K: New project with some initial questions ==

Hello! I'm creating a new learning resource on ''Wikiversity''. The respective project is based on my transcription of a classical dictionary from 1849 by Anthony Rich. For more information about the project see its [[User:CalRis25/RICH: Description|description page]] (see also that page for why not ''Wikisource'' or ''Wikibooks''). The project's scope is fairly big: 3205 article-pages plus 304 REDIRECT-pages. The images (scanned by myself from an original copy) have been uploaded to ''Commons''. I have some initial technical questions (more of these and more detailed ones will follow):

* '''Upload''': Due to the large number of pages it is not realistic to create these manually. Is it possible to bulk-upload these in some way (the Wikitext of the pages is created using a Python-script with one file per article/page)? Is it possible to upload these to a test-environment first where any problems (hopefully none) can be identified and dealt with more easily than on the production-version of ''Wikiversity''?
* '''(Technical) Structure''': I am planning to set up this project at ''&lt;nowiki&gt;https://en.wikiversity.org/wiki/RICH-2K&lt;/nowiki&gt;'' as the main page and anything else as subpages: ''RICH-2K/Subpage_1 ... RICH-2K/Subpage_n''. However, these subpages fall into two categories: 1. Article-pages (content) and 2. Meta/Administrative pages. This project requires search capability restricted to the ''RICH-2K''-namespace. The ''Mediawiki''-software seems to supply a ''Search''-input field with the possibility to restrict the search to some namespace. I would like, however, to restrict the search further to the first group of pages, namely the articles. Is that possible, perhaps by use of (hidden) categories?
* '''External links''': This project will need many external links, and yes, I have read the relevant ''Wikiversity''-pages, but this specific project needs them. The ''Recommended Editions''-page (used for recommended online editions, to which to link when citing texts) alone probably will require several hundred external links. However, only relatively few [[w:Second-level domain|second-level domains]] will be involved, and most of these should be trustworthy (Perseus Digital library, digital collections of universities etc., in some cases, however, also ''Archive.org''). Perhaps there is a list of web-sites, for which external links are generally allowed? And who is allowed to create external links on ''Wikiversity''-pages (I haven't found the relevant policy)?
* '''Categories''': This project requires quite a few of its own categories, which belong to two large groups: 1. Categories (2 levels) of the ''Classed Index'' (about 170 categories), a thematic index of some (but not all) of the articles. 2. Administrative categories. Is there a recommended way to distinguish between different classes of categories within a project (category name or other method)? What about naming conventions for project-specific categories?

I am looking forward to your input. If you think that it's preferable we can move the discussions to the [[User_talk:CalRis25/RICH:_Description|Talk-page]] of the project's description. Thank you in advance. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 05:29, 20 September 2024 (UTC)

:*Admins have access to [[Special:Import]] and can bulk import XML pages. You can create pages in your sandbox if you'd like and make an indefinite amount of them at pages like [[User:CalRis25/sandbox]]. What can and cannot be hosted in user namespace is very loose, but still has to follow in principle Wikiversity's scope.
:*Using subpages is in principle a good way to organize these various resources. Please do not name them after a user name or something obscure. I personally think that &quot;RICH-2K&quot; is a not optimal name. I may recommend something like [[Anthony Rich Dictionary Project]] or [[21st-Century Anthony Rich Dictionary]] or something more obviously intelligible. While we have very few actual policies and guidelines, see [[Wikiversity:Naming conventions]] for a rough consensus of what is probably best practice for naming pages.
:*External linking generally does not use an allowed list (a.k.a. whitelist model), but a disallow (a.k.a. blacklist) model. See [[MediaWiki:Spam-blacklist]] and [[Special:BlockedExternalDomains]] (which is currently empty but is another method of listing blocked domains). It's perfectly fine to aggregate external links in learning resources.
:*I'm not 100% sure what the distinction is that you're drawing, but you can freely arrange categories underneath a main category that has the same name as your larger project. So, following the suggestions I gave, you could have a category like [[:Category:Anthony Rich Dictionary Project]] and then create any number of subcategories that logically help users navigate all these pages. Please make sure the main category you create is itself categorized under some relevant category(ies). If you need help, please ask.
:I think this answers your questions, please let me know if I'm unclear or you have more. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:11, 20 September 2024 (UTC)
::Hello Justin!
::* '''Upload:''' Creating the project in sandbox pages of my User-namespace defeats the purpose, as this is an ''open'' project. Also that would not solve, as such, the problem of having to manually create thousands of pages. I wonder, does ''Wikiversity'' support creation of pages using its API. ''Mediawiki's'' [[mw:API:Main_page|API-description]] seems to imply that it ought to be possible. If that's the case, I should be able to create a Python-script which automatically creates the pages (of course, a few trial pages first).
::* '''(Technical) Structure''': You may be right, here. RICH-2K is, for now, merely a technical name to make a clear but not too verbose distinction between the original text and the current project. I'll give this more thought.
::* '''External links''': I brought this up mainly because when I first edited my ''Wikiversity''-page, I got a message that I was not allowed to create external links. However, I just now tested creating an external link on my user-page and got no error, so this problem seems to be solved.
::* '''Categories''': I think I know what you mean. I'll create a category structure and maybe ask some specific questions once I am ready to do so.
::Thank you for your quick help. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 18:51, 20 September 2024 (UTC)
:::re: upload, I'm just suggesting your sandbox(es) as you asked about &quot;a test-environment&quot;. Anyone can edit someone else's sandboxes, but you typically defer to other users to control what's in their own subpages as a collegial thing. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:39, 23 September 2024 (UTC)

Hello! I have two further questions:
# I created a category-structure for the project. Could you (or someone else) have a look at it ([[User:CalRis25/RICH: Categories]]) and answer the questions in the section [[User:CalRis25/RICH:_Categories#Questions|Questions]]? I gave it some thought and believe that this would work fine for the project.
# ''Project boxes'' (see [[Help:Tour of project boxes]]): It is unclear to me, whether these belong only on the main page of the project (that makes the most sense to me), or on every single subpage.
Thanks in advance for your help. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 17:51, 24 September 2024 (UTC)

:To answer your questions here:
:*No, you are not contravening any policies we have.
:*A leading &quot;The&quot; is acceptable, but if you want it to sort alphabetically, you will have to use &lt;nowiki&gt;{{DEFAULTSORT:}}&lt;/nowiki&gt;. E.g. to get Category:The Best Stuff to sort under &quot;B&quot;, insert &quot;&lt;nowiki&gt;{{DEFAULTSORT:Best Stuff, The}}&lt;/nowiki&gt;.
:*Trailing &quot;etc.&quot; is acceptable.
:*An accent in a category title is acceptable.
:I'll also note that it looks like you have in mind some tracking categories that are redundant. Pages such as [[Special:LonelyPages]] and [[Special:DeadendPages]] already do automatically what you're proposing to do manually.
:As for project boxes, it's typically the case that the subjects are only placed on the main resource, but as you may imagine, [[Help:Tour of project boxes/1|status completion ones]] may vary from subpage to subpage. As with most things at Wikiversity, there are very few actual rules, so it's pretty much the wild west, even tho this project has been around for almost 20&amp;nbsp;years. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:18, 25 September 2024 (UTC)
::Hello Justin, thanks for the DEFAULTSORT-hint for categories beginning with ''The''. I will restrict the project boxes to the main page. As for the the orphaned/dead-end-categories, I prefer these to be project-specific. Once the project is up and running, putting articles &quot;on the map&quot; (making them accessible from other articles and creating links to other articles) is one of the first tasks to be dealt with. I already know which articles are involved and will add these categories to these articles. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 16:51, 25 September 2024 (UTC)

== Your wiki will be in read-only soon ==

&lt;section begin=&quot;server-switch&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;

[[:m:Special:MyLanguage/Tech/Server switch|Read this message in another language]] • [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-Tech%2FServer+switch&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]

The [[foundation:|Wikimedia Foundation]] will switch the traffic between its data centers. This will make sure that Wikipedia and the other Wikimedia wikis can stay online even after a disaster.

All traffic will switch on '''{{#time:j xg|2024-09-25|en}}'''. The switch will start at '''[https://zonestamp.toolforge.org/{{#time:U|2024-09-25T15:00|en}} {{#time:H:i e|2024-09-25T15:00}}]'''.

Unfortunately, because of some limitations in [[mw:Special:MyLanguage/Manual:What is MediaWiki?|MediaWiki]], all editing must stop while the switch is made. We apologize for this disruption, and we are working to minimize it in the future. 

A banner will be displayed on all wikis 30 minutes before this operation happens. This banner will remain visible until the end of the operation. 

'''You will be able to read, but not edit, all wikis for a short period of time.'''

*You will not be able to edit for up to an hour on {{#time:l j xg Y|2024-09-25|en}}.
*If you try to edit or save during these times, you will see an error message. We hope that no edits will be lost during these minutes, but we can't guarantee it. If you see the error message, then please wait until everything is back to normal. Then you should be able to save your edit. But, we recommend that you make a copy of your changes first, just in case.

''Other effects'':

*Background jobs will be slower and some may be dropped. Red links might not be updated as quickly as normal. If you create an article that is already linked somewhere else, the link will stay red longer than usual. Some long-running scripts will have to be stopped.
* We expect the code deployments to happen as any other week. However, some case-by-case code freezes could punctually happen if the operation require them afterwards.
* [[mw:Special:MyLanguage/GitLab|GitLab]] will be unavailable for about 90 minutes.

This project may be postponed if necessary. You can [[wikitech:Switch_Datacenter|read the schedule at wikitech.wikimedia.org]]. Any changes will be announced in the schedule. 

'''Please share this information with your community.'''&lt;/div&gt;&lt;section end=&quot;server-switch&quot;/&gt;

[[User:Trizek_(WMF)|Trizek_(WMF)]], 09:37, 20 September 2024 (UTC)
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== 'Wikidata item' link is moving. Find out where... ==

&lt;div lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot;&gt;&lt;i&gt;Apologies for cross-posting in English. Please consider translating this message.&lt;/i&gt;{{tracked|T66315}}

Hello everyone, a small change will soon be coming to the user-interface of your Wikimedia project.
The [[d:Q16222597|Wikidata item]] [[w:|sitelink]] currently found under the &lt;span style=&quot;color: #54595d;&quot;&gt;&lt;u&gt;''General''&lt;/u&gt;&lt;/span&gt; section of the '''Tools''' sidebar menu will move into the &lt;span style=&quot;color: #54595d;&quot;&gt;&lt;u&gt;''In Other Projects''&lt;/u&gt;&lt;/span&gt; section.

We would like the Wiki communities feedback so please let us know or ask questions on the   [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Discussion page]] before we enable the change which can take place October 4 2024, circa 15:00 UTC+2.
More information can be found on [[m:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|the project page]].&lt;br&gt;&lt;br&gt;We welcome your feedback and questions.&lt;br&gt; [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 18:56, 27 September 2024 (UTC)
&lt;/div&gt;
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==Download as PDF==
[[Phabricator:T376438]]: &quot;Download to PDF&quot; on en.wv is returning error: &quot;{&quot;name&quot;:&quot;HTTPError&quot;,&quot;message&quot;:&quot;500&quot;,&quot;status&quot;:500,&quot;detail&quot;:&quot;Internal Server Error&quot;}&quot;

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:36, 3 October 2024 (UTC)

:I just downloaded this page as a PDF and it worked just fine. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:04, 3 October 2024 (UTC)

== Protected template bug for Pp ==

It seems that templates derivative of {{tlx|Pp}} (compiled in {{tlx|Protection templates}}) are being sorted into protection categories using the name 'Wikipedia' instead of 'Wikiversity' (e.g., [[:Category:Wikipedia pages with incorrect protection templates]]). From what I can tell, it is not in the publicly accessible source code of any of the templates. The only other impacted pages are modules which call {{tlx|pp}}-derivatives (e.g., [[Module:Navbar/styles.css]]).

This does not seem to affect any other pages in [[:Category:Wikiversity protected templates]]. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 18:59, 4 October 2024 (UTC)

:The problem is that &quot;Wikipedia&quot; is [https://en.wikiversity.org/w/index.php?title=Special%3ASearch&amp;limit=500&amp;offset=0&amp;ns828=1&amp;search=Wikipedia&amp;searchToken=9svkpqlxxoquoq7bnkt55ugts mentioned in several modules that were copied over from en.wp]; many of these are legit and many of them need to be replaced with &quot;Wikiversity&quot; ([https://en.wikiversity.org/w/index.php?title=Module%3APp-move-indef&amp;diff=2662815&amp;oldid=1944984 e.g.]) This particular change ''may'' fix all of these issues...? But 1.) it will take time to propagate across the site and 2.) there are still many more &quot;Wikipedia&quot;s that need to be changed, so I'll go thru a few more, but if you want to give me an assist, if you can just check this one week from now and ping me if the problem persists, that would be nice. Sometimes, I make calendar reminders to follow up on these, but I'm not a perfect person. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:55, 5 October 2024 (UTC)

== Invitation to Participate in Wiki Loves Ramadan Community Engagement Survey ==

Dear all,


We are excited to announce the upcoming [[m:Wiki Loves Ramadan|Wiki Loves Ramadan]] event, a global initiative aimed at celebrating Ramadan by enriching Wikipedia and its sister projects with content related to this significant time of year. As we plan to organize this event globally, your insights and experiences are crucial in shaping the best possible participation experience for the community.

To ensure that Wiki Loves Ramadan is engaging, inclusive, and impactful, we kindly invite you to participate in our community engagement survey. Your feedback will help us understand the needs of the community, set the event's focus, and guide our strategies for organizing this global event.

Survey link: https://forms.gle/f66MuzjcPpwzVymu5

Please take a few minutes to share your thoughts. Your input will make a difference!

Thank you for being a part of our journey to make Wiki Loves Ramadan a success.


Warm regards,

User:ZI Jony 03:19, 6 October 2024 (UTC)

Wiki Loves Ramadan Organizing Team
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== 'Edit to my talk page' notification bug? ==

This may belong at the bug tracker, but does anyone else have an issue disabling ''email'' notifications upon  an 'Edit to my talk page' in [[Special:GlobalPreferences]]? Oddly I ''am'' able to disable the global preference on Wikipedia, MediaWiki, etc, but not here. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 09:23, 7 October 2024 (UTC)

:I have not experienced this, but to be clear, do you also have the option to get emails when items on your talk page are edited turned on? —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:39, 7 October 2024 (UTC)
::The only (non-grayed out) options I have enabled for email are 'Failed login attempts' and 'Login from an unfamiliar device'. 'Edit to my talk page' re-checks after every save. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 09:54, 7 October 2024 (UTC)
:::That does sound like a [[phab:]] issue, with the caveat that I don't 100% recall how global preferences work and if they override local ones, etc. If you have parsed that and still have this issue, you'll probably need to file a ticket. Maybe someone else has this issue. Wish I could help. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:57, 7 October 2024 (UTC)
::::[[phab:T376601|Off 'n away]] 🫡 [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 10:35, 7 October 2024 (UTC)

== [[Portal:Computer Science]] ➝ [[Portal:Information sciences]] ==

Seeking consensus to complete the merge into the broader portal. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 06:28, 8 October 2024 (UTC)

:Why should it be merged? Computer Science seems well-enough designed. What is the incentive to collapse it into a broader field of study? —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:18, 8 October 2024 (UTC)
::Portals as top level organizations allowing for content to be best centralized. Also note that I did not start the merge, just offering to finish it. Perhaps a {{tlx|prod}} instead? [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 07:20, 8 October 2024 (UTC)
:::I have no objections, personally. If it gets done, please use a redirect and should someone want to come along to resurrect it later, it will be easier. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:21, 8 October 2024 (UTC)
: Is computer science really a branch of information sciences? I would not think so, but what do I know. Do we have some external resources/links confirm this idea? [[W:Information science]] currently says: &quot;Information science, documentology[1] or informatology[2][3] is an academic field which is primarily concerned with analysis, collection, classification, manipulation, storage, retrieval, movement, dissemination, and protection of information.&quot; --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:49, 11 October 2024 (UTC)
::Looking through [https://stackoverflow.com/q/1047014/22673230] [https://businessdegrees.uab.edu/mis-degree-bachelors/resources/computer-information-systems-vs-computer-science/] [https://www.si.umich.edu/student-experience/what-information-science] a few top (not necessarily RS) searches I'm inclined to agree. I am more familiar with the grafted [[:w:Information and computer science|information ''and'' computer science]] which makes an effort to merge the disciplines, but it does not seem like reaching to say that IS is presented as more applications-concerned (certainly with no lack of theoretical abstraction), whereas CS can be more freely associated with any and all 'science related to computers'. It is easy to reason about the connection between the fields, but I think it is clear academia maintains this taxonomy for a good reason.
::With these considerations, I think I will ''stop'' the process of merging in favor of expanding the existing [[School:Library and Information Science]]. 

::Let me know if there is not consensus to redirect [[Portal:Information sciences]] to [[School:Library and Information Science]] (with enough expansion it can generalize away from just library sciences). [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 16:16, 11 October 2024 (UTC)
::: I do not see that a merge of a ''portal'' to a ''school'' is a good thing. Do you have a clear idea of the concepts of school and portal and how they relate to each other? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:34, 11 October 2024 (UTC)
::::Found [[:Category:Information sciences]]; there are enough existing resources in there to make my other proposed merge excessive. I will simply continue developing the existing [[Portal:Information sciences]] instead. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 17:05, 11 October 2024 (UTC)
::::: Frankly, I would ideally see [[Portal:Information sciences]] deleted: I don't see what it does that a category would not do well enough. There does not seem to be any material specific to &quot;Information sciences&quot; (whatever that is) in that portal at all. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:11, 11 October 2024 (UTC)
::::::Tacked a {{tlx|prod}} for an eventual deletion, but I may still try to develop it as proof of concept at some point. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 17:33, 11 October 2024 (UTC)

== [[:Category:Occupational Epidemiology]] ==

I propose moving the pages in this category (without leaving redirects) to their equivalent under the parent resource [[Occupational Health Risk Surveillance]]. Also due to the number of subpages, it seems &lt;code&gt;|filing=deep&lt;/code&gt; would be a justified. (Also [[Special:PrefixIndex/Occupational_Epidemiology|there are quite a few]] untagged subpages.) [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 05:11, 9 October 2024 (UTC)
: I above all think that the content should be ''moved out of the mainspace'': I do not see readers learning anything from e.g. [[Occupational Epidemiology/Research tools/Reading of scientific articles for learning epidemiology and biostatstics]] or [[Occupational Epidemiology/Research tools/Ongoing projects/Risk Communication in Seafaring/Writing the article guideline IMRAD]]. Wikiversity can be kind enough to host that material in, say, subspace of [[User:Saltrabook]], but more should not be asked, I think. Let us recall that per [[WV:Deletions]], &quot;Resources may be eligible for proposed deletion when education objectives and learning outcomes are scarce, and objections to deletion are unlikely&quot;; I do not see how learning outcomes can be anything but scarce. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 15:04, 11 October 2024 (UTC)
::thank you, agree @ [[User:Saltrabook|Saltrabook]] ([[User talk:Saltrabook|discuss]] • [[Special:Contributions/Saltrabook|contribs]]) 21:03, 13 November 2024 (UTC)

== Active editors ==

It is interesting to observe the stats on [https://stats.wikimedia.org/#/en.wikiversity.org/contributing/active-editors/normal|line|all|(page_type)~content*non-content|monthly active editors] through the project's history. October is our month! [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 20:44, 8 October 2024 (UTC)

:Odd. Maybe related to the school year? —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 02:10, 9 October 2024 (UTC)
::I wonder how many are [[User:Jtneill|Jtneill]]'s crowd... the number is in the hundreds though, so that is one chunky cohort —[[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:16, 9 October 2024 (UTC)
:::Yes, [[Motivation and emotion/Book]] involves ~100-150 students editing most intensely during October each year. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:20, 9 October 2024 (UTC)
::::Neat, that still leaves around ~50-100 other students from other avenues each year since 2021. I also wonder which projects were involved in the COVID enrollment spike. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:26, 9 October 2024 (UTC)
:::::Personally I can admit that my editing is much more active during the school season vs. the summer break, so I'm in the same boat as Jtneill's students. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:24, 13 November 2024 (UTC)
:@[[User:Tule-hog|Tule-hog]] This is an interesting topic, but it is not clear to me as an outsider what you and other participants in this discussion find interesting. I  find this graph not very meaningful because it does not tell me if the number of  Active editors has gone up or down during the period covered, which I think was 2000-now. 
:I can see a big jump between 2000 and 2007, but what happened since then? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 15:45, 16 December 2024 (UTC)

== Intentionally incorrect resource ==

There is a [[Special:Diff/2583464|disclaimer inserted onto a resource]] (by not the original author) that:
&lt;blockquote&gt;I am merely [making this page false] to show you (The viewer) that Wikipedia and this page 'Wikiversity' is bull sh*t and it will not give you the reliability you need when writing an academic piece of writing.&lt;/blockquote&gt;
However, that IP has [[Special:Contributions/86.22.73.151|not made any other edits]], so unless they vandalized via a sock, the intent went un-realized and only that portion need be removed. Bumping here in case there is some obvious jumbo in that essay that someone else can catch. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 16:58, 9 October 2024 (UTC)

:Removed that portion, which was obviously vandalism. No perspective on the rest of the essay. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:38, 9 October 2024 (UTC)

== [[:Category:Proposed guidelines]] ==

Noting for future editors that WV has collapsed all proposals into [[:Category:Proposed policies|proposed policies]]. Seeking consensus to further collapse [[:Category:Wikiversity proposals]] into the former, or to restore [[:Category:Proposed guidelines]]. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 19:19, 9 October 2024 (UTC)

== [[Around Wikiversity in 80 Seconds|Broken 80-second tour]] ==

Bumping a [[Talk:Around_Wikiversity_in_80_Seconds|comment]] on the ''Wikiversity in 80 seconds'' tour. Appears wikisuite is not working with the Vector 2022 appearance. 

Also see [[:w:Wikipedia:Miscellany_for_deletion/Wikiversuite_pages|this thread]] on the Wikiversal package - may not be relevant to Wikiversity, but FYC. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 00:26, 10 October 2024 (UTC)
: I would just delete the material; I do not see value in it. If others agree, I would try to articulate why I think it should be deleted (or move to author user space). --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:57, 13 October 2024 (UTC)
::Just mark as {{tl|historical}}. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:39, 13 October 2024 (UTC)
::: This thing was created by [[User:Planotse]]. His creations are now being discussed in Wikibooks for deletion: [[B:Wikibooks:Requests for deletion#Wikiversal generated pages]]. It seems he used some kind of tool that is no longer available (the above mentioned &quot;Wikiversal&quot; package) to create this kind of slideshow-like material (believing the Wikibooks discussion). I do not see value of this in the mainspace, not even as historical (I am okay with userspace, but maybe even that is not the best option?). A look at the source code of [[Around Wikiversity in 80 Seconds/Introduction]] confirms the words of Omphalographer, namely that &quot;the HTML-heavy markup generated by Wikiversal makes them [the pages] unreasonably difficult to edit.&quot;
::: I went ahead and marked the page for proposed deletion. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:35, 14 October 2024 (UTC)

== Preliminary results of the 2024 Wikimedia Foundation Board of Trustees elections ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
Hello all,

Thank you to everyone who participated in the [[m:Special:MyLanguage/Wikimedia Foundation elections/2024|2024 Wikimedia Foundation Board of Trustees election]]. Close to 6000 community members from more than 180 wiki projects have voted.

The following four candidates were the most voted:

# [[User:Kritzolina|Christel Steigenberger]]
# [[User:Nadzik|Maciej Artur Nadzikiewicz]]
# [[User:Victoria|Victoria Doronina]]
# [[User:Laurentius|Lorenzo Losa]]

While these candidates have been ranked through the vote, they still need to be appointed to the Board of Trustees. They need to pass a successful background check and meet the qualifications outlined in the Bylaws. New trustees will be appointed at the next Board meeting in December 2024.

[[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Results|Learn more about the results on Meta-Wiki.]]

Best regards,

The Elections Committee and Board Selection Working Group

&lt;section end=&quot;announcement-content&quot; /&gt;

[[User:MPossoupe_(WMF)|MPossoupe_(WMF)]] 08:26, 14 October 2024 (UTC)
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== Seeking volunteers to join several of the movement’s committees ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
Each year, typically from October through December, several of the movement’s committees seek new volunteers. 

Read more about the committees on their Meta-wiki pages:
* [[m:Special:MyLanguage/Affiliations_Committee|Affiliations Committee (AffCom)]]
* [[m:Special:MyLanguage/Ombuds_commission|Ombuds commission (OC)]]
* [[m:Special:MyLanguage/Wikimedia Foundation/Legal/Community Resilience and Sustainability/Trust and Safety/Case Review Committee|Case Review Committee (CRC)]]

Applications for the committees open on 16 October 2024. Applications for the Affiliations Committee close on 18 November 2024, and applications for the Ombuds commission and the Case Review Committee close on 2 December 2024. Learn how to apply by [[m:Special:MyLanguage/Wikimedia_Foundation/Legal/Committee_appointments|visiting the appointment page on Meta-wiki]]. Post to the talk page or email [mailto:cst@wikimedia.org cst@wikimedia.org] with any questions you may have.

For the Committee Support team,
&lt;section end=&quot;announcement-content&quot; /&gt;

-- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 23:09, 16 October 2024 (UTC)
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== Interactive elements ==
Can we use interactive elements on Wikiversity? I'd like to add JavaScript to a page. If it's not possible now, where can I suggest this feature? I have a safe integration idea.
[[User:Отец Никифор|Отец Никифор]] ([[User talk:Отец Никифор|discuss]] • [[Special:Contributions/Отец Никифор|contribs]]) 12:10, 17 October 2024 (UTC)
: This is beyond my technical knowledge, but have you checked out:
:* https://www.mediawiki.org/wiki/Manual:Interface/JavaScript?
:* [[Wikipedia:WikiProject JavaScript]]
:* [[MediaWiki:Common.js]]

:What sort of interactive elements are you thinking about?

: Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:39, 18 October 2024 (UTC)
::I was thinking about adding something like a graph with adjustable controls, where users can interact with it and see how different changes affect the outcome. It seems like this could be a useful feature. There might already be discussions about enhancing Wikiversity or similar platforms—perhaps on a relevant talk page or in a Discord group. Do you know where such discussions might be happening? [[User:Отец Никифор|Отец Никифор]] ([[User talk:Отец Никифор|discuss]] • [[Special:Contributions/Отец Никифор|contribs]]) 19:47, 18 October 2024 (UTC)
:::From a quick look, maybe check out:
:::* [[mw:Extension:Graph]]
:::* [[phab:tag/graphs]]
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:40, 18 October 2024 (UTC)
:::: mw:Extension:Graph is currently disabled on Wikipedia etc. wikis, for security reasons, and seems unlikely to be enabled again. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:30, 19 October 2024 (UTC)

== An unexplained spurt of Wikiversity page views ==
The [https://pageviews.wmcloud.org/siteviews/?platform=all-access&amp;source=pageviews&amp;agent=user&amp;start=2024-06-01&amp;end=2024-10-18&amp;sites=en.wikiversity.org|en.wikibooks.org|en.wikiquote.org|en.wikisource.org page view report] shows an unexplained spurt of Wikiversity page views, reaching over 4 times the baseline and then falling back again. Does anyone have any idea what is going on? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:01, 19 October 2024 (UTC)

:Interesting. I wonder why only the English wikiquote and wikiversity and not Wikisource or wikibooks? How reliable do you think those stats are? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 15:44, 8 December 2024 (UTC)

== Center tempate failed on a contributors phone... ==

See the edit comment here - https://en.wikiversity.org/w/index.php?title=Wikiphilosophers&amp;diff=prev&amp;oldid=2673962. I'm puzzled as this is the first failure of this, I've noted recently. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 08:45, 19 October 2024 (UTC)

== Essay-like page in user space that makes little sense and seems incoherent ==
The page [[User:TheoYalur/Illusions]] seems to match the description, at least by my assessment. My understanding is that since the page is only in user space and not in the mainspace, it can stay there even if it has those disqualifying qualities. But if I am wrong and the page belongs deleted, please correct me and let me know. I do not know which policy or guideline, if any, guides the case. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:30, 21 October 2024 (UTC)

== 'Wikidata item' link is moving, finally. ==

Hello everyone, I previously wrote on the 27th September to advise that the ''Wikidata item'' sitelink will change places in the sidebar menu, moving from the '''General''' section into the '''In Other Projects''' section. The scheduled rollout date of 04.10.2024 was delayed due to a necessary request for Mobile/MinervaNeue skin. I am happy to inform that the global rollout can now proceed and will occur later today, 22.10.2024 at 15:00 UTC-2. [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Please let us know]] if you notice any problems or bugs after this change. There should be no need for null-edits or purging cache for the changes to occur. Kind regards, -[[m:User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] 11:28, 22 October 2024 (UTC)
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:Hi @[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]]: I Just noticed your post above, and it is timely.
:I have been participating in the English WikiUniversity for a few years, much less often recently. I seems like something in the way the site displays is different, but I cannot put my finger on it. Your posting gave me a clue. Can you please tell me where  the link to wikidata items has moved to? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:23, 11 December 2024 (UTC)
::Hello @[[User:Ottawahitech|Ottawahitech]], sure, I would be happy to. The button/sitelink name didn't change, just its position. You should find it in the sidebar-menu under the section '''In other projects''' (where the links to all other Wikimedia Projects are displayed). If you do not see it, please reach out to us on the [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Move Wikidata item - Discussion page]]. Thank you, -[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] ([[User talk:Danny Benjafield (WMDE)|discuss]] • [[Special:Contributions/Danny Benjafield (WMDE)|contribs]]) 09:24, 12 December 2024 (UTC)
:::@[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]], thank you for responding. I intend to followup on the ''Move Wikidata item - Discussion page'' as per your post above by putting it on my ever growing todo list.
:::I don't know about others on this wiki, as I said I have not been visiting here frequently, but for me the constant changes are a big distraction. I have been around wikimedia projects since 2007, so why do I have to spend so much time learning and re-learning how to find what I came here for? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:41, 12 December 2024 (UTC)
::::Hi @[[User:Ottawahitech|Ottawahitech]], thanks for you thoughts. Your input whether positive or critical helps us understand the impacts to editors so we welcome your further thoughts when you reach us in your To Do List :)
::::I can't speak about the other changes you've experienced here but I do hope they are made with a spirit of improvement for the community as a whole. -[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] ([[User talk:Danny Benjafield (WMDE)|discuss]] • [[Special:Contributions/Danny Benjafield (WMDE)|contribs]]) 10:43, 16 December 2024 (UTC)

== Final Reminder: Join us in Making Wiki Loves Ramadan Success ==

Dear all,

We’re thrilled to announce the Wiki Loves Ramadan event, a global initiative to celebrate Ramadan by enhancing Wikipedia and its sister projects with valuable content related to this special time of year. As we organize this event globally, we need your valuable input to make it a memorable experience for the community.

Last Call to Participate in Our Survey: To ensure that Wiki Loves Ramadan is inclusive and impactful, we kindly request you to complete our community engagement survey. Your feedback will shape the event’s focus and guide our organizing strategies to better meet community needs.

* Survey Link: [https://docs.google.com/forms/d/e/1FAIpQLSffN4prPtR5DRSq9nH-t1z8hG3jZFBbySrv32YoxV8KbTwxig/viewform?usp=sf_link Complete the Survey]
* Deadline: November 10, 2024
Please take a few minutes to share your thoughts. Your input will truly make a difference!

'''Volunteer Opportunity''': Join the Wiki Loves Ramadan Team! We’re seeking dedicated volunteers for key team roles essential to the success of this initiative. If you’re interested in volunteer roles, we invite you to apply.
* Application Link: [https://docs.google.com/forms/d/e/1FAIpQLSfXiox_eEDH4yJ0gxVBgtL7jPe41TINAWYtpNp1JHSk8zhdgw/viewform?usp=sf_link Apply Here]
* Application Deadline: October 31, 2024
Explore Open Positions: For a detailed list of roles and their responsibilities, please refer to the position descriptions here: [https://docs.google.com/document/d/1oy0_tilC6kow5GGf6cEuFvdFpekcubCqJlaxkxh-jT4/ Position Descriptions]

Thank you for being part of this journey. We look forward to working together to make Wiki Loves Ramadan a success!


Warm regards,&lt;br&gt;
The Wiki Loves Ramadan Organizing Team 05:11, 29 October 2024 (UTC)
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== Android app for Wikiversity ==

Hi, is there an Android app for Wikiversity? How does it work? I have been advised that there is no infrastructure for push notifications for Android apps for sister wikis and I would be interested to know more. Related: [[:phab:T378545]]. Thanks! [[User:Gryllida|Gryllida]] 23:15, 29 October 2024 (UTC)

:Thanks for suggesting this - I agree that it would be useful. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:56, 31 October 2024 (UTC)
:@[[User:Gryllida|Gryllida]]: Would you explain your terminology  for those of us not in the know. What does ''push notifications'' mean? I use [https://www.mediawiki.org/wiki/Help:Notifications notifications] when I am communicating on wikimedia projects, but have never heard this term before. [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:13, 11 December 2024 (UTC)

à

== Import Resource From Wikibooks? ==

Hello! [[wikibooks:Character_List_for_Baxter&amp;Sagart|Character List for Baxter&amp;Sagart]] and related titles [[wikibooks:Wikibooks:Requests_for_deletion#Character_List_for_Baxter&amp;Sagart|are up for deletion at Wikibooks]] because WB policy does not allow dictionaries like them. However, because they are useful as learning tools, I am wondering if they might have a home here at Wikiversity. Pinging @[[User:Tibetologist|Tibetologist]] here to link them in to this discussion, since they are the affected user. Thank you! —[[User:Kittycataclysm|Kittycataclysm]] ([[User talk:Kittycataclysm|discuss]] • [[Special:Contributions/Kittycataclysm|contribs]]) 18:18, 1 November 2024 (UTC)

:Sure, I can do it. That said, as mentioned there, it does seem like something like this is ideally suited for Wiktionary in the Appendix namespace, but I'm not very familiar with CJK characters and languages. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:23, 3 November 2024 (UTC)
::Oh man, these pages are too big to import and while I've already tried a half-dozen times, it will constantly fail. Strictly speaking, we don't have to use the import feature for licensing purposes. We can just copy and paste the contents and list the usernames or on the talk page. I think that's the solution. {{Ping|Tibetologist}}, are you interested in doing that? If you just copied and pasted these pages and then added [[:Category:Chinese]] and maybe include a couple of links to the pages, that would probably be ideal. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:31, 3 November 2024 (UTC)

== Language translation requests? ==

Is there anywhere on Wikiversity to request translation, for example, requesting Latin or French translation? I would be asking from the context as a student, so I would be interested in translation explanation as well. [[User:Indexcard88|Indexcard88]] ([[User talk:Indexcard88|discuss]] • [[Special:Contributions/Indexcard88|contribs]]) 04:56, 20 November 2024 (UTC)

== Sign up for the language community meeting on November 29th, 16:00 UTC ==

Hello everyone,

The next language community meeting is coming up next week, on November 29th, at 16:00 UTC (Zonestamp! For your timezone &lt;https://zonestamp.toolforge.org/1732896000&gt;). If you're interested in joining, you can sign up on this wiki page: &lt;https://www.mediawiki.org/wiki/Wikimedia_Language_and_Product_Localization/Community_meetings#29_November_2024&gt;.

This participant-driven meeting will be organized by the Wikimedia Foundation’s Language Product Localization team and the Language Diversity Hub. There will be presentations on topics like developing language keyboards, the creation of the Moore Wikipedia, and the language support track at Wiki Indaba. We will also have members from the Wayuunaiki community joining us to share their experiences with the Incubator and as a new community within our movement. This meeting will have a Spanish interpretation.

Looking forward to seeing you at the language community meeting! Cheers, [[User:SSethi (WMF)|Srishti]] 19:55, 21 November 2024 (UTC)
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== Events on Wikiversity ==

Since Wikipedia and Wikivoyage are having their &quot;Asian Month&quot; editathon, I was thinking if we could start up a Wikiversity version of that. This would be an &quot;Asian Month&quot; as well, but it would be about creating resources based on Asia and its culture. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 17:57, 6 December 2024 (UTC)

:Not immediately opposed, but the question is, do we have an active enough community to facilitate this? —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:31, 6 December 2024 (UTC)
::I'm not too sure. As long as we get enough traffic, this could happen. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 08:45, 7 December 2024 (UTC)
:::This is to increase traffic on Wikiversity, which is promoted amongst other communities. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 10:47, 7 December 2024 (UTC)
:Hi @[[User:RockTransport|RockTransport]], This is a good idea, but will it also involve users who are not &quot;professors and scientists&quot;. Just curious. cheers, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:30, 9 December 2024 (UTC)
::Yes, considering the fact that Wikiversity is for everyone, and not just for specific users. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 09:09, 10 December 2024 (UTC)
:::because I'm personally not a &quot;professor&quot; or a &quot;scientist&quot; and because '''anyone''' can create resources on Wikiversity. We want to make Wikiversity open for everyone, and not just for certain users. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 09:10, 10 December 2024 (UTC)
::::I am also not a professor or a scientist, but it seems to me that as result I am viewed here as a visitor rather than someone who can contribute. Just my $.02. [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:05, 12 December 2024 (UTC)

== Wikiversity - Newsletters ==

Hello All,

I wanted to create a newsletter on Wikiversity, which would highlight what is going on in certain months and events on Wikiversity; which would bolster engagement by many people. This would be on the website and would have its dedicated 'Newsletter' tab.

I hope you acknowledge this idea. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 21:05, 8 December 2024 (UTC)

:@[[User:RockTransport|RockTransport]], What sort of things do you plan to include in your newsletter? Will they be different than what is currently in [[Main Page/News]]? Just curious.
:I am also wondering about your motive which I think is: to bolster engagement by many people. I am asking because I wonder if others who are currently active here also think this I is desirable? Have you asked them? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:34, 11 December 2024 (UTC)
::Not yet, which was why I was asking this on the colloquium. I plan to include things that many people have created on Wikiversity over the month, as it is a monthly newsletter. It would be somewhere on the website here. It will be more frequent that the ones seen on [[Main Page/News]]. We will include people's resources to essentially promote them. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 06:50, 12 December 2024 (UTC)
:::@[[User:RockTransport|RockTransport]], I Think what you are saying is that ''Main Page/News'' does not update frequently enough? 
:::If this is the reason, why not start small by simply increasing the frequency of posting news on the main page, instead of trying to start a newsletter?
:::If there is more, can you articulate what else is missing. Thanks in advance, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:51, 12 December 2024 (UTC)
::::I meant going to detail into topics covered in that month, rather than just giving a few points. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 16:53, 12 December 2024 (UTC)
:::::What sort of details did you have in mind? You can pick one of the links provided in [[Main Page/News]] to illustrate. cheers, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 15:29, 16 December 2024 (UTC)
::::::I'm thinking of the community entering their projects, and discussing those in the newsletter. It depends on what they want, though. There would be a dedicated page for giving the information about their projects [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 17:24, 16 December 2024 (UTC)</text>
      <sha1>mjuisqvu2yvsdtmi6jwpuwm856tq212</sha1>
    </revision>
    <revision>
      <id>2692261</id>
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      <timestamp>2024-12-17T09:59:50Z</timestamp>
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        <id>2651</id>
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      <comment>/* Events on Wikiversity */ Reply</comment>
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      <text bytes="78010" sha1="26u7tx3daruv4ueyyy39tl70jpjztc9" xml:space="preserve">{{Wikiversity:Colloquium/Header}}
&lt;!-- MESSAGES GO BELOW --&gt;
== Reminder! Vote closing soon to fill vacancies of the first U4C ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
:''[[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement – reminder to vote|You can find this message translated into additional languages on Meta-wiki.]] [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement – reminder to vote}}&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]''

Dear all,

The voting period for the Universal Code of Conduct Coordinating Committee (U4C) is closing soon. It is open through 10 August 2024. Read the information on [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2024_Special_Election#Voting|the voting page on Meta-wiki to learn more about voting and voter eligibility]]. If you are eligible to vote and have not voted in this special election, it is important that you vote now.

'''Why should you vote?''' The U4C is a global group dedicated to providing an equitable and consistent implementation of the UCoC. Community input into the committee membership is critical to the success of the UCoC.

Please share this message with members of your community so they can participate as well.

In cooperation with the U4C,&lt;section end=&quot;announcement-content&quot; /&gt;

-- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 15:30, 6 August 2024 (UTC)
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== User group for Wikiversians ==

Was there ever a discussion about the possibility of establishing a user group in the sense of an affiliated organization that would defend the interests of professors and scientists on Wikiversity and possibly actively develop some projects? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:21, 8 August 2024 (UTC)

:Not that I'm aware of. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:20, 8 August 2024 (UTC)
:It's a pleasure to talk to a scientist on Wikiversity. I am a historian of technics and I would like to publish the following biography either on Wikiversity or on Wikipedia: 
:https://en.wikiversity.org/wiki/User:Rbmn/Arthur_Constantin_KREBS_(1850-1935):_Military_engineer,_Automotive_industrialist,_Great_projects_manager
:What would be your advice? [[User:Rbmn|Rbmn]] ([[User talk:Rbmn|discuss]] • [[Special:Contributions/Rbmn|contribs]]) 15:44, 6 October 2024 (UTC)
::The content appears to be largely biographical/encyclopedic, so I think it is likely best suited to Wikipedia. Consider improving/incorporating this content into the existing page: [[w:Arthur Constantin Krebs]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:05, 7 October 2024 (UTC)
::Please do not link to the Wikiversity [[wv:userspace|Userspace]] in Wikipedia articles. You will want to wait until you have a page in the [[wv:mainspace|Wikiversity mainspace]]. You'll also want to use the &lt;code&gt;{{[[:w:Template:Wikiversity|Wikiversity]]}}&lt;/code&gt; template (on Wikipedia) rather than embedding a photo with a link. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:21, 7 October 2024 (UTC)
:I haven't heard anything about this topic. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 21:06, 8 December 2024 (UTC)

== Rich's ''Illustrated Companion'' at Wikiversity: Right place? ==

Hello! I am creating a Wiki-version of a classical glossary (''Illustrated Companion to the Latin Dictionary, and Greek Lexicon'' by Anthony Rich, 1849), which explains the meaning of Latin headwords, primarily those &quot;representing visible objects connected with the arts, manufactures, and every-day life of the Greeks and Romans.&quot; The aim is to help understand what a (classical) Latin text is actually about, instead of merely translating it. I already transcribed the entire text and scanned the images (about 1900) from an original 1849-edition. I am currently working on uploading the images to ''Mediawiki Commons'', which probably will take some time. In the meantime I want to prepare the other aspects of the project (more than 3000 articles, already with many internal links). The important thing: this is ''not'' a ''might exist''-project.  {{Color|red|My question: Is ''Wikiversity'' the proper place for it?}} Although I created an exact rendition of the original text, ''Wikisource'' is not applicable, because the project has a broader scope (adding content to the articles, e. g. links to online editions for quotations, adding images, but also adding entirely new articles). Neither is ''Wikibooks'', because this is not a textbook and may otherwise breach its scope. For more about the project see [[w:User:CalRis25/Temp-RICH-Prospectus|my user-page]] at en.wikipedia. {{Color|Red|So, is Wikiversity the right place for it?}} [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 09:15, 17 August 2024 (UTC)

:Thanks for asking. To be clear, it ''is'' acceptable to make [[:s:en:Category:Wikisource annotations|annotated editions]] of texts at Wikisource and Wikibooks does host at least one [[:b:en:Annotations of The Complete Peanuts|annotated guide to a copyright-protected work]]. So if what you're looking to do is to include inline annotations to a public domain text, you certainly can put that on Wikisource. If you have a textbook or guidebook that is a companion, that would go at Wikibooks. If you have some other kind of learning resources (like maintaining a list of relevant links, organizing a book reading group, etc.), that could go here. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:26, 17 August 2024 (UTC)
::Thank you for your quick answer. Actually, ''Wikibooks'' was my first thought. However, this project is not merely an annotated edition. Although at first it ''will'' be a faithful copy of the original text, I want the project to be &quot;open&quot;, i. e. adding articles should be possible. And the project should enable to do a lot more than mere inline annotation. See section [[w:User:CalRis25/Temp-RICH-Prospectus#Improving_RICH|Improving Rich]] in the project description a my user-page (en.Wikipedia). No ''Mediawiki''-project (Wikisource, Wikibooks, Wikipedia, Wiktionary) seemed to be a sufficiently applicable &quot;fit&quot; for the project, so I thought of Wikiversity as a last resort, because it is supposed to be home to all sorts of &quot;learning resources&quot;. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 09:57, 17 August 2024 (UTC)
:::The scope of Wikiversity ''is'' pretty catch-all and would allow for a pretty flexible place to host most learning resources that don't fit elsewhere.
:::Also, as nitpick, &quot;MediaWiki&quot; is the software that is the basis of these wikis (wikis being collections of interlinked documents that can be edited) and &quot;Wikimedia Foundation&quot; is the non-profit who owns the trademarks and hosts these projects like Wiktionary and Wikivoyage. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 10:06, 17 August 2024 (UTC)
::::Hello Justin, thank you for the reply. '''I think that settles it. I will create this project at ''Wikiversity''.''' Just for additional clarification, why I do so. Let's imagine a full transcription of the original 1849-edition of the ''Illustrated Companion'' by Anthony Rich and call it ''RICH-1849''. We shall call my project, for brevity sake, RICH-2K. And now, let's have a look at the article about the Roman toga (a piece of attire). In ''RICH-1849'' we can can call it ''RICH-1849/Toga'', and it contains ''exactly'' the content of the 1849-book. Now, let's look at the article ''RICH-2K/Toga''. At the beginning its only content would be the article ''RICH-1849/Toga''. Does that make ''RICH-2K/Toga'' and ''RICH-1849/Toga'' the same? Not at all, because in truth ''RICH-2K/Toga'' is a &quot;container&quot; which initially contains only the article ''RICH-1849/Toga'' but later on may include more stuff: images, external links, article text which builds on or extends ''RICH-1849/Toga'' and information from other sources of information (Wikipedia, specialized books). By the way, this added article information would not be a mere copy of the text at en.Wikipedia, because the information needs to looked at through the eyes of someone reading the original text (more citations with direct links to these etc.). [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 11:39, 17 August 2024 (UTC)

== Coming soon: A new sub-referencing feature – try it! ==

&lt;section begin=&quot;Sub-referencing&quot;/&gt;
[[File:Sub-referencing reuse visual.png|{{#ifeq:{{#dir}}|ltr|right|left}}|400px]]
Hello. For many years, community members have requested an easy way to re-use references with different details. Now, a MediaWiki solution is coming: The new sub-referencing feature will work for wikitext and Visual Editor and will enhance the existing reference system. You can continue to use different ways of referencing, but you will probably encounter sub-references in articles written by other users. More information on [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing|the project page]].

'''We want your feedback''' to make sure this feature works well for you:

* [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing#Test|Please try]] the current state of development on beta wiki and [[m:Talk:WMDE Technical Wishes/Sub-referencing|let us know what you think]]. 
* [[m:WMDE Technical Wishes/Sub-referencing/Sign-up|Sign up here]] to get updates and/or invites to participate in user research activities.

[[m:Special:MyLanguage/Wikimedia Deutschland|Wikimedia Deutschland]]’s [[m:Special:MyLanguage/WMDE Technical Wishes|Technical Wishes]] team is planning to bring this feature to Wikimedia wikis later this year. We will reach out to creators/maintainers of tools and templates related to references beforehand.

Please help us spread the message. --[[m:User:Johannes Richter (WMDE)|Johannes Richter (WMDE)]] ([[m:User talk:Johannes Richter (WMDE)|talk]]) 10:36, 19 August 2024 (UTC)
&lt;section end=&quot;Sub-referencing&quot;/&gt;
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== New [[Template:Form]] ==

Hi! Today I was bold and created [[Template:Form]] (which calls [[Module:WikiForm]] and [[MediaWiki:Gadget-WikiForm.js]]). The template allows to create user-friendly forms that can create pages or add content to existing pages. My motivation and first use case was [[Wikidebate/New|this form]] to create new [[wikidebates]], but I suspect the template can be useful elsewhere on Wikiversity. Let me know if you notice any issues or have any requests or concerns. Kind regards, [[User:Sophivorus|Sophivorus]] ([[User talk:Sophivorus|discuss]] • [[Special:Contributions/Sophivorus|contribs]]) 15:21, 21 August 2024 (UTC)

== Sign up for the language community meeting on August 30th, 15:00 UTC ==

Hi all,

The next language community meeting is scheduled in a few weeks—on August 30th at 15:00 UTC. If you're interested in joining, you can [https://www.mediawiki.org/wiki/Wikimedia_Language_and_Product_Localization/Community_meetings#30_August_2024 sign up on this wiki page].

This participant-driven meeting will focus on sharing language-specific updates related to various projects, discussing technical issues related to language wikis, and working together to find possible solutions. For example, in the last meeting, topics included the Language Converter, the state of language research, updates on the Incubator conversations, and technical challenges around external links not working with special characters on Bengali sites.

Do you have any ideas for topics to share technical updates or discuss challenges? Please add agenda items to the document [https://etherpad.wikimedia.org/p/language-community-meeting-aug-2024 here] and reach out to ssethi(__AT__)wikimedia.org. We look forward to your participation!

[[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 23:20, 22 August 2024 (UTC)
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== Template consolidation: User talk page block notice ==
Wondering if someone who likes templates could have a go at consolidating or helping decide between use of:
* [[Template:Block]]
* [[Template:Blocked]]

Unless I'm missing something, it seems like we don't need both?

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 07:16, 23 August 2024 (UTC)
: I tried to figure out a Wikidata item with most links to projects. I found this: [[Wikidata:Q6379131]], which is Template:Uw-block. There is even a corresponding Wikiversity template, [[Template:Uw-block1]] (not used anywhere).
: My impression is that of the three templates, we only need one. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:43, 13 September 2024 (UTC)

== Announcing the Universal Code of Conduct Coordinating Committee ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
:''[https://lists.wikimedia.org/hyperkitty/list/board-elections@lists.wikimedia.org/thread/OKCCN2CANIH2K7DXJOL2GPVDFWL27R7C/ Original message at wikimedia-l]. [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement - results|You can find this message translated into additional languages on Meta-wiki.]] [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement - results}}&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]''

Hello all,

The scrutineers have finished reviewing the vote and the [[m:Special:MyLanguage/Elections Committee|Elections Committee]] have certified the [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Results|results]] for the [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election|Universal Code of Conduct Coordinating Committee (U4C) special election]].

I am pleased to announce the following individual as regional members of the U4C, who will fulfill a term until 15 June 2026:

* North America (USA and Canada)
** Ajraddatz

The following seats were not filled during this special election:

* Latin America and Caribbean
* Central and East Europe (CEE)
* Sub-Saharan Africa
* South Asia
* The four remaining Community-At-Large seats

Thank you again to everyone who participated in this process and much appreciation to the candidates for your leadership and dedication to the Wikimedia movement and community.

Over the next few weeks, the U4C will begin meeting and planning the 2024-25 year in supporting the implementation and review of the UCoC and Enforcement Guidelines. You can follow their work on [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee|Meta-Wiki]].

On behalf of the U4C and the Elections Committee,&lt;section end=&quot;announcement-content&quot; /&gt;

[[m:User:RamzyM (WMF)|RamzyM (WMF)]] 14:07, 2 September 2024 (UTC)
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== Re: The Vector 2022 skin as the default in two weeks? ==

[[File:Vector 2022 video-en.webm|thumb|A two minute-long video about Vector 2022]]
Hello everyone, I'm reaching out on behalf of the [[mediawikiwiki:Reading/Web|Wikimedia Foundation Web team]] responsible for the MediaWiki skins. I'd like to revisit the topic of making Vector 2022 the default here on English Wikiversity. I [[Wikiversity:Colloquium/archives/September 2022#The Vector 2022 skin as the default in two weeks?|did post a message about this almost two years ago]] (where you can find all the details about the skin), but we didn't finalize it back then.

What happened in the meantime? We built [[mw:Reading/Web/Accessibility for reading|dark mode and different options for font sizes]], and made Vector 2022 the default on most wikis, including all other Wikiversities. With the not-so-new V22 skin being the default, existing and coming features, like dark mode and [[mw:Trust and Safety Product/Temporary Accounts|temporary accounts]] respectively, will become available for logged-out users here.

So, if no large concerns are raised, we will deploy Vector 2022 here in two weeks, in the week of September 16. Do let me know if you have any questions. Thank you! [[User:SGrabarczuk (WMF)|SGrabarczuk (WMF)]] ([[User talk:SGrabarczuk (WMF)|discuss]] • [[Special:Contributions/SGrabarczuk (WMF)|contribs]]) 21:48, 2 September 2024 (UTC)

:Sounds good, Szymon - we look forward to the upcoming change of skin {{smile}} Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 07:35, 13 September 2024 (UTC)
* I for one oppose a switch to Vector 2022. I do not find it preferable. Here is a staggering evidence of user refusal of Vector 2022 once it was deployed: [[W:en:Wikipedia:Requests for comment/Rollback of Vector 2022]], Junuary 2023. 355 voters supported rollback to Vector 2010 whereas 64 opposed, yielding 84.7% support, as clear a supermajority as one may wish. These people opposing Vector 2022 feel the same way as I do. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:48, 13 September 2024 (UTC)
*:Hey @[[User:Dan Polansky|Dan Polansky]]. Thanks for your comment. I'm open to discussion about problems with our software, and I hope we can maintain a respectful tone.
*:I understand that there are users who prefer Vector legacy or other skins, just as there are people who still stick to Monobook. Such people are active across many wikis. They can keep Vector legacy, although non-default skins don't have the support the default ones do. We are rolling out for technical reasons, as I mentioned above, with benefit to not logged-in users.
*:Regarding the rollback RfC on Wikipedia, two neutral users stated that there was no consensus for rollback, RfC is not a vote, and the numbers were different (355:226:24). I believe this all is pretty easy to verify.
*:So to sum up, Vector 2022 needs to become the default, tons and tons of comments were made about the skin and related stuff, and we have taken many ideas into account, and it's totally OK if you stick to Vector legacy.
*:Thanks! [[User:SGrabarczuk (WMF)|SGrabarczuk (WMF)]] ([[User talk:SGrabarczuk (WMF)|discuss]] • [[Special:Contributions/SGrabarczuk (WMF)|contribs]]) 19:30, 16 September 2024 (UTC)
*:: Today, I visited Wikiversity and found it switched to Vector 2022. I changed my preference settings to Vector 2010. From what I understand, non-registered visitors are now defaulted to Vector 2022 despite its unpopularity in [[W:en:Wikipedia:Requests for comment/Rollback of Vector 2022]]. I have not seen any evidence that users prefer Vector 2022, and given the evidence in the linked RfC, I tentatively conclude that the decision to switch has made the site experience worse for the majority of users. The logic of &quot;you can switch&quot; surely applies to Vector 2022 as well: those who prefer it can switch to it. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:08, 17 September 2024 (UTC)

== Have your say: Vote for the 2024 Board of Trustees! ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
Hello all,

The voting period for the [[m:Special:MyLanguage/Wikimedia Foundation elections/2024|2024 Board of Trustees election]] is now open. There are twelve (12) candidates running for four (4) seats on the Board.

Learn more about the candidates by [[m:Special:MyLanguage/Wikimedia Foundation elections/2024/Candidates|reading their statements]] and their [[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Questions_for_candidates|answers to community questions]].

When you are ready, go to the [[Special:SecurePoll/vote/400|SecurePoll]] voting page to vote. '''The vote is open from September 3rd at 00:00 UTC to September 17th at 23:59 UTC'''.

To check your voter eligibility, please visit the [[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Voter_eligibility_guidelines|voter eligibility page]].

Best regards,

The Elections Committee and Board Selection Working Group&lt;section end=&quot;announcement-content&quot; /&gt;

[[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 12:15, 3 September 2024 (UTC)
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== Separate page for hyperbola. ==

Good morning,

I notice that a search for &quot;hyperbola&quot; redirects to &quot;Conic sections&quot;.

At present there is a separate page for &quot;ellipse&quot;. Therefore a separate page for &quot;hyperbola&quot; seems to be justified.

Could this redirection be changed so that search for &quot;hyperbola&quot; goes to a separate page for &quot;hyperbola&quot;?

Many thanks,

[[User:ThaniosAkro|ThaniosAkro]] ([[User talk:ThaniosAkro|discuss]] • [[Special:Contributions/ThaniosAkro|contribs]]) 12:04, 15 September 2024 (UTC)

:It is true that ellipses are covered at [[Conic sections]] (along with hyperbolas, parabolas, etc.) and there is a separate page for [[ellipse]]s that elaborates. We certainly ''could'' have a page about [[hyperbola]]s that is separate, but no one has written sufficient content to spin it off yet. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 12:17, 15 September 2024 (UTC)

== I hereby request for your Unblocking IP address and just reviewed and received a reverted rec ==

Hi there. {{unsigned|Ishmael Raphasha}}
:No one has any clue what you're talking about. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:53, 18 September 2024 (UTC)

== RICH-2K: New project with some initial questions ==

Hello! I'm creating a new learning resource on ''Wikiversity''. The respective project is based on my transcription of a classical dictionary from 1849 by Anthony Rich. For more information about the project see its [[User:CalRis25/RICH: Description|description page]] (see also that page for why not ''Wikisource'' or ''Wikibooks''). The project's scope is fairly big: 3205 article-pages plus 304 REDIRECT-pages. The images (scanned by myself from an original copy) have been uploaded to ''Commons''. I have some initial technical questions (more of these and more detailed ones will follow):

* '''Upload''': Due to the large number of pages it is not realistic to create these manually. Is it possible to bulk-upload these in some way (the Wikitext of the pages is created using a Python-script with one file per article/page)? Is it possible to upload these to a test-environment first where any problems (hopefully none) can be identified and dealt with more easily than on the production-version of ''Wikiversity''?
* '''(Technical) Structure''': I am planning to set up this project at ''&lt;nowiki&gt;https://en.wikiversity.org/wiki/RICH-2K&lt;/nowiki&gt;'' as the main page and anything else as subpages: ''RICH-2K/Subpage_1 ... RICH-2K/Subpage_n''. However, these subpages fall into two categories: 1. Article-pages (content) and 2. Meta/Administrative pages. This project requires search capability restricted to the ''RICH-2K''-namespace. The ''Mediawiki''-software seems to supply a ''Search''-input field with the possibility to restrict the search to some namespace. I would like, however, to restrict the search further to the first group of pages, namely the articles. Is that possible, perhaps by use of (hidden) categories?
* '''External links''': This project will need many external links, and yes, I have read the relevant ''Wikiversity''-pages, but this specific project needs them. The ''Recommended Editions''-page (used for recommended online editions, to which to link when citing texts) alone probably will require several hundred external links. However, only relatively few [[w:Second-level domain|second-level domains]] will be involved, and most of these should be trustworthy (Perseus Digital library, digital collections of universities etc., in some cases, however, also ''Archive.org''). Perhaps there is a list of web-sites, for which external links are generally allowed? And who is allowed to create external links on ''Wikiversity''-pages (I haven't found the relevant policy)?
* '''Categories''': This project requires quite a few of its own categories, which belong to two large groups: 1. Categories (2 levels) of the ''Classed Index'' (about 170 categories), a thematic index of some (but not all) of the articles. 2. Administrative categories. Is there a recommended way to distinguish between different classes of categories within a project (category name or other method)? What about naming conventions for project-specific categories?

I am looking forward to your input. If you think that it's preferable we can move the discussions to the [[User_talk:CalRis25/RICH:_Description|Talk-page]] of the project's description. Thank you in advance. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 05:29, 20 September 2024 (UTC)

:*Admins have access to [[Special:Import]] and can bulk import XML pages. You can create pages in your sandbox if you'd like and make an indefinite amount of them at pages like [[User:CalRis25/sandbox]]. What can and cannot be hosted in user namespace is very loose, but still has to follow in principle Wikiversity's scope.
:*Using subpages is in principle a good way to organize these various resources. Please do not name them after a user name or something obscure. I personally think that &quot;RICH-2K&quot; is a not optimal name. I may recommend something like [[Anthony Rich Dictionary Project]] or [[21st-Century Anthony Rich Dictionary]] or something more obviously intelligible. While we have very few actual policies and guidelines, see [[Wikiversity:Naming conventions]] for a rough consensus of what is probably best practice for naming pages.
:*External linking generally does not use an allowed list (a.k.a. whitelist model), but a disallow (a.k.a. blacklist) model. See [[MediaWiki:Spam-blacklist]] and [[Special:BlockedExternalDomains]] (which is currently empty but is another method of listing blocked domains). It's perfectly fine to aggregate external links in learning resources.
:*I'm not 100% sure what the distinction is that you're drawing, but you can freely arrange categories underneath a main category that has the same name as your larger project. So, following the suggestions I gave, you could have a category like [[:Category:Anthony Rich Dictionary Project]] and then create any number of subcategories that logically help users navigate all these pages. Please make sure the main category you create is itself categorized under some relevant category(ies). If you need help, please ask.
:I think this answers your questions, please let me know if I'm unclear or you have more. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:11, 20 September 2024 (UTC)
::Hello Justin!
::* '''Upload:''' Creating the project in sandbox pages of my User-namespace defeats the purpose, as this is an ''open'' project. Also that would not solve, as such, the problem of having to manually create thousands of pages. I wonder, does ''Wikiversity'' support creation of pages using its API. ''Mediawiki's'' [[mw:API:Main_page|API-description]] seems to imply that it ought to be possible. If that's the case, I should be able to create a Python-script which automatically creates the pages (of course, a few trial pages first).
::* '''(Technical) Structure''': You may be right, here. RICH-2K is, for now, merely a technical name to make a clear but not too verbose distinction between the original text and the current project. I'll give this more thought.
::* '''External links''': I brought this up mainly because when I first edited my ''Wikiversity''-page, I got a message that I was not allowed to create external links. However, I just now tested creating an external link on my user-page and got no error, so this problem seems to be solved.
::* '''Categories''': I think I know what you mean. I'll create a category structure and maybe ask some specific questions once I am ready to do so.
::Thank you for your quick help. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 18:51, 20 September 2024 (UTC)
:::re: upload, I'm just suggesting your sandbox(es) as you asked about &quot;a test-environment&quot;. Anyone can edit someone else's sandboxes, but you typically defer to other users to control what's in their own subpages as a collegial thing. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:39, 23 September 2024 (UTC)

Hello! I have two further questions:
# I created a category-structure for the project. Could you (or someone else) have a look at it ([[User:CalRis25/RICH: Categories]]) and answer the questions in the section [[User:CalRis25/RICH:_Categories#Questions|Questions]]? I gave it some thought and believe that this would work fine for the project.
# ''Project boxes'' (see [[Help:Tour of project boxes]]): It is unclear to me, whether these belong only on the main page of the project (that makes the most sense to me), or on every single subpage.
Thanks in advance for your help. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 17:51, 24 September 2024 (UTC)

:To answer your questions here:
:*No, you are not contravening any policies we have.
:*A leading &quot;The&quot; is acceptable, but if you want it to sort alphabetically, you will have to use &lt;nowiki&gt;{{DEFAULTSORT:}}&lt;/nowiki&gt;. E.g. to get Category:The Best Stuff to sort under &quot;B&quot;, insert &quot;&lt;nowiki&gt;{{DEFAULTSORT:Best Stuff, The}}&lt;/nowiki&gt;.
:*Trailing &quot;etc.&quot; is acceptable.
:*An accent in a category title is acceptable.
:I'll also note that it looks like you have in mind some tracking categories that are redundant. Pages such as [[Special:LonelyPages]] and [[Special:DeadendPages]] already do automatically what you're proposing to do manually.
:As for project boxes, it's typically the case that the subjects are only placed on the main resource, but as you may imagine, [[Help:Tour of project boxes/1|status completion ones]] may vary from subpage to subpage. As with most things at Wikiversity, there are very few actual rules, so it's pretty much the wild west, even tho this project has been around for almost 20&amp;nbsp;years. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:18, 25 September 2024 (UTC)
::Hello Justin, thanks for the DEFAULTSORT-hint for categories beginning with ''The''. I will restrict the project boxes to the main page. As for the the orphaned/dead-end-categories, I prefer these to be project-specific. Once the project is up and running, putting articles &quot;on the map&quot; (making them accessible from other articles and creating links to other articles) is one of the first tasks to be dealt with. I already know which articles are involved and will add these categories to these articles. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 16:51, 25 September 2024 (UTC)

== Your wiki will be in read-only soon ==

&lt;section begin=&quot;server-switch&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;

[[:m:Special:MyLanguage/Tech/Server switch|Read this message in another language]] • [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-Tech%2FServer+switch&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]

The [[foundation:|Wikimedia Foundation]] will switch the traffic between its data centers. This will make sure that Wikipedia and the other Wikimedia wikis can stay online even after a disaster.

All traffic will switch on '''{{#time:j xg|2024-09-25|en}}'''. The switch will start at '''[https://zonestamp.toolforge.org/{{#time:U|2024-09-25T15:00|en}} {{#time:H:i e|2024-09-25T15:00}}]'''.

Unfortunately, because of some limitations in [[mw:Special:MyLanguage/Manual:What is MediaWiki?|MediaWiki]], all editing must stop while the switch is made. We apologize for this disruption, and we are working to minimize it in the future. 

A banner will be displayed on all wikis 30 minutes before this operation happens. This banner will remain visible until the end of the operation. 

'''You will be able to read, but not edit, all wikis for a short period of time.'''

*You will not be able to edit for up to an hour on {{#time:l j xg Y|2024-09-25|en}}.
*If you try to edit or save during these times, you will see an error message. We hope that no edits will be lost during these minutes, but we can't guarantee it. If you see the error message, then please wait until everything is back to normal. Then you should be able to save your edit. But, we recommend that you make a copy of your changes first, just in case.

''Other effects'':

*Background jobs will be slower and some may be dropped. Red links might not be updated as quickly as normal. If you create an article that is already linked somewhere else, the link will stay red longer than usual. Some long-running scripts will have to be stopped.
* We expect the code deployments to happen as any other week. However, some case-by-case code freezes could punctually happen if the operation require them afterwards.
* [[mw:Special:MyLanguage/GitLab|GitLab]] will be unavailable for about 90 minutes.

This project may be postponed if necessary. You can [[wikitech:Switch_Datacenter|read the schedule at wikitech.wikimedia.org]]. Any changes will be announced in the schedule. 

'''Please share this information with your community.'''&lt;/div&gt;&lt;section end=&quot;server-switch&quot;/&gt;

[[User:Trizek_(WMF)|Trizek_(WMF)]], 09:37, 20 September 2024 (UTC)
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== 'Wikidata item' link is moving. Find out where... ==

&lt;div lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot;&gt;&lt;i&gt;Apologies for cross-posting in English. Please consider translating this message.&lt;/i&gt;{{tracked|T66315}}

Hello everyone, a small change will soon be coming to the user-interface of your Wikimedia project.
The [[d:Q16222597|Wikidata item]] [[w:|sitelink]] currently found under the &lt;span style=&quot;color: #54595d;&quot;&gt;&lt;u&gt;''General''&lt;/u&gt;&lt;/span&gt; section of the '''Tools''' sidebar menu will move into the &lt;span style=&quot;color: #54595d;&quot;&gt;&lt;u&gt;''In Other Projects''&lt;/u&gt;&lt;/span&gt; section.

We would like the Wiki communities feedback so please let us know or ask questions on the   [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Discussion page]] before we enable the change which can take place October 4 2024, circa 15:00 UTC+2.
More information can be found on [[m:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|the project page]].&lt;br&gt;&lt;br&gt;We welcome your feedback and questions.&lt;br&gt; [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 18:56, 27 September 2024 (UTC)
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==Download as PDF==
[[Phabricator:T376438]]: &quot;Download to PDF&quot; on en.wv is returning error: &quot;{&quot;name&quot;:&quot;HTTPError&quot;,&quot;message&quot;:&quot;500&quot;,&quot;status&quot;:500,&quot;detail&quot;:&quot;Internal Server Error&quot;}&quot;

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:36, 3 October 2024 (UTC)

:I just downloaded this page as a PDF and it worked just fine. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:04, 3 October 2024 (UTC)

== Protected template bug for Pp ==

It seems that templates derivative of {{tlx|Pp}} (compiled in {{tlx|Protection templates}}) are being sorted into protection categories using the name 'Wikipedia' instead of 'Wikiversity' (e.g., [[:Category:Wikipedia pages with incorrect protection templates]]). From what I can tell, it is not in the publicly accessible source code of any of the templates. The only other impacted pages are modules which call {{tlx|pp}}-derivatives (e.g., [[Module:Navbar/styles.css]]).

This does not seem to affect any other pages in [[:Category:Wikiversity protected templates]]. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 18:59, 4 October 2024 (UTC)

:The problem is that &quot;Wikipedia&quot; is [https://en.wikiversity.org/w/index.php?title=Special%3ASearch&amp;limit=500&amp;offset=0&amp;ns828=1&amp;search=Wikipedia&amp;searchToken=9svkpqlxxoquoq7bnkt55ugts mentioned in several modules that were copied over from en.wp]; many of these are legit and many of them need to be replaced with &quot;Wikiversity&quot; ([https://en.wikiversity.org/w/index.php?title=Module%3APp-move-indef&amp;diff=2662815&amp;oldid=1944984 e.g.]) This particular change ''may'' fix all of these issues...? But 1.) it will take time to propagate across the site and 2.) there are still many more &quot;Wikipedia&quot;s that need to be changed, so I'll go thru a few more, but if you want to give me an assist, if you can just check this one week from now and ping me if the problem persists, that would be nice. Sometimes, I make calendar reminders to follow up on these, but I'm not a perfect person. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:55, 5 October 2024 (UTC)

== Invitation to Participate in Wiki Loves Ramadan Community Engagement Survey ==

Dear all,


We are excited to announce the upcoming [[m:Wiki Loves Ramadan|Wiki Loves Ramadan]] event, a global initiative aimed at celebrating Ramadan by enriching Wikipedia and its sister projects with content related to this significant time of year. As we plan to organize this event globally, your insights and experiences are crucial in shaping the best possible participation experience for the community.

To ensure that Wiki Loves Ramadan is engaging, inclusive, and impactful, we kindly invite you to participate in our community engagement survey. Your feedback will help us understand the needs of the community, set the event's focus, and guide our strategies for organizing this global event.

Survey link: https://forms.gle/f66MuzjcPpwzVymu5

Please take a few minutes to share your thoughts. Your input will make a difference!

Thank you for being a part of our journey to make Wiki Loves Ramadan a success.


Warm regards,

User:ZI Jony 03:19, 6 October 2024 (UTC)

Wiki Loves Ramadan Organizing Team
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== 'Edit to my talk page' notification bug? ==

This may belong at the bug tracker, but does anyone else have an issue disabling ''email'' notifications upon  an 'Edit to my talk page' in [[Special:GlobalPreferences]]? Oddly I ''am'' able to disable the global preference on Wikipedia, MediaWiki, etc, but not here. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 09:23, 7 October 2024 (UTC)

:I have not experienced this, but to be clear, do you also have the option to get emails when items on your talk page are edited turned on? —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:39, 7 October 2024 (UTC)
::The only (non-grayed out) options I have enabled for email are 'Failed login attempts' and 'Login from an unfamiliar device'. 'Edit to my talk page' re-checks after every save. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 09:54, 7 October 2024 (UTC)
:::That does sound like a [[phab:]] issue, with the caveat that I don't 100% recall how global preferences work and if they override local ones, etc. If you have parsed that and still have this issue, you'll probably need to file a ticket. Maybe someone else has this issue. Wish I could help. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:57, 7 October 2024 (UTC)
::::[[phab:T376601|Off 'n away]] 🫡 [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 10:35, 7 October 2024 (UTC)

== [[Portal:Computer Science]] ➝ [[Portal:Information sciences]] ==

Seeking consensus to complete the merge into the broader portal. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 06:28, 8 October 2024 (UTC)

:Why should it be merged? Computer Science seems well-enough designed. What is the incentive to collapse it into a broader field of study? —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:18, 8 October 2024 (UTC)
::Portals as top level organizations allowing for content to be best centralized. Also note that I did not start the merge, just offering to finish it. Perhaps a {{tlx|prod}} instead? [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 07:20, 8 October 2024 (UTC)
:::I have no objections, personally. If it gets done, please use a redirect and should someone want to come along to resurrect it later, it will be easier. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:21, 8 October 2024 (UTC)
: Is computer science really a branch of information sciences? I would not think so, but what do I know. Do we have some external resources/links confirm this idea? [[W:Information science]] currently says: &quot;Information science, documentology[1] or informatology[2][3] is an academic field which is primarily concerned with analysis, collection, classification, manipulation, storage, retrieval, movement, dissemination, and protection of information.&quot; --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:49, 11 October 2024 (UTC)
::Looking through [https://stackoverflow.com/q/1047014/22673230] [https://businessdegrees.uab.edu/mis-degree-bachelors/resources/computer-information-systems-vs-computer-science/] [https://www.si.umich.edu/student-experience/what-information-science] a few top (not necessarily RS) searches I'm inclined to agree. I am more familiar with the grafted [[:w:Information and computer science|information ''and'' computer science]] which makes an effort to merge the disciplines, but it does not seem like reaching to say that IS is presented as more applications-concerned (certainly with no lack of theoretical abstraction), whereas CS can be more freely associated with any and all 'science related to computers'. It is easy to reason about the connection between the fields, but I think it is clear academia maintains this taxonomy for a good reason.
::With these considerations, I think I will ''stop'' the process of merging in favor of expanding the existing [[School:Library and Information Science]]. 

::Let me know if there is not consensus to redirect [[Portal:Information sciences]] to [[School:Library and Information Science]] (with enough expansion it can generalize away from just library sciences). [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 16:16, 11 October 2024 (UTC)
::: I do not see that a merge of a ''portal'' to a ''school'' is a good thing. Do you have a clear idea of the concepts of school and portal and how they relate to each other? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:34, 11 October 2024 (UTC)
::::Found [[:Category:Information sciences]]; there are enough existing resources in there to make my other proposed merge excessive. I will simply continue developing the existing [[Portal:Information sciences]] instead. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 17:05, 11 October 2024 (UTC)
::::: Frankly, I would ideally see [[Portal:Information sciences]] deleted: I don't see what it does that a category would not do well enough. There does not seem to be any material specific to &quot;Information sciences&quot; (whatever that is) in that portal at all. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:11, 11 October 2024 (UTC)
::::::Tacked a {{tlx|prod}} for an eventual deletion, but I may still try to develop it as proof of concept at some point. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 17:33, 11 October 2024 (UTC)

== [[:Category:Occupational Epidemiology]] ==

I propose moving the pages in this category (without leaving redirects) to their equivalent under the parent resource [[Occupational Health Risk Surveillance]]. Also due to the number of subpages, it seems &lt;code&gt;|filing=deep&lt;/code&gt; would be a justified. (Also [[Special:PrefixIndex/Occupational_Epidemiology|there are quite a few]] untagged subpages.) [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 05:11, 9 October 2024 (UTC)
: I above all think that the content should be ''moved out of the mainspace'': I do not see readers learning anything from e.g. [[Occupational Epidemiology/Research tools/Reading of scientific articles for learning epidemiology and biostatstics]] or [[Occupational Epidemiology/Research tools/Ongoing projects/Risk Communication in Seafaring/Writing the article guideline IMRAD]]. Wikiversity can be kind enough to host that material in, say, subspace of [[User:Saltrabook]], but more should not be asked, I think. Let us recall that per [[WV:Deletions]], &quot;Resources may be eligible for proposed deletion when education objectives and learning outcomes are scarce, and objections to deletion are unlikely&quot;; I do not see how learning outcomes can be anything but scarce. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 15:04, 11 October 2024 (UTC)
::thank you, agree @ [[User:Saltrabook|Saltrabook]] ([[User talk:Saltrabook|discuss]] • [[Special:Contributions/Saltrabook|contribs]]) 21:03, 13 November 2024 (UTC)

== Active editors ==

It is interesting to observe the stats on [https://stats.wikimedia.org/#/en.wikiversity.org/contributing/active-editors/normal|line|all|(page_type)~content*non-content|monthly active editors] through the project's history. October is our month! [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 20:44, 8 October 2024 (UTC)

:Odd. Maybe related to the school year? —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 02:10, 9 October 2024 (UTC)
::I wonder how many are [[User:Jtneill|Jtneill]]'s crowd... the number is in the hundreds though, so that is one chunky cohort —[[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:16, 9 October 2024 (UTC)
:::Yes, [[Motivation and emotion/Book]] involves ~100-150 students editing most intensely during October each year. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:20, 9 October 2024 (UTC)
::::Neat, that still leaves around ~50-100 other students from other avenues each year since 2021. I also wonder which projects were involved in the COVID enrollment spike. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:26, 9 October 2024 (UTC)
:::::Personally I can admit that my editing is much more active during the school season vs. the summer break, so I'm in the same boat as Jtneill's students. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:24, 13 November 2024 (UTC)
:@[[User:Tule-hog|Tule-hog]] This is an interesting topic, but it is not clear to me as an outsider what you and other participants in this discussion find interesting. I  find this graph not very meaningful because it does not tell me if the number of  Active editors has gone up or down during the period covered, which I think was 2000-now. 
:I can see a big jump between 2000 and 2007, but what happened since then? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 15:45, 16 December 2024 (UTC)

== Intentionally incorrect resource ==

There is a [[Special:Diff/2583464|disclaimer inserted onto a resource]] (by not the original author) that:
&lt;blockquote&gt;I am merely [making this page false] to show you (The viewer) that Wikipedia and this page 'Wikiversity' is bull sh*t and it will not give you the reliability you need when writing an academic piece of writing.&lt;/blockquote&gt;
However, that IP has [[Special:Contributions/86.22.73.151|not made any other edits]], so unless they vandalized via a sock, the intent went un-realized and only that portion need be removed. Bumping here in case there is some obvious jumbo in that essay that someone else can catch. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 16:58, 9 October 2024 (UTC)

:Removed that portion, which was obviously vandalism. No perspective on the rest of the essay. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:38, 9 October 2024 (UTC)

== [[:Category:Proposed guidelines]] ==

Noting for future editors that WV has collapsed all proposals into [[:Category:Proposed policies|proposed policies]]. Seeking consensus to further collapse [[:Category:Wikiversity proposals]] into the former, or to restore [[:Category:Proposed guidelines]]. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 19:19, 9 October 2024 (UTC)

== [[Around Wikiversity in 80 Seconds|Broken 80-second tour]] ==

Bumping a [[Talk:Around_Wikiversity_in_80_Seconds|comment]] on the ''Wikiversity in 80 seconds'' tour. Appears wikisuite is not working with the Vector 2022 appearance. 

Also see [[:w:Wikipedia:Miscellany_for_deletion/Wikiversuite_pages|this thread]] on the Wikiversal package - may not be relevant to Wikiversity, but FYC. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 00:26, 10 October 2024 (UTC)
: I would just delete the material; I do not see value in it. If others agree, I would try to articulate why I think it should be deleted (or move to author user space). --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:57, 13 October 2024 (UTC)
::Just mark as {{tl|historical}}. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:39, 13 October 2024 (UTC)
::: This thing was created by [[User:Planotse]]. His creations are now being discussed in Wikibooks for deletion: [[B:Wikibooks:Requests for deletion#Wikiversal generated pages]]. It seems he used some kind of tool that is no longer available (the above mentioned &quot;Wikiversal&quot; package) to create this kind of slideshow-like material (believing the Wikibooks discussion). I do not see value of this in the mainspace, not even as historical (I am okay with userspace, but maybe even that is not the best option?). A look at the source code of [[Around Wikiversity in 80 Seconds/Introduction]] confirms the words of Omphalographer, namely that &quot;the HTML-heavy markup generated by Wikiversal makes them [the pages] unreasonably difficult to edit.&quot;
::: I went ahead and marked the page for proposed deletion. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:35, 14 October 2024 (UTC)

== Preliminary results of the 2024 Wikimedia Foundation Board of Trustees elections ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
Hello all,

Thank you to everyone who participated in the [[m:Special:MyLanguage/Wikimedia Foundation elections/2024|2024 Wikimedia Foundation Board of Trustees election]]. Close to 6000 community members from more than 180 wiki projects have voted.

The following four candidates were the most voted:

# [[User:Kritzolina|Christel Steigenberger]]
# [[User:Nadzik|Maciej Artur Nadzikiewicz]]
# [[User:Victoria|Victoria Doronina]]
# [[User:Laurentius|Lorenzo Losa]]

While these candidates have been ranked through the vote, they still need to be appointed to the Board of Trustees. They need to pass a successful background check and meet the qualifications outlined in the Bylaws. New trustees will be appointed at the next Board meeting in December 2024.

[[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Results|Learn more about the results on Meta-Wiki.]]

Best regards,

The Elections Committee and Board Selection Working Group

&lt;section end=&quot;announcement-content&quot; /&gt;

[[User:MPossoupe_(WMF)|MPossoupe_(WMF)]] 08:26, 14 October 2024 (UTC)
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== Seeking volunteers to join several of the movement’s committees ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
Each year, typically from October through December, several of the movement’s committees seek new volunteers. 

Read more about the committees on their Meta-wiki pages:
* [[m:Special:MyLanguage/Affiliations_Committee|Affiliations Committee (AffCom)]]
* [[m:Special:MyLanguage/Ombuds_commission|Ombuds commission (OC)]]
* [[m:Special:MyLanguage/Wikimedia Foundation/Legal/Community Resilience and Sustainability/Trust and Safety/Case Review Committee|Case Review Committee (CRC)]]

Applications for the committees open on 16 October 2024. Applications for the Affiliations Committee close on 18 November 2024, and applications for the Ombuds commission and the Case Review Committee close on 2 December 2024. Learn how to apply by [[m:Special:MyLanguage/Wikimedia_Foundation/Legal/Committee_appointments|visiting the appointment page on Meta-wiki]]. Post to the talk page or email [mailto:cst@wikimedia.org cst@wikimedia.org] with any questions you may have.

For the Committee Support team,
&lt;section end=&quot;announcement-content&quot; /&gt;

-- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 23:09, 16 October 2024 (UTC)
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== Interactive elements ==
Can we use interactive elements on Wikiversity? I'd like to add JavaScript to a page. If it's not possible now, where can I suggest this feature? I have a safe integration idea.
[[User:Отец Никифор|Отец Никифор]] ([[User talk:Отец Никифор|discuss]] • [[Special:Contributions/Отец Никифор|contribs]]) 12:10, 17 October 2024 (UTC)
: This is beyond my technical knowledge, but have you checked out:
:* https://www.mediawiki.org/wiki/Manual:Interface/JavaScript?
:* [[Wikipedia:WikiProject JavaScript]]
:* [[MediaWiki:Common.js]]

:What sort of interactive elements are you thinking about?

: Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:39, 18 October 2024 (UTC)
::I was thinking about adding something like a graph with adjustable controls, where users can interact with it and see how different changes affect the outcome. It seems like this could be a useful feature. There might already be discussions about enhancing Wikiversity or similar platforms—perhaps on a relevant talk page or in a Discord group. Do you know where such discussions might be happening? [[User:Отец Никифор|Отец Никифор]] ([[User talk:Отец Никифор|discuss]] • [[Special:Contributions/Отец Никифор|contribs]]) 19:47, 18 October 2024 (UTC)
:::From a quick look, maybe check out:
:::* [[mw:Extension:Graph]]
:::* [[phab:tag/graphs]]
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:40, 18 October 2024 (UTC)
:::: mw:Extension:Graph is currently disabled on Wikipedia etc. wikis, for security reasons, and seems unlikely to be enabled again. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:30, 19 October 2024 (UTC)

== An unexplained spurt of Wikiversity page views ==
The [https://pageviews.wmcloud.org/siteviews/?platform=all-access&amp;source=pageviews&amp;agent=user&amp;start=2024-06-01&amp;end=2024-10-18&amp;sites=en.wikiversity.org|en.wikibooks.org|en.wikiquote.org|en.wikisource.org page view report] shows an unexplained spurt of Wikiversity page views, reaching over 4 times the baseline and then falling back again. Does anyone have any idea what is going on? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:01, 19 October 2024 (UTC)

:Interesting. I wonder why only the English wikiquote and wikiversity and not Wikisource or wikibooks? How reliable do you think those stats are? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 15:44, 8 December 2024 (UTC)

== Center tempate failed on a contributors phone... ==

See the edit comment here - https://en.wikiversity.org/w/index.php?title=Wikiphilosophers&amp;diff=prev&amp;oldid=2673962. I'm puzzled as this is the first failure of this, I've noted recently. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 08:45, 19 October 2024 (UTC)

== Essay-like page in user space that makes little sense and seems incoherent ==
The page [[User:TheoYalur/Illusions]] seems to match the description, at least by my assessment. My understanding is that since the page is only in user space and not in the mainspace, it can stay there even if it has those disqualifying qualities. But if I am wrong and the page belongs deleted, please correct me and let me know. I do not know which policy or guideline, if any, guides the case. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:30, 21 October 2024 (UTC)

== 'Wikidata item' link is moving, finally. ==

Hello everyone, I previously wrote on the 27th September to advise that the ''Wikidata item'' sitelink will change places in the sidebar menu, moving from the '''General''' section into the '''In Other Projects''' section. The scheduled rollout date of 04.10.2024 was delayed due to a necessary request for Mobile/MinervaNeue skin. I am happy to inform that the global rollout can now proceed and will occur later today, 22.10.2024 at 15:00 UTC-2. [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Please let us know]] if you notice any problems or bugs after this change. There should be no need for null-edits or purging cache for the changes to occur. Kind regards, -[[m:User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] 11:28, 22 October 2024 (UTC)
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:Hi @[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]]: I Just noticed your post above, and it is timely.
:I have been participating in the English WikiUniversity for a few years, much less often recently. I seems like something in the way the site displays is different, but I cannot put my finger on it. Your posting gave me a clue. Can you please tell me where  the link to wikidata items has moved to? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:23, 11 December 2024 (UTC)
::Hello @[[User:Ottawahitech|Ottawahitech]], sure, I would be happy to. The button/sitelink name didn't change, just its position. You should find it in the sidebar-menu under the section '''In other projects''' (where the links to all other Wikimedia Projects are displayed). If you do not see it, please reach out to us on the [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Move Wikidata item - Discussion page]]. Thank you, -[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] ([[User talk:Danny Benjafield (WMDE)|discuss]] • [[Special:Contributions/Danny Benjafield (WMDE)|contribs]]) 09:24, 12 December 2024 (UTC)
:::@[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]], thank you for responding. I intend to followup on the ''Move Wikidata item - Discussion page'' as per your post above by putting it on my ever growing todo list.
:::I don't know about others on this wiki, as I said I have not been visiting here frequently, but for me the constant changes are a big distraction. I have been around wikimedia projects since 2007, so why do I have to spend so much time learning and re-learning how to find what I came here for? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:41, 12 December 2024 (UTC)
::::Hi @[[User:Ottawahitech|Ottawahitech]], thanks for you thoughts. Your input whether positive or critical helps us understand the impacts to editors so we welcome your further thoughts when you reach us in your To Do List :)
::::I can't speak about the other changes you've experienced here but I do hope they are made with a spirit of improvement for the community as a whole. -[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] ([[User talk:Danny Benjafield (WMDE)|discuss]] • [[Special:Contributions/Danny Benjafield (WMDE)|contribs]]) 10:43, 16 December 2024 (UTC)

== Final Reminder: Join us in Making Wiki Loves Ramadan Success ==

Dear all,

We’re thrilled to announce the Wiki Loves Ramadan event, a global initiative to celebrate Ramadan by enhancing Wikipedia and its sister projects with valuable content related to this special time of year. As we organize this event globally, we need your valuable input to make it a memorable experience for the community.

Last Call to Participate in Our Survey: To ensure that Wiki Loves Ramadan is inclusive and impactful, we kindly request you to complete our community engagement survey. Your feedback will shape the event’s focus and guide our organizing strategies to better meet community needs.

* Survey Link: [https://docs.google.com/forms/d/e/1FAIpQLSffN4prPtR5DRSq9nH-t1z8hG3jZFBbySrv32YoxV8KbTwxig/viewform?usp=sf_link Complete the Survey]
* Deadline: November 10, 2024
Please take a few minutes to share your thoughts. Your input will truly make a difference!

'''Volunteer Opportunity''': Join the Wiki Loves Ramadan Team! We’re seeking dedicated volunteers for key team roles essential to the success of this initiative. If you’re interested in volunteer roles, we invite you to apply.
* Application Link: [https://docs.google.com/forms/d/e/1FAIpQLSfXiox_eEDH4yJ0gxVBgtL7jPe41TINAWYtpNp1JHSk8zhdgw/viewform?usp=sf_link Apply Here]
* Application Deadline: October 31, 2024
Explore Open Positions: For a detailed list of roles and their responsibilities, please refer to the position descriptions here: [https://docs.google.com/document/d/1oy0_tilC6kow5GGf6cEuFvdFpekcubCqJlaxkxh-jT4/ Position Descriptions]

Thank you for being part of this journey. We look forward to working together to make Wiki Loves Ramadan a success!


Warm regards,&lt;br&gt;
The Wiki Loves Ramadan Organizing Team 05:11, 29 October 2024 (UTC)
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== Android app for Wikiversity ==

Hi, is there an Android app for Wikiversity? How does it work? I have been advised that there is no infrastructure for push notifications for Android apps for sister wikis and I would be interested to know more. Related: [[:phab:T378545]]. Thanks! [[User:Gryllida|Gryllida]] 23:15, 29 October 2024 (UTC)

:Thanks for suggesting this - I agree that it would be useful. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:56, 31 October 2024 (UTC)
:@[[User:Gryllida|Gryllida]]: Would you explain your terminology  for those of us not in the know. What does ''push notifications'' mean? I use [https://www.mediawiki.org/wiki/Help:Notifications notifications] when I am communicating on wikimedia projects, but have never heard this term before. [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:13, 11 December 2024 (UTC)

à

== Import Resource From Wikibooks? ==

Hello! [[wikibooks:Character_List_for_Baxter&amp;Sagart|Character List for Baxter&amp;Sagart]] and related titles [[wikibooks:Wikibooks:Requests_for_deletion#Character_List_for_Baxter&amp;Sagart|are up for deletion at Wikibooks]] because WB policy does not allow dictionaries like them. However, because they are useful as learning tools, I am wondering if they might have a home here at Wikiversity. Pinging @[[User:Tibetologist|Tibetologist]] here to link them in to this discussion, since they are the affected user. Thank you! —[[User:Kittycataclysm|Kittycataclysm]] ([[User talk:Kittycataclysm|discuss]] • [[Special:Contributions/Kittycataclysm|contribs]]) 18:18, 1 November 2024 (UTC)

:Sure, I can do it. That said, as mentioned there, it does seem like something like this is ideally suited for Wiktionary in the Appendix namespace, but I'm not very familiar with CJK characters and languages. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:23, 3 November 2024 (UTC)
::Oh man, these pages are too big to import and while I've already tried a half-dozen times, it will constantly fail. Strictly speaking, we don't have to use the import feature for licensing purposes. We can just copy and paste the contents and list the usernames or on the talk page. I think that's the solution. {{Ping|Tibetologist}}, are you interested in doing that? If you just copied and pasted these pages and then added [[:Category:Chinese]] and maybe include a couple of links to the pages, that would probably be ideal. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:31, 3 November 2024 (UTC)

== Language translation requests? ==

Is there anywhere on Wikiversity to request translation, for example, requesting Latin or French translation? I would be asking from the context as a student, so I would be interested in translation explanation as well. [[User:Indexcard88|Indexcard88]] ([[User talk:Indexcard88|discuss]] • [[Special:Contributions/Indexcard88|contribs]]) 04:56, 20 November 2024 (UTC)

== Sign up for the language community meeting on November 29th, 16:00 UTC ==

Hello everyone,

The next language community meeting is coming up next week, on November 29th, at 16:00 UTC (Zonestamp! For your timezone &lt;https://zonestamp.toolforge.org/1732896000&gt;). If you're interested in joining, you can sign up on this wiki page: &lt;https://www.mediawiki.org/wiki/Wikimedia_Language_and_Product_Localization/Community_meetings#29_November_2024&gt;.

This participant-driven meeting will be organized by the Wikimedia Foundation’s Language Product Localization team and the Language Diversity Hub. There will be presentations on topics like developing language keyboards, the creation of the Moore Wikipedia, and the language support track at Wiki Indaba. We will also have members from the Wayuunaiki community joining us to share their experiences with the Incubator and as a new community within our movement. This meeting will have a Spanish interpretation.

Looking forward to seeing you at the language community meeting! Cheers, [[User:SSethi (WMF)|Srishti]] 19:55, 21 November 2024 (UTC)
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== Events on Wikiversity ==

Since Wikipedia and Wikivoyage are having their &quot;Asian Month&quot; editathon, I was thinking if we could start up a Wikiversity version of that. This would be an &quot;Asian Month&quot; as well, but it would be about creating resources based on Asia and its culture. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 17:57, 6 December 2024 (UTC)

:Not immediately opposed, but the question is, do we have an active enough community to facilitate this? —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:31, 6 December 2024 (UTC)
::I'm not too sure. As long as we get enough traffic, this could happen. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 08:45, 7 December 2024 (UTC)
:::This is to increase traffic on Wikiversity, which is promoted amongst other communities. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 10:47, 7 December 2024 (UTC)
:Hi @[[User:RockTransport|RockTransport]], This is a good idea, but will it also involve users who are not &quot;professors and scientists&quot;. Just curious. cheers, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:30, 9 December 2024 (UTC)
::Yes, considering the fact that Wikiversity is for everyone, and not just for specific users. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 09:09, 10 December 2024 (UTC)
:::because I'm personally not a &quot;professor&quot; or a &quot;scientist&quot; and because '''anyone''' can create resources on Wikiversity. We want to make Wikiversity open for everyone, and not just for certain users. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 09:10, 10 December 2024 (UTC)
::::I am also not a professor or a scientist, but it seems to me that as result I am viewed here as a visitor rather than someone who can contribute. Just my $.02. [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:05, 12 December 2024 (UTC)
:I am affraid, that creation of educational resources on certain topic is way harder then wikipedia. Secondly while wikipedia stub does not matter, education resource stub is uselless completly. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:59, 17 December 2024 (UTC)

== Wikiversity - Newsletters ==

Hello All,

I wanted to create a newsletter on Wikiversity, which would highlight what is going on in certain months and events on Wikiversity; which would bolster engagement by many people. This would be on the website and would have its dedicated 'Newsletter' tab.

I hope you acknowledge this idea. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 21:05, 8 December 2024 (UTC)

:@[[User:RockTransport|RockTransport]], What sort of things do you plan to include in your newsletter? Will they be different than what is currently in [[Main Page/News]]? Just curious.
:I am also wondering about your motive which I think is: to bolster engagement by many people. I am asking because I wonder if others who are currently active here also think this I is desirable? Have you asked them? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:34, 11 December 2024 (UTC)
::Not yet, which was why I was asking this on the colloquium. I plan to include things that many people have created on Wikiversity over the month, as it is a monthly newsletter. It would be somewhere on the website here. It will be more frequent that the ones seen on [[Main Page/News]]. We will include people's resources to essentially promote them. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 06:50, 12 December 2024 (UTC)
:::@[[User:RockTransport|RockTransport]], I Think what you are saying is that ''Main Page/News'' does not update frequently enough? 
:::If this is the reason, why not start small by simply increasing the frequency of posting news on the main page, instead of trying to start a newsletter?
:::If there is more, can you articulate what else is missing. Thanks in advance, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:51, 12 December 2024 (UTC)
::::I meant going to detail into topics covered in that month, rather than just giving a few points. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 16:53, 12 December 2024 (UTC)
:::::What sort of details did you have in mind? You can pick one of the links provided in [[Main Page/News]] to illustrate. cheers, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 15:29, 16 December 2024 (UTC)
::::::I'm thinking of the community entering their projects, and discussing those in the newsletter. It depends on what they want, though. There would be a dedicated page for giving the information about their projects [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 17:24, 16 December 2024 (UTC)</text>
      <sha1>26u7tx3daruv4ueyyy39tl70jpjztc9</sha1>
    </revision>
    <revision>
      <id>2692262</id>
      <parentid>2692261</parentid>
      <timestamp>2024-12-17T10:01:18Z</timestamp>
      <contributor>
        <username>Juandev</username>
        <id>2651</id>
      </contributor>
      <comment>/* Android app for Wikiversity */ Reply</comment>
      <origin>2692262</origin>
      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="78176" sha1="ac7eeekjck0j4xof10eeziruso5pfh1" xml:space="preserve">{{Wikiversity:Colloquium/Header}}
&lt;!-- MESSAGES GO BELOW --&gt;
== Reminder! Vote closing soon to fill vacancies of the first U4C ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
:''[[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement – reminder to vote|You can find this message translated into additional languages on Meta-wiki.]] [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement – reminder to vote}}&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]''

Dear all,

The voting period for the Universal Code of Conduct Coordinating Committee (U4C) is closing soon. It is open through 10 August 2024. Read the information on [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2024_Special_Election#Voting|the voting page on Meta-wiki to learn more about voting and voter eligibility]]. If you are eligible to vote and have not voted in this special election, it is important that you vote now.

'''Why should you vote?''' The U4C is a global group dedicated to providing an equitable and consistent implementation of the UCoC. Community input into the committee membership is critical to the success of the UCoC.

Please share this message with members of your community so they can participate as well.

In cooperation with the U4C,&lt;section end=&quot;announcement-content&quot; /&gt;

-- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 15:30, 6 August 2024 (UTC)
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== User group for Wikiversians ==

Was there ever a discussion about the possibility of establishing a user group in the sense of an affiliated organization that would defend the interests of professors and scientists on Wikiversity and possibly actively develop some projects? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:21, 8 August 2024 (UTC)

:Not that I'm aware of. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:20, 8 August 2024 (UTC)
:It's a pleasure to talk to a scientist on Wikiversity. I am a historian of technics and I would like to publish the following biography either on Wikiversity or on Wikipedia: 
:https://en.wikiversity.org/wiki/User:Rbmn/Arthur_Constantin_KREBS_(1850-1935):_Military_engineer,_Automotive_industrialist,_Great_projects_manager
:What would be your advice? [[User:Rbmn|Rbmn]] ([[User talk:Rbmn|discuss]] • [[Special:Contributions/Rbmn|contribs]]) 15:44, 6 October 2024 (UTC)
::The content appears to be largely biographical/encyclopedic, so I think it is likely best suited to Wikipedia. Consider improving/incorporating this content into the existing page: [[w:Arthur Constantin Krebs]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:05, 7 October 2024 (UTC)
::Please do not link to the Wikiversity [[wv:userspace|Userspace]] in Wikipedia articles. You will want to wait until you have a page in the [[wv:mainspace|Wikiversity mainspace]]. You'll also want to use the &lt;code&gt;{{[[:w:Template:Wikiversity|Wikiversity]]}}&lt;/code&gt; template (on Wikipedia) rather than embedding a photo with a link. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:21, 7 October 2024 (UTC)
:I haven't heard anything about this topic. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 21:06, 8 December 2024 (UTC)

== Rich's ''Illustrated Companion'' at Wikiversity: Right place? ==

Hello! I am creating a Wiki-version of a classical glossary (''Illustrated Companion to the Latin Dictionary, and Greek Lexicon'' by Anthony Rich, 1849), which explains the meaning of Latin headwords, primarily those &quot;representing visible objects connected with the arts, manufactures, and every-day life of the Greeks and Romans.&quot; The aim is to help understand what a (classical) Latin text is actually about, instead of merely translating it. I already transcribed the entire text and scanned the images (about 1900) from an original 1849-edition. I am currently working on uploading the images to ''Mediawiki Commons'', which probably will take some time. In the meantime I want to prepare the other aspects of the project (more than 3000 articles, already with many internal links). The important thing: this is ''not'' a ''might exist''-project.  {{Color|red|My question: Is ''Wikiversity'' the proper place for it?}} Although I created an exact rendition of the original text, ''Wikisource'' is not applicable, because the project has a broader scope (adding content to the articles, e. g. links to online editions for quotations, adding images, but also adding entirely new articles). Neither is ''Wikibooks'', because this is not a textbook and may otherwise breach its scope. For more about the project see [[w:User:CalRis25/Temp-RICH-Prospectus|my user-page]] at en.wikipedia. {{Color|Red|So, is Wikiversity the right place for it?}} [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 09:15, 17 August 2024 (UTC)

:Thanks for asking. To be clear, it ''is'' acceptable to make [[:s:en:Category:Wikisource annotations|annotated editions]] of texts at Wikisource and Wikibooks does host at least one [[:b:en:Annotations of The Complete Peanuts|annotated guide to a copyright-protected work]]. So if what you're looking to do is to include inline annotations to a public domain text, you certainly can put that on Wikisource. If you have a textbook or guidebook that is a companion, that would go at Wikibooks. If you have some other kind of learning resources (like maintaining a list of relevant links, organizing a book reading group, etc.), that could go here. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:26, 17 August 2024 (UTC)
::Thank you for your quick answer. Actually, ''Wikibooks'' was my first thought. However, this project is not merely an annotated edition. Although at first it ''will'' be a faithful copy of the original text, I want the project to be &quot;open&quot;, i. e. adding articles should be possible. And the project should enable to do a lot more than mere inline annotation. See section [[w:User:CalRis25/Temp-RICH-Prospectus#Improving_RICH|Improving Rich]] in the project description a my user-page (en.Wikipedia). No ''Mediawiki''-project (Wikisource, Wikibooks, Wikipedia, Wiktionary) seemed to be a sufficiently applicable &quot;fit&quot; for the project, so I thought of Wikiversity as a last resort, because it is supposed to be home to all sorts of &quot;learning resources&quot;. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 09:57, 17 August 2024 (UTC)
:::The scope of Wikiversity ''is'' pretty catch-all and would allow for a pretty flexible place to host most learning resources that don't fit elsewhere.
:::Also, as nitpick, &quot;MediaWiki&quot; is the software that is the basis of these wikis (wikis being collections of interlinked documents that can be edited) and &quot;Wikimedia Foundation&quot; is the non-profit who owns the trademarks and hosts these projects like Wiktionary and Wikivoyage. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 10:06, 17 August 2024 (UTC)
::::Hello Justin, thank you for the reply. '''I think that settles it. I will create this project at ''Wikiversity''.''' Just for additional clarification, why I do so. Let's imagine a full transcription of the original 1849-edition of the ''Illustrated Companion'' by Anthony Rich and call it ''RICH-1849''. We shall call my project, for brevity sake, RICH-2K. And now, let's have a look at the article about the Roman toga (a piece of attire). In ''RICH-1849'' we can can call it ''RICH-1849/Toga'', and it contains ''exactly'' the content of the 1849-book. Now, let's look at the article ''RICH-2K/Toga''. At the beginning its only content would be the article ''RICH-1849/Toga''. Does that make ''RICH-2K/Toga'' and ''RICH-1849/Toga'' the same? Not at all, because in truth ''RICH-2K/Toga'' is a &quot;container&quot; which initially contains only the article ''RICH-1849/Toga'' but later on may include more stuff: images, external links, article text which builds on or extends ''RICH-1849/Toga'' and information from other sources of information (Wikipedia, specialized books). By the way, this added article information would not be a mere copy of the text at en.Wikipedia, because the information needs to looked at through the eyes of someone reading the original text (more citations with direct links to these etc.). [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 11:39, 17 August 2024 (UTC)

== Coming soon: A new sub-referencing feature – try it! ==

&lt;section begin=&quot;Sub-referencing&quot;/&gt;
[[File:Sub-referencing reuse visual.png|{{#ifeq:{{#dir}}|ltr|right|left}}|400px]]
Hello. For many years, community members have requested an easy way to re-use references with different details. Now, a MediaWiki solution is coming: The new sub-referencing feature will work for wikitext and Visual Editor and will enhance the existing reference system. You can continue to use different ways of referencing, but you will probably encounter sub-references in articles written by other users. More information on [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing|the project page]].

'''We want your feedback''' to make sure this feature works well for you:

* [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing#Test|Please try]] the current state of development on beta wiki and [[m:Talk:WMDE Technical Wishes/Sub-referencing|let us know what you think]]. 
* [[m:WMDE Technical Wishes/Sub-referencing/Sign-up|Sign up here]] to get updates and/or invites to participate in user research activities.

[[m:Special:MyLanguage/Wikimedia Deutschland|Wikimedia Deutschland]]’s [[m:Special:MyLanguage/WMDE Technical Wishes|Technical Wishes]] team is planning to bring this feature to Wikimedia wikis later this year. We will reach out to creators/maintainers of tools and templates related to references beforehand.

Please help us spread the message. --[[m:User:Johannes Richter (WMDE)|Johannes Richter (WMDE)]] ([[m:User talk:Johannes Richter (WMDE)|talk]]) 10:36, 19 August 2024 (UTC)
&lt;section end=&quot;Sub-referencing&quot;/&gt;
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== New [[Template:Form]] ==

Hi! Today I was bold and created [[Template:Form]] (which calls [[Module:WikiForm]] and [[MediaWiki:Gadget-WikiForm.js]]). The template allows to create user-friendly forms that can create pages or add content to existing pages. My motivation and first use case was [[Wikidebate/New|this form]] to create new [[wikidebates]], but I suspect the template can be useful elsewhere on Wikiversity. Let me know if you notice any issues or have any requests or concerns. Kind regards, [[User:Sophivorus|Sophivorus]] ([[User talk:Sophivorus|discuss]] • [[Special:Contributions/Sophivorus|contribs]]) 15:21, 21 August 2024 (UTC)

== Sign up for the language community meeting on August 30th, 15:00 UTC ==

Hi all,

The next language community meeting is scheduled in a few weeks—on August 30th at 15:00 UTC. If you're interested in joining, you can [https://www.mediawiki.org/wiki/Wikimedia_Language_and_Product_Localization/Community_meetings#30_August_2024 sign up on this wiki page].

This participant-driven meeting will focus on sharing language-specific updates related to various projects, discussing technical issues related to language wikis, and working together to find possible solutions. For example, in the last meeting, topics included the Language Converter, the state of language research, updates on the Incubator conversations, and technical challenges around external links not working with special characters on Bengali sites.

Do you have any ideas for topics to share technical updates or discuss challenges? Please add agenda items to the document [https://etherpad.wikimedia.org/p/language-community-meeting-aug-2024 here] and reach out to ssethi(__AT__)wikimedia.org. We look forward to your participation!

[[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 23:20, 22 August 2024 (UTC)
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== Template consolidation: User talk page block notice ==
Wondering if someone who likes templates could have a go at consolidating or helping decide between use of:
* [[Template:Block]]
* [[Template:Blocked]]

Unless I'm missing something, it seems like we don't need both?

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 07:16, 23 August 2024 (UTC)
: I tried to figure out a Wikidata item with most links to projects. I found this: [[Wikidata:Q6379131]], which is Template:Uw-block. There is even a corresponding Wikiversity template, [[Template:Uw-block1]] (not used anywhere).
: My impression is that of the three templates, we only need one. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:43, 13 September 2024 (UTC)

== Announcing the Universal Code of Conduct Coordinating Committee ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
:''[https://lists.wikimedia.org/hyperkitty/list/board-elections@lists.wikimedia.org/thread/OKCCN2CANIH2K7DXJOL2GPVDFWL27R7C/ Original message at wikimedia-l]. [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement - results|You can find this message translated into additional languages on Meta-wiki.]] [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement - results}}&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]''

Hello all,

The scrutineers have finished reviewing the vote and the [[m:Special:MyLanguage/Elections Committee|Elections Committee]] have certified the [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Results|results]] for the [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election|Universal Code of Conduct Coordinating Committee (U4C) special election]].

I am pleased to announce the following individual as regional members of the U4C, who will fulfill a term until 15 June 2026:

* North America (USA and Canada)
** Ajraddatz

The following seats were not filled during this special election:

* Latin America and Caribbean
* Central and East Europe (CEE)
* Sub-Saharan Africa
* South Asia
* The four remaining Community-At-Large seats

Thank you again to everyone who participated in this process and much appreciation to the candidates for your leadership and dedication to the Wikimedia movement and community.

Over the next few weeks, the U4C will begin meeting and planning the 2024-25 year in supporting the implementation and review of the UCoC and Enforcement Guidelines. You can follow their work on [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee|Meta-Wiki]].

On behalf of the U4C and the Elections Committee,&lt;section end=&quot;announcement-content&quot; /&gt;

[[m:User:RamzyM (WMF)|RamzyM (WMF)]] 14:07, 2 September 2024 (UTC)
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== Re: The Vector 2022 skin as the default in two weeks? ==

[[File:Vector 2022 video-en.webm|thumb|A two minute-long video about Vector 2022]]
Hello everyone, I'm reaching out on behalf of the [[mediawikiwiki:Reading/Web|Wikimedia Foundation Web team]] responsible for the MediaWiki skins. I'd like to revisit the topic of making Vector 2022 the default here on English Wikiversity. I [[Wikiversity:Colloquium/archives/September 2022#The Vector 2022 skin as the default in two weeks?|did post a message about this almost two years ago]] (where you can find all the details about the skin), but we didn't finalize it back then.

What happened in the meantime? We built [[mw:Reading/Web/Accessibility for reading|dark mode and different options for font sizes]], and made Vector 2022 the default on most wikis, including all other Wikiversities. With the not-so-new V22 skin being the default, existing and coming features, like dark mode and [[mw:Trust and Safety Product/Temporary Accounts|temporary accounts]] respectively, will become available for logged-out users here.

So, if no large concerns are raised, we will deploy Vector 2022 here in two weeks, in the week of September 16. Do let me know if you have any questions. Thank you! [[User:SGrabarczuk (WMF)|SGrabarczuk (WMF)]] ([[User talk:SGrabarczuk (WMF)|discuss]] • [[Special:Contributions/SGrabarczuk (WMF)|contribs]]) 21:48, 2 September 2024 (UTC)

:Sounds good, Szymon - we look forward to the upcoming change of skin {{smile}} Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 07:35, 13 September 2024 (UTC)
* I for one oppose a switch to Vector 2022. I do not find it preferable. Here is a staggering evidence of user refusal of Vector 2022 once it was deployed: [[W:en:Wikipedia:Requests for comment/Rollback of Vector 2022]], Junuary 2023. 355 voters supported rollback to Vector 2010 whereas 64 opposed, yielding 84.7% support, as clear a supermajority as one may wish. These people opposing Vector 2022 feel the same way as I do. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:48, 13 September 2024 (UTC)
*:Hey @[[User:Dan Polansky|Dan Polansky]]. Thanks for your comment. I'm open to discussion about problems with our software, and I hope we can maintain a respectful tone.
*:I understand that there are users who prefer Vector legacy or other skins, just as there are people who still stick to Monobook. Such people are active across many wikis. They can keep Vector legacy, although non-default skins don't have the support the default ones do. We are rolling out for technical reasons, as I mentioned above, with benefit to not logged-in users.
*:Regarding the rollback RfC on Wikipedia, two neutral users stated that there was no consensus for rollback, RfC is not a vote, and the numbers were different (355:226:24). I believe this all is pretty easy to verify.
*:So to sum up, Vector 2022 needs to become the default, tons and tons of comments were made about the skin and related stuff, and we have taken many ideas into account, and it's totally OK if you stick to Vector legacy.
*:Thanks! [[User:SGrabarczuk (WMF)|SGrabarczuk (WMF)]] ([[User talk:SGrabarczuk (WMF)|discuss]] • [[Special:Contributions/SGrabarczuk (WMF)|contribs]]) 19:30, 16 September 2024 (UTC)
*:: Today, I visited Wikiversity and found it switched to Vector 2022. I changed my preference settings to Vector 2010. From what I understand, non-registered visitors are now defaulted to Vector 2022 despite its unpopularity in [[W:en:Wikipedia:Requests for comment/Rollback of Vector 2022]]. I have not seen any evidence that users prefer Vector 2022, and given the evidence in the linked RfC, I tentatively conclude that the decision to switch has made the site experience worse for the majority of users. The logic of &quot;you can switch&quot; surely applies to Vector 2022 as well: those who prefer it can switch to it. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:08, 17 September 2024 (UTC)

== Have your say: Vote for the 2024 Board of Trustees! ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
Hello all,

The voting period for the [[m:Special:MyLanguage/Wikimedia Foundation elections/2024|2024 Board of Trustees election]] is now open. There are twelve (12) candidates running for four (4) seats on the Board.

Learn more about the candidates by [[m:Special:MyLanguage/Wikimedia Foundation elections/2024/Candidates|reading their statements]] and their [[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Questions_for_candidates|answers to community questions]].

When you are ready, go to the [[Special:SecurePoll/vote/400|SecurePoll]] voting page to vote. '''The vote is open from September 3rd at 00:00 UTC to September 17th at 23:59 UTC'''.

To check your voter eligibility, please visit the [[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Voter_eligibility_guidelines|voter eligibility page]].

Best regards,

The Elections Committee and Board Selection Working Group&lt;section end=&quot;announcement-content&quot; /&gt;

[[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 12:15, 3 September 2024 (UTC)
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== Separate page for hyperbola. ==

Good morning,

I notice that a search for &quot;hyperbola&quot; redirects to &quot;Conic sections&quot;.

At present there is a separate page for &quot;ellipse&quot;. Therefore a separate page for &quot;hyperbola&quot; seems to be justified.

Could this redirection be changed so that search for &quot;hyperbola&quot; goes to a separate page for &quot;hyperbola&quot;?

Many thanks,

[[User:ThaniosAkro|ThaniosAkro]] ([[User talk:ThaniosAkro|discuss]] • [[Special:Contributions/ThaniosAkro|contribs]]) 12:04, 15 September 2024 (UTC)

:It is true that ellipses are covered at [[Conic sections]] (along with hyperbolas, parabolas, etc.) and there is a separate page for [[ellipse]]s that elaborates. We certainly ''could'' have a page about [[hyperbola]]s that is separate, but no one has written sufficient content to spin it off yet. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 12:17, 15 September 2024 (UTC)

== I hereby request for your Unblocking IP address and just reviewed and received a reverted rec ==

Hi there. {{unsigned|Ishmael Raphasha}}
:No one has any clue what you're talking about. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:53, 18 September 2024 (UTC)

== RICH-2K: New project with some initial questions ==

Hello! I'm creating a new learning resource on ''Wikiversity''. The respective project is based on my transcription of a classical dictionary from 1849 by Anthony Rich. For more information about the project see its [[User:CalRis25/RICH: Description|description page]] (see also that page for why not ''Wikisource'' or ''Wikibooks''). The project's scope is fairly big: 3205 article-pages plus 304 REDIRECT-pages. The images (scanned by myself from an original copy) have been uploaded to ''Commons''. I have some initial technical questions (more of these and more detailed ones will follow):

* '''Upload''': Due to the large number of pages it is not realistic to create these manually. Is it possible to bulk-upload these in some way (the Wikitext of the pages is created using a Python-script with one file per article/page)? Is it possible to upload these to a test-environment first where any problems (hopefully none) can be identified and dealt with more easily than on the production-version of ''Wikiversity''?
* '''(Technical) Structure''': I am planning to set up this project at ''&lt;nowiki&gt;https://en.wikiversity.org/wiki/RICH-2K&lt;/nowiki&gt;'' as the main page and anything else as subpages: ''RICH-2K/Subpage_1 ... RICH-2K/Subpage_n''. However, these subpages fall into two categories: 1. Article-pages (content) and 2. Meta/Administrative pages. This project requires search capability restricted to the ''RICH-2K''-namespace. The ''Mediawiki''-software seems to supply a ''Search''-input field with the possibility to restrict the search to some namespace. I would like, however, to restrict the search further to the first group of pages, namely the articles. Is that possible, perhaps by use of (hidden) categories?
* '''External links''': This project will need many external links, and yes, I have read the relevant ''Wikiversity''-pages, but this specific project needs them. The ''Recommended Editions''-page (used for recommended online editions, to which to link when citing texts) alone probably will require several hundred external links. However, only relatively few [[w:Second-level domain|second-level domains]] will be involved, and most of these should be trustworthy (Perseus Digital library, digital collections of universities etc., in some cases, however, also ''Archive.org''). Perhaps there is a list of web-sites, for which external links are generally allowed? And who is allowed to create external links on ''Wikiversity''-pages (I haven't found the relevant policy)?
* '''Categories''': This project requires quite a few of its own categories, which belong to two large groups: 1. Categories (2 levels) of the ''Classed Index'' (about 170 categories), a thematic index of some (but not all) of the articles. 2. Administrative categories. Is there a recommended way to distinguish between different classes of categories within a project (category name or other method)? What about naming conventions for project-specific categories?

I am looking forward to your input. If you think that it's preferable we can move the discussions to the [[User_talk:CalRis25/RICH:_Description|Talk-page]] of the project's description. Thank you in advance. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 05:29, 20 September 2024 (UTC)

:*Admins have access to [[Special:Import]] and can bulk import XML pages. You can create pages in your sandbox if you'd like and make an indefinite amount of them at pages like [[User:CalRis25/sandbox]]. What can and cannot be hosted in user namespace is very loose, but still has to follow in principle Wikiversity's scope.
:*Using subpages is in principle a good way to organize these various resources. Please do not name them after a user name or something obscure. I personally think that &quot;RICH-2K&quot; is a not optimal name. I may recommend something like [[Anthony Rich Dictionary Project]] or [[21st-Century Anthony Rich Dictionary]] or something more obviously intelligible. While we have very few actual policies and guidelines, see [[Wikiversity:Naming conventions]] for a rough consensus of what is probably best practice for naming pages.
:*External linking generally does not use an allowed list (a.k.a. whitelist model), but a disallow (a.k.a. blacklist) model. See [[MediaWiki:Spam-blacklist]] and [[Special:BlockedExternalDomains]] (which is currently empty but is another method of listing blocked domains). It's perfectly fine to aggregate external links in learning resources.
:*I'm not 100% sure what the distinction is that you're drawing, but you can freely arrange categories underneath a main category that has the same name as your larger project. So, following the suggestions I gave, you could have a category like [[:Category:Anthony Rich Dictionary Project]] and then create any number of subcategories that logically help users navigate all these pages. Please make sure the main category you create is itself categorized under some relevant category(ies). If you need help, please ask.
:I think this answers your questions, please let me know if I'm unclear or you have more. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:11, 20 September 2024 (UTC)
::Hello Justin!
::* '''Upload:''' Creating the project in sandbox pages of my User-namespace defeats the purpose, as this is an ''open'' project. Also that would not solve, as such, the problem of having to manually create thousands of pages. I wonder, does ''Wikiversity'' support creation of pages using its API. ''Mediawiki's'' [[mw:API:Main_page|API-description]] seems to imply that it ought to be possible. If that's the case, I should be able to create a Python-script which automatically creates the pages (of course, a few trial pages first).
::* '''(Technical) Structure''': You may be right, here. RICH-2K is, for now, merely a technical name to make a clear but not too verbose distinction between the original text and the current project. I'll give this more thought.
::* '''External links''': I brought this up mainly because when I first edited my ''Wikiversity''-page, I got a message that I was not allowed to create external links. However, I just now tested creating an external link on my user-page and got no error, so this problem seems to be solved.
::* '''Categories''': I think I know what you mean. I'll create a category structure and maybe ask some specific questions once I am ready to do so.
::Thank you for your quick help. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 18:51, 20 September 2024 (UTC)
:::re: upload, I'm just suggesting your sandbox(es) as you asked about &quot;a test-environment&quot;. Anyone can edit someone else's sandboxes, but you typically defer to other users to control what's in their own subpages as a collegial thing. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:39, 23 September 2024 (UTC)

Hello! I have two further questions:
# I created a category-structure for the project. Could you (or someone else) have a look at it ([[User:CalRis25/RICH: Categories]]) and answer the questions in the section [[User:CalRis25/RICH:_Categories#Questions|Questions]]? I gave it some thought and believe that this would work fine for the project.
# ''Project boxes'' (see [[Help:Tour of project boxes]]): It is unclear to me, whether these belong only on the main page of the project (that makes the most sense to me), or on every single subpage.
Thanks in advance for your help. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 17:51, 24 September 2024 (UTC)

:To answer your questions here:
:*No, you are not contravening any policies we have.
:*A leading &quot;The&quot; is acceptable, but if you want it to sort alphabetically, you will have to use &lt;nowiki&gt;{{DEFAULTSORT:}}&lt;/nowiki&gt;. E.g. to get Category:The Best Stuff to sort under &quot;B&quot;, insert &quot;&lt;nowiki&gt;{{DEFAULTSORT:Best Stuff, The}}&lt;/nowiki&gt;.
:*Trailing &quot;etc.&quot; is acceptable.
:*An accent in a category title is acceptable.
:I'll also note that it looks like you have in mind some tracking categories that are redundant. Pages such as [[Special:LonelyPages]] and [[Special:DeadendPages]] already do automatically what you're proposing to do manually.
:As for project boxes, it's typically the case that the subjects are only placed on the main resource, but as you may imagine, [[Help:Tour of project boxes/1|status completion ones]] may vary from subpage to subpage. As with most things at Wikiversity, there are very few actual rules, so it's pretty much the wild west, even tho this project has been around for almost 20&amp;nbsp;years. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:18, 25 September 2024 (UTC)
::Hello Justin, thanks for the DEFAULTSORT-hint for categories beginning with ''The''. I will restrict the project boxes to the main page. As for the the orphaned/dead-end-categories, I prefer these to be project-specific. Once the project is up and running, putting articles &quot;on the map&quot; (making them accessible from other articles and creating links to other articles) is one of the first tasks to be dealt with. I already know which articles are involved and will add these categories to these articles. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 16:51, 25 September 2024 (UTC)

== Your wiki will be in read-only soon ==

&lt;section begin=&quot;server-switch&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;

[[:m:Special:MyLanguage/Tech/Server switch|Read this message in another language]] • [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-Tech%2FServer+switch&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]

The [[foundation:|Wikimedia Foundation]] will switch the traffic between its data centers. This will make sure that Wikipedia and the other Wikimedia wikis can stay online even after a disaster.

All traffic will switch on '''{{#time:j xg|2024-09-25|en}}'''. The switch will start at '''[https://zonestamp.toolforge.org/{{#time:U|2024-09-25T15:00|en}} {{#time:H:i e|2024-09-25T15:00}}]'''.

Unfortunately, because of some limitations in [[mw:Special:MyLanguage/Manual:What is MediaWiki?|MediaWiki]], all editing must stop while the switch is made. We apologize for this disruption, and we are working to minimize it in the future. 

A banner will be displayed on all wikis 30 minutes before this operation happens. This banner will remain visible until the end of the operation. 

'''You will be able to read, but not edit, all wikis for a short period of time.'''

*You will not be able to edit for up to an hour on {{#time:l j xg Y|2024-09-25|en}}.
*If you try to edit or save during these times, you will see an error message. We hope that no edits will be lost during these minutes, but we can't guarantee it. If you see the error message, then please wait until everything is back to normal. Then you should be able to save your edit. But, we recommend that you make a copy of your changes first, just in case.

''Other effects'':

*Background jobs will be slower and some may be dropped. Red links might not be updated as quickly as normal. If you create an article that is already linked somewhere else, the link will stay red longer than usual. Some long-running scripts will have to be stopped.
* We expect the code deployments to happen as any other week. However, some case-by-case code freezes could punctually happen if the operation require them afterwards.
* [[mw:Special:MyLanguage/GitLab|GitLab]] will be unavailable for about 90 minutes.

This project may be postponed if necessary. You can [[wikitech:Switch_Datacenter|read the schedule at wikitech.wikimedia.org]]. Any changes will be announced in the schedule. 

'''Please share this information with your community.'''&lt;/div&gt;&lt;section end=&quot;server-switch&quot;/&gt;

[[User:Trizek_(WMF)|Trizek_(WMF)]], 09:37, 20 September 2024 (UTC)
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== 'Wikidata item' link is moving. Find out where... ==

&lt;div lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot;&gt;&lt;i&gt;Apologies for cross-posting in English. Please consider translating this message.&lt;/i&gt;{{tracked|T66315}}

Hello everyone, a small change will soon be coming to the user-interface of your Wikimedia project.
The [[d:Q16222597|Wikidata item]] [[w:|sitelink]] currently found under the &lt;span style=&quot;color: #54595d;&quot;&gt;&lt;u&gt;''General''&lt;/u&gt;&lt;/span&gt; section of the '''Tools''' sidebar menu will move into the &lt;span style=&quot;color: #54595d;&quot;&gt;&lt;u&gt;''In Other Projects''&lt;/u&gt;&lt;/span&gt; section.

We would like the Wiki communities feedback so please let us know or ask questions on the   [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Discussion page]] before we enable the change which can take place October 4 2024, circa 15:00 UTC+2.
More information can be found on [[m:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|the project page]].&lt;br&gt;&lt;br&gt;We welcome your feedback and questions.&lt;br&gt; [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 18:56, 27 September 2024 (UTC)
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==Download as PDF==
[[Phabricator:T376438]]: &quot;Download to PDF&quot; on en.wv is returning error: &quot;{&quot;name&quot;:&quot;HTTPError&quot;,&quot;message&quot;:&quot;500&quot;,&quot;status&quot;:500,&quot;detail&quot;:&quot;Internal Server Error&quot;}&quot;

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:36, 3 October 2024 (UTC)

:I just downloaded this page as a PDF and it worked just fine. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:04, 3 October 2024 (UTC)

== Protected template bug for Pp ==

It seems that templates derivative of {{tlx|Pp}} (compiled in {{tlx|Protection templates}}) are being sorted into protection categories using the name 'Wikipedia' instead of 'Wikiversity' (e.g., [[:Category:Wikipedia pages with incorrect protection templates]]). From what I can tell, it is not in the publicly accessible source code of any of the templates. The only other impacted pages are modules which call {{tlx|pp}}-derivatives (e.g., [[Module:Navbar/styles.css]]).

This does not seem to affect any other pages in [[:Category:Wikiversity protected templates]]. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 18:59, 4 October 2024 (UTC)

:The problem is that &quot;Wikipedia&quot; is [https://en.wikiversity.org/w/index.php?title=Special%3ASearch&amp;limit=500&amp;offset=0&amp;ns828=1&amp;search=Wikipedia&amp;searchToken=9svkpqlxxoquoq7bnkt55ugts mentioned in several modules that were copied over from en.wp]; many of these are legit and many of them need to be replaced with &quot;Wikiversity&quot; ([https://en.wikiversity.org/w/index.php?title=Module%3APp-move-indef&amp;diff=2662815&amp;oldid=1944984 e.g.]) This particular change ''may'' fix all of these issues...? But 1.) it will take time to propagate across the site and 2.) there are still many more &quot;Wikipedia&quot;s that need to be changed, so I'll go thru a few more, but if you want to give me an assist, if you can just check this one week from now and ping me if the problem persists, that would be nice. Sometimes, I make calendar reminders to follow up on these, but I'm not a perfect person. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:55, 5 October 2024 (UTC)

== Invitation to Participate in Wiki Loves Ramadan Community Engagement Survey ==

Dear all,


We are excited to announce the upcoming [[m:Wiki Loves Ramadan|Wiki Loves Ramadan]] event, a global initiative aimed at celebrating Ramadan by enriching Wikipedia and its sister projects with content related to this significant time of year. As we plan to organize this event globally, your insights and experiences are crucial in shaping the best possible participation experience for the community.

To ensure that Wiki Loves Ramadan is engaging, inclusive, and impactful, we kindly invite you to participate in our community engagement survey. Your feedback will help us understand the needs of the community, set the event's focus, and guide our strategies for organizing this global event.

Survey link: https://forms.gle/f66MuzjcPpwzVymu5

Please take a few minutes to share your thoughts. Your input will make a difference!

Thank you for being a part of our journey to make Wiki Loves Ramadan a success.


Warm regards,

User:ZI Jony 03:19, 6 October 2024 (UTC)

Wiki Loves Ramadan Organizing Team
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== 'Edit to my talk page' notification bug? ==

This may belong at the bug tracker, but does anyone else have an issue disabling ''email'' notifications upon  an 'Edit to my talk page' in [[Special:GlobalPreferences]]? Oddly I ''am'' able to disable the global preference on Wikipedia, MediaWiki, etc, but not here. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 09:23, 7 October 2024 (UTC)

:I have not experienced this, but to be clear, do you also have the option to get emails when items on your talk page are edited turned on? —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:39, 7 October 2024 (UTC)
::The only (non-grayed out) options I have enabled for email are 'Failed login attempts' and 'Login from an unfamiliar device'. 'Edit to my talk page' re-checks after every save. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 09:54, 7 October 2024 (UTC)
:::That does sound like a [[phab:]] issue, with the caveat that I don't 100% recall how global preferences work and if they override local ones, etc. If you have parsed that and still have this issue, you'll probably need to file a ticket. Maybe someone else has this issue. Wish I could help. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:57, 7 October 2024 (UTC)
::::[[phab:T376601|Off 'n away]] 🫡 [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 10:35, 7 October 2024 (UTC)

== [[Portal:Computer Science]] ➝ [[Portal:Information sciences]] ==

Seeking consensus to complete the merge into the broader portal. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 06:28, 8 October 2024 (UTC)

:Why should it be merged? Computer Science seems well-enough designed. What is the incentive to collapse it into a broader field of study? —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:18, 8 October 2024 (UTC)
::Portals as top level organizations allowing for content to be best centralized. Also note that I did not start the merge, just offering to finish it. Perhaps a {{tlx|prod}} instead? [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 07:20, 8 October 2024 (UTC)
:::I have no objections, personally. If it gets done, please use a redirect and should someone want to come along to resurrect it later, it will be easier. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:21, 8 October 2024 (UTC)
: Is computer science really a branch of information sciences? I would not think so, but what do I know. Do we have some external resources/links confirm this idea? [[W:Information science]] currently says: &quot;Information science, documentology[1] or informatology[2][3] is an academic field which is primarily concerned with analysis, collection, classification, manipulation, storage, retrieval, movement, dissemination, and protection of information.&quot; --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:49, 11 October 2024 (UTC)
::Looking through [https://stackoverflow.com/q/1047014/22673230] [https://businessdegrees.uab.edu/mis-degree-bachelors/resources/computer-information-systems-vs-computer-science/] [https://www.si.umich.edu/student-experience/what-information-science] a few top (not necessarily RS) searches I'm inclined to agree. I am more familiar with the grafted [[:w:Information and computer science|information ''and'' computer science]] which makes an effort to merge the disciplines, but it does not seem like reaching to say that IS is presented as more applications-concerned (certainly with no lack of theoretical abstraction), whereas CS can be more freely associated with any and all 'science related to computers'. It is easy to reason about the connection between the fields, but I think it is clear academia maintains this taxonomy for a good reason.
::With these considerations, I think I will ''stop'' the process of merging in favor of expanding the existing [[School:Library and Information Science]]. 

::Let me know if there is not consensus to redirect [[Portal:Information sciences]] to [[School:Library and Information Science]] (with enough expansion it can generalize away from just library sciences). [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 16:16, 11 October 2024 (UTC)
::: I do not see that a merge of a ''portal'' to a ''school'' is a good thing. Do you have a clear idea of the concepts of school and portal and how they relate to each other? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:34, 11 October 2024 (UTC)
::::Found [[:Category:Information sciences]]; there are enough existing resources in there to make my other proposed merge excessive. I will simply continue developing the existing [[Portal:Information sciences]] instead. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 17:05, 11 October 2024 (UTC)
::::: Frankly, I would ideally see [[Portal:Information sciences]] deleted: I don't see what it does that a category would not do well enough. There does not seem to be any material specific to &quot;Information sciences&quot; (whatever that is) in that portal at all. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:11, 11 October 2024 (UTC)
::::::Tacked a {{tlx|prod}} for an eventual deletion, but I may still try to develop it as proof of concept at some point. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 17:33, 11 October 2024 (UTC)

== [[:Category:Occupational Epidemiology]] ==

I propose moving the pages in this category (without leaving redirects) to their equivalent under the parent resource [[Occupational Health Risk Surveillance]]. Also due to the number of subpages, it seems &lt;code&gt;|filing=deep&lt;/code&gt; would be a justified. (Also [[Special:PrefixIndex/Occupational_Epidemiology|there are quite a few]] untagged subpages.) [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 05:11, 9 October 2024 (UTC)
: I above all think that the content should be ''moved out of the mainspace'': I do not see readers learning anything from e.g. [[Occupational Epidemiology/Research tools/Reading of scientific articles for learning epidemiology and biostatstics]] or [[Occupational Epidemiology/Research tools/Ongoing projects/Risk Communication in Seafaring/Writing the article guideline IMRAD]]. Wikiversity can be kind enough to host that material in, say, subspace of [[User:Saltrabook]], but more should not be asked, I think. Let us recall that per [[WV:Deletions]], &quot;Resources may be eligible for proposed deletion when education objectives and learning outcomes are scarce, and objections to deletion are unlikely&quot;; I do not see how learning outcomes can be anything but scarce. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 15:04, 11 October 2024 (UTC)
::thank you, agree @ [[User:Saltrabook|Saltrabook]] ([[User talk:Saltrabook|discuss]] • [[Special:Contributions/Saltrabook|contribs]]) 21:03, 13 November 2024 (UTC)

== Active editors ==

It is interesting to observe the stats on [https://stats.wikimedia.org/#/en.wikiversity.org/contributing/active-editors/normal|line|all|(page_type)~content*non-content|monthly active editors] through the project's history. October is our month! [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 20:44, 8 October 2024 (UTC)

:Odd. Maybe related to the school year? —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 02:10, 9 October 2024 (UTC)
::I wonder how many are [[User:Jtneill|Jtneill]]'s crowd... the number is in the hundreds though, so that is one chunky cohort —[[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:16, 9 October 2024 (UTC)
:::Yes, [[Motivation and emotion/Book]] involves ~100-150 students editing most intensely during October each year. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:20, 9 October 2024 (UTC)
::::Neat, that still leaves around ~50-100 other students from other avenues each year since 2021. I also wonder which projects were involved in the COVID enrollment spike. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:26, 9 October 2024 (UTC)
:::::Personally I can admit that my editing is much more active during the school season vs. the summer break, so I'm in the same boat as Jtneill's students. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:24, 13 November 2024 (UTC)
:@[[User:Tule-hog|Tule-hog]] This is an interesting topic, but it is not clear to me as an outsider what you and other participants in this discussion find interesting. I  find this graph not very meaningful because it does not tell me if the number of  Active editors has gone up or down during the period covered, which I think was 2000-now. 
:I can see a big jump between 2000 and 2007, but what happened since then? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 15:45, 16 December 2024 (UTC)

== Intentionally incorrect resource ==

There is a [[Special:Diff/2583464|disclaimer inserted onto a resource]] (by not the original author) that:
&lt;blockquote&gt;I am merely [making this page false] to show you (The viewer) that Wikipedia and this page 'Wikiversity' is bull sh*t and it will not give you the reliability you need when writing an academic piece of writing.&lt;/blockquote&gt;
However, that IP has [[Special:Contributions/86.22.73.151|not made any other edits]], so unless they vandalized via a sock, the intent went un-realized and only that portion need be removed. Bumping here in case there is some obvious jumbo in that essay that someone else can catch. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 16:58, 9 October 2024 (UTC)

:Removed that portion, which was obviously vandalism. No perspective on the rest of the essay. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:38, 9 October 2024 (UTC)

== [[:Category:Proposed guidelines]] ==

Noting for future editors that WV has collapsed all proposals into [[:Category:Proposed policies|proposed policies]]. Seeking consensus to further collapse [[:Category:Wikiversity proposals]] into the former, or to restore [[:Category:Proposed guidelines]]. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 19:19, 9 October 2024 (UTC)

== [[Around Wikiversity in 80 Seconds|Broken 80-second tour]] ==

Bumping a [[Talk:Around_Wikiversity_in_80_Seconds|comment]] on the ''Wikiversity in 80 seconds'' tour. Appears wikisuite is not working with the Vector 2022 appearance. 

Also see [[:w:Wikipedia:Miscellany_for_deletion/Wikiversuite_pages|this thread]] on the Wikiversal package - may not be relevant to Wikiversity, but FYC. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 00:26, 10 October 2024 (UTC)
: I would just delete the material; I do not see value in it. If others agree, I would try to articulate why I think it should be deleted (or move to author user space). --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:57, 13 October 2024 (UTC)
::Just mark as {{tl|historical}}. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:39, 13 October 2024 (UTC)
::: This thing was created by [[User:Planotse]]. His creations are now being discussed in Wikibooks for deletion: [[B:Wikibooks:Requests for deletion#Wikiversal generated pages]]. It seems he used some kind of tool that is no longer available (the above mentioned &quot;Wikiversal&quot; package) to create this kind of slideshow-like material (believing the Wikibooks discussion). I do not see value of this in the mainspace, not even as historical (I am okay with userspace, but maybe even that is not the best option?). A look at the source code of [[Around Wikiversity in 80 Seconds/Introduction]] confirms the words of Omphalographer, namely that &quot;the HTML-heavy markup generated by Wikiversal makes them [the pages] unreasonably difficult to edit.&quot;
::: I went ahead and marked the page for proposed deletion. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:35, 14 October 2024 (UTC)

== Preliminary results of the 2024 Wikimedia Foundation Board of Trustees elections ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
Hello all,

Thank you to everyone who participated in the [[m:Special:MyLanguage/Wikimedia Foundation elections/2024|2024 Wikimedia Foundation Board of Trustees election]]. Close to 6000 community members from more than 180 wiki projects have voted.

The following four candidates were the most voted:

# [[User:Kritzolina|Christel Steigenberger]]
# [[User:Nadzik|Maciej Artur Nadzikiewicz]]
# [[User:Victoria|Victoria Doronina]]
# [[User:Laurentius|Lorenzo Losa]]

While these candidates have been ranked through the vote, they still need to be appointed to the Board of Trustees. They need to pass a successful background check and meet the qualifications outlined in the Bylaws. New trustees will be appointed at the next Board meeting in December 2024.

[[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Results|Learn more about the results on Meta-Wiki.]]

Best regards,

The Elections Committee and Board Selection Working Group

&lt;section end=&quot;announcement-content&quot; /&gt;

[[User:MPossoupe_(WMF)|MPossoupe_(WMF)]] 08:26, 14 October 2024 (UTC)
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== Seeking volunteers to join several of the movement’s committees ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
Each year, typically from October through December, several of the movement’s committees seek new volunteers. 

Read more about the committees on their Meta-wiki pages:
* [[m:Special:MyLanguage/Affiliations_Committee|Affiliations Committee (AffCom)]]
* [[m:Special:MyLanguage/Ombuds_commission|Ombuds commission (OC)]]
* [[m:Special:MyLanguage/Wikimedia Foundation/Legal/Community Resilience and Sustainability/Trust and Safety/Case Review Committee|Case Review Committee (CRC)]]

Applications for the committees open on 16 October 2024. Applications for the Affiliations Committee close on 18 November 2024, and applications for the Ombuds commission and the Case Review Committee close on 2 December 2024. Learn how to apply by [[m:Special:MyLanguage/Wikimedia_Foundation/Legal/Committee_appointments|visiting the appointment page on Meta-wiki]]. Post to the talk page or email [mailto:cst@wikimedia.org cst@wikimedia.org] with any questions you may have.

For the Committee Support team,
&lt;section end=&quot;announcement-content&quot; /&gt;

-- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 23:09, 16 October 2024 (UTC)
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== Interactive elements ==
Can we use interactive elements on Wikiversity? I'd like to add JavaScript to a page. If it's not possible now, where can I suggest this feature? I have a safe integration idea.
[[User:Отец Никифор|Отец Никифор]] ([[User talk:Отец Никифор|discuss]] • [[Special:Contributions/Отец Никифор|contribs]]) 12:10, 17 October 2024 (UTC)
: This is beyond my technical knowledge, but have you checked out:
:* https://www.mediawiki.org/wiki/Manual:Interface/JavaScript?
:* [[Wikipedia:WikiProject JavaScript]]
:* [[MediaWiki:Common.js]]

:What sort of interactive elements are you thinking about?

: Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:39, 18 October 2024 (UTC)
::I was thinking about adding something like a graph with adjustable controls, where users can interact with it and see how different changes affect the outcome. It seems like this could be a useful feature. There might already be discussions about enhancing Wikiversity or similar platforms—perhaps on a relevant talk page or in a Discord group. Do you know where such discussions might be happening? [[User:Отец Никифор|Отец Никифор]] ([[User talk:Отец Никифор|discuss]] • [[Special:Contributions/Отец Никифор|contribs]]) 19:47, 18 October 2024 (UTC)
:::From a quick look, maybe check out:
:::* [[mw:Extension:Graph]]
:::* [[phab:tag/graphs]]
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:40, 18 October 2024 (UTC)
:::: mw:Extension:Graph is currently disabled on Wikipedia etc. wikis, for security reasons, and seems unlikely to be enabled again. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:30, 19 October 2024 (UTC)

== An unexplained spurt of Wikiversity page views ==
The [https://pageviews.wmcloud.org/siteviews/?platform=all-access&amp;source=pageviews&amp;agent=user&amp;start=2024-06-01&amp;end=2024-10-18&amp;sites=en.wikiversity.org|en.wikibooks.org|en.wikiquote.org|en.wikisource.org page view report] shows an unexplained spurt of Wikiversity page views, reaching over 4 times the baseline and then falling back again. Does anyone have any idea what is going on? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:01, 19 October 2024 (UTC)

:Interesting. I wonder why only the English wikiquote and wikiversity and not Wikisource or wikibooks? How reliable do you think those stats are? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 15:44, 8 December 2024 (UTC)

== Center tempate failed on a contributors phone... ==

See the edit comment here - https://en.wikiversity.org/w/index.php?title=Wikiphilosophers&amp;diff=prev&amp;oldid=2673962. I'm puzzled as this is the first failure of this, I've noted recently. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 08:45, 19 October 2024 (UTC)

== Essay-like page in user space that makes little sense and seems incoherent ==
The page [[User:TheoYalur/Illusions]] seems to match the description, at least by my assessment. My understanding is that since the page is only in user space and not in the mainspace, it can stay there even if it has those disqualifying qualities. But if I am wrong and the page belongs deleted, please correct me and let me know. I do not know which policy or guideline, if any, guides the case. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:30, 21 October 2024 (UTC)

== 'Wikidata item' link is moving, finally. ==

Hello everyone, I previously wrote on the 27th September to advise that the ''Wikidata item'' sitelink will change places in the sidebar menu, moving from the '''General''' section into the '''In Other Projects''' section. The scheduled rollout date of 04.10.2024 was delayed due to a necessary request for Mobile/MinervaNeue skin. I am happy to inform that the global rollout can now proceed and will occur later today, 22.10.2024 at 15:00 UTC-2. [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Please let us know]] if you notice any problems or bugs after this change. There should be no need for null-edits or purging cache for the changes to occur. Kind regards, -[[m:User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] 11:28, 22 October 2024 (UTC)
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:Hi @[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]]: I Just noticed your post above, and it is timely.
:I have been participating in the English WikiUniversity for a few years, much less often recently. I seems like something in the way the site displays is different, but I cannot put my finger on it. Your posting gave me a clue. Can you please tell me where  the link to wikidata items has moved to? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:23, 11 December 2024 (UTC)
::Hello @[[User:Ottawahitech|Ottawahitech]], sure, I would be happy to. The button/sitelink name didn't change, just its position. You should find it in the sidebar-menu under the section '''In other projects''' (where the links to all other Wikimedia Projects are displayed). If you do not see it, please reach out to us on the [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Move Wikidata item - Discussion page]]. Thank you, -[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] ([[User talk:Danny Benjafield (WMDE)|discuss]] • [[Special:Contributions/Danny Benjafield (WMDE)|contribs]]) 09:24, 12 December 2024 (UTC)
:::@[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]], thank you for responding. I intend to followup on the ''Move Wikidata item - Discussion page'' as per your post above by putting it on my ever growing todo list.
:::I don't know about others on this wiki, as I said I have not been visiting here frequently, but for me the constant changes are a big distraction. I have been around wikimedia projects since 2007, so why do I have to spend so much time learning and re-learning how to find what I came here for? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:41, 12 December 2024 (UTC)
::::Hi @[[User:Ottawahitech|Ottawahitech]], thanks for you thoughts. Your input whether positive or critical helps us understand the impacts to editors so we welcome your further thoughts when you reach us in your To Do List :)
::::I can't speak about the other changes you've experienced here but I do hope they are made with a spirit of improvement for the community as a whole. -[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] ([[User talk:Danny Benjafield (WMDE)|discuss]] • [[Special:Contributions/Danny Benjafield (WMDE)|contribs]]) 10:43, 16 December 2024 (UTC)

== Final Reminder: Join us in Making Wiki Loves Ramadan Success ==

Dear all,

We’re thrilled to announce the Wiki Loves Ramadan event, a global initiative to celebrate Ramadan by enhancing Wikipedia and its sister projects with valuable content related to this special time of year. As we organize this event globally, we need your valuable input to make it a memorable experience for the community.

Last Call to Participate in Our Survey: To ensure that Wiki Loves Ramadan is inclusive and impactful, we kindly request you to complete our community engagement survey. Your feedback will shape the event’s focus and guide our organizing strategies to better meet community needs.

* Survey Link: [https://docs.google.com/forms/d/e/1FAIpQLSffN4prPtR5DRSq9nH-t1z8hG3jZFBbySrv32YoxV8KbTwxig/viewform?usp=sf_link Complete the Survey]
* Deadline: November 10, 2024
Please take a few minutes to share your thoughts. Your input will truly make a difference!

'''Volunteer Opportunity''': Join the Wiki Loves Ramadan Team! We’re seeking dedicated volunteers for key team roles essential to the success of this initiative. If you’re interested in volunteer roles, we invite you to apply.
* Application Link: [https://docs.google.com/forms/d/e/1FAIpQLSfXiox_eEDH4yJ0gxVBgtL7jPe41TINAWYtpNp1JHSk8zhdgw/viewform?usp=sf_link Apply Here]
* Application Deadline: October 31, 2024
Explore Open Positions: For a detailed list of roles and their responsibilities, please refer to the position descriptions here: [https://docs.google.com/document/d/1oy0_tilC6kow5GGf6cEuFvdFpekcubCqJlaxkxh-jT4/ Position Descriptions]

Thank you for being part of this journey. We look forward to working together to make Wiki Loves Ramadan a success!


Warm regards,&lt;br&gt;
The Wiki Loves Ramadan Organizing Team 05:11, 29 October 2024 (UTC)
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== Android app for Wikiversity ==

Hi, is there an Android app for Wikiversity? How does it work? I have been advised that there is no infrastructure for push notifications for Android apps for sister wikis and I would be interested to know more. Related: [[:phab:T378545]]. Thanks! [[User:Gryllida|Gryllida]] 23:15, 29 October 2024 (UTC)

:Thanks for suggesting this - I agree that it would be useful. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:56, 31 October 2024 (UTC)
:@[[User:Gryllida|Gryllida]]: Would you explain your terminology  for those of us not in the know. What does ''push notifications'' mean? I use [https://www.mediawiki.org/wiki/Help:Notifications notifications] when I am communicating on wikimedia projects, but have never heard this term before. [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:13, 11 December 2024 (UTC)
:I dont think there is an app.  [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:01, 17 December 2024 (UTC)

à

== Import Resource From Wikibooks? ==

Hello! [[wikibooks:Character_List_for_Baxter&amp;Sagart|Character List for Baxter&amp;Sagart]] and related titles [[wikibooks:Wikibooks:Requests_for_deletion#Character_List_for_Baxter&amp;Sagart|are up for deletion at Wikibooks]] because WB policy does not allow dictionaries like them. However, because they are useful as learning tools, I am wondering if they might have a home here at Wikiversity. Pinging @[[User:Tibetologist|Tibetologist]] here to link them in to this discussion, since they are the affected user. Thank you! —[[User:Kittycataclysm|Kittycataclysm]] ([[User talk:Kittycataclysm|discuss]] • [[Special:Contributions/Kittycataclysm|contribs]]) 18:18, 1 November 2024 (UTC)

:Sure, I can do it. That said, as mentioned there, it does seem like something like this is ideally suited for Wiktionary in the Appendix namespace, but I'm not very familiar with CJK characters and languages. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:23, 3 November 2024 (UTC)
::Oh man, these pages are too big to import and while I've already tried a half-dozen times, it will constantly fail. Strictly speaking, we don't have to use the import feature for licensing purposes. We can just copy and paste the contents and list the usernames or on the talk page. I think that's the solution. {{Ping|Tibetologist}}, are you interested in doing that? If you just copied and pasted these pages and then added [[:Category:Chinese]] and maybe include a couple of links to the pages, that would probably be ideal. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:31, 3 November 2024 (UTC)

== Language translation requests? ==

Is there anywhere on Wikiversity to request translation, for example, requesting Latin or French translation? I would be asking from the context as a student, so I would be interested in translation explanation as well. [[User:Indexcard88|Indexcard88]] ([[User talk:Indexcard88|discuss]] • [[Special:Contributions/Indexcard88|contribs]]) 04:56, 20 November 2024 (UTC)

== Sign up for the language community meeting on November 29th, 16:00 UTC ==

Hello everyone,

The next language community meeting is coming up next week, on November 29th, at 16:00 UTC (Zonestamp! For your timezone &lt;https://zonestamp.toolforge.org/1732896000&gt;). If you're interested in joining, you can sign up on this wiki page: &lt;https://www.mediawiki.org/wiki/Wikimedia_Language_and_Product_Localization/Community_meetings#29_November_2024&gt;.

This participant-driven meeting will be organized by the Wikimedia Foundation’s Language Product Localization team and the Language Diversity Hub. There will be presentations on topics like developing language keyboards, the creation of the Moore Wikipedia, and the language support track at Wiki Indaba. We will also have members from the Wayuunaiki community joining us to share their experiences with the Incubator and as a new community within our movement. This meeting will have a Spanish interpretation.

Looking forward to seeing you at the language community meeting! Cheers, [[User:SSethi (WMF)|Srishti]] 19:55, 21 November 2024 (UTC)
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== Events on Wikiversity ==

Since Wikipedia and Wikivoyage are having their &quot;Asian Month&quot; editathon, I was thinking if we could start up a Wikiversity version of that. This would be an &quot;Asian Month&quot; as well, but it would be about creating resources based on Asia and its culture. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 17:57, 6 December 2024 (UTC)

:Not immediately opposed, but the question is, do we have an active enough community to facilitate this? —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:31, 6 December 2024 (UTC)
::I'm not too sure. As long as we get enough traffic, this could happen. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 08:45, 7 December 2024 (UTC)
:::This is to increase traffic on Wikiversity, which is promoted amongst other communities. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 10:47, 7 December 2024 (UTC)
:Hi @[[User:RockTransport|RockTransport]], This is a good idea, but will it also involve users who are not &quot;professors and scientists&quot;. Just curious. cheers, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:30, 9 December 2024 (UTC)
::Yes, considering the fact that Wikiversity is for everyone, and not just for specific users. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 09:09, 10 December 2024 (UTC)
:::because I'm personally not a &quot;professor&quot; or a &quot;scientist&quot; and because '''anyone''' can create resources on Wikiversity. We want to make Wikiversity open for everyone, and not just for certain users. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 09:10, 10 December 2024 (UTC)
::::I am also not a professor or a scientist, but it seems to me that as result I am viewed here as a visitor rather than someone who can contribute. Just my $.02. [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:05, 12 December 2024 (UTC)
:I am affraid, that creation of educational resources on certain topic is way harder then wikipedia. Secondly while wikipedia stub does not matter, education resource stub is uselless completly. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:59, 17 December 2024 (UTC)

== Wikiversity - Newsletters ==

Hello All,

I wanted to create a newsletter on Wikiversity, which would highlight what is going on in certain months and events on Wikiversity; which would bolster engagement by many people. This would be on the website and would have its dedicated 'Newsletter' tab.

I hope you acknowledge this idea. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 21:05, 8 December 2024 (UTC)

:@[[User:RockTransport|RockTransport]], What sort of things do you plan to include in your newsletter? Will they be different than what is currently in [[Main Page/News]]? Just curious.
:I am also wondering about your motive which I think is: to bolster engagement by many people. I am asking because I wonder if others who are currently active here also think this I is desirable? Have you asked them? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:34, 11 December 2024 (UTC)
::Not yet, which was why I was asking this on the colloquium. I plan to include things that many people have created on Wikiversity over the month, as it is a monthly newsletter. It would be somewhere on the website here. It will be more frequent that the ones seen on [[Main Page/News]]. We will include people's resources to essentially promote them. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 06:50, 12 December 2024 (UTC)
:::@[[User:RockTransport|RockTransport]], I Think what you are saying is that ''Main Page/News'' does not update frequently enough? 
:::If this is the reason, why not start small by simply increasing the frequency of posting news on the main page, instead of trying to start a newsletter?
:::If there is more, can you articulate what else is missing. Thanks in advance, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:51, 12 December 2024 (UTC)
::::I meant going to detail into topics covered in that month, rather than just giving a few points. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 16:53, 12 December 2024 (UTC)
:::::What sort of details did you have in mind? You can pick one of the links provided in [[Main Page/News]] to illustrate. cheers, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 15:29, 16 December 2024 (UTC)
::::::I'm thinking of the community entering their projects, and discussing those in the newsletter. It depends on what they want, though. There would be a dedicated page for giving the information about their projects [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 17:24, 16 December 2024 (UTC)</text>
      <sha1>ac7eeekjck0j4xof10eeziruso5pfh1</sha1>
    </revision>
    <revision>
      <id>2692263</id>
      <parentid>2692262</parentid>
      <timestamp>2024-12-17T10:02:56Z</timestamp>
      <contributor>
        <username>Juandev</username>
        <id>2651</id>
      </contributor>
      <comment>/* An unexplained spurt of Wikiversity page views */ Reply</comment>
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      <text bytes="78414" sha1="b34tlib6gxejn2udrvf8btq8twjtorz" xml:space="preserve">{{Wikiversity:Colloquium/Header}}
&lt;!-- MESSAGES GO BELOW --&gt;
== Reminder! Vote closing soon to fill vacancies of the first U4C ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
:''[[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement – reminder to vote|You can find this message translated into additional languages on Meta-wiki.]] [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement – reminder to vote}}&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]''

Dear all,

The voting period for the Universal Code of Conduct Coordinating Committee (U4C) is closing soon. It is open through 10 August 2024. Read the information on [[m:Special:MyLanguage/Universal_Code_of_Conduct/Coordinating_Committee/Election/2024_Special_Election#Voting|the voting page on Meta-wiki to learn more about voting and voter eligibility]]. If you are eligible to vote and have not voted in this special election, it is important that you vote now.

'''Why should you vote?''' The U4C is a global group dedicated to providing an equitable and consistent implementation of the UCoC. Community input into the committee membership is critical to the success of the UCoC.

Please share this message with members of your community so they can participate as well.

In cooperation with the U4C,&lt;section end=&quot;announcement-content&quot; /&gt;

-- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 15:30, 6 August 2024 (UTC)
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== User group for Wikiversians ==

Was there ever a discussion about the possibility of establishing a user group in the sense of an affiliated organization that would defend the interests of professors and scientists on Wikiversity and possibly actively develop some projects? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 20:21, 8 August 2024 (UTC)

:Not that I'm aware of. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 23:20, 8 August 2024 (UTC)
:It's a pleasure to talk to a scientist on Wikiversity. I am a historian of technics and I would like to publish the following biography either on Wikiversity or on Wikipedia: 
:https://en.wikiversity.org/wiki/User:Rbmn/Arthur_Constantin_KREBS_(1850-1935):_Military_engineer,_Automotive_industrialist,_Great_projects_manager
:What would be your advice? [[User:Rbmn|Rbmn]] ([[User talk:Rbmn|discuss]] • [[Special:Contributions/Rbmn|contribs]]) 15:44, 6 October 2024 (UTC)
::The content appears to be largely biographical/encyclopedic, so I think it is likely best suited to Wikipedia. Consider improving/incorporating this content into the existing page: [[w:Arthur Constantin Krebs]]. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:05, 7 October 2024 (UTC)
::Please do not link to the Wikiversity [[wv:userspace|Userspace]] in Wikipedia articles. You will want to wait until you have a page in the [[wv:mainspace|Wikiversity mainspace]]. You'll also want to use the &lt;code&gt;{{[[:w:Template:Wikiversity|Wikiversity]]}}&lt;/code&gt; template (on Wikipedia) rather than embedding a photo with a link. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:21, 7 October 2024 (UTC)
:I haven't heard anything about this topic. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 21:06, 8 December 2024 (UTC)

== Rich's ''Illustrated Companion'' at Wikiversity: Right place? ==

Hello! I am creating a Wiki-version of a classical glossary (''Illustrated Companion to the Latin Dictionary, and Greek Lexicon'' by Anthony Rich, 1849), which explains the meaning of Latin headwords, primarily those &quot;representing visible objects connected with the arts, manufactures, and every-day life of the Greeks and Romans.&quot; The aim is to help understand what a (classical) Latin text is actually about, instead of merely translating it. I already transcribed the entire text and scanned the images (about 1900) from an original 1849-edition. I am currently working on uploading the images to ''Mediawiki Commons'', which probably will take some time. In the meantime I want to prepare the other aspects of the project (more than 3000 articles, already with many internal links). The important thing: this is ''not'' a ''might exist''-project.  {{Color|red|My question: Is ''Wikiversity'' the proper place for it?}} Although I created an exact rendition of the original text, ''Wikisource'' is not applicable, because the project has a broader scope (adding content to the articles, e. g. links to online editions for quotations, adding images, but also adding entirely new articles). Neither is ''Wikibooks'', because this is not a textbook and may otherwise breach its scope. For more about the project see [[w:User:CalRis25/Temp-RICH-Prospectus|my user-page]] at en.wikipedia. {{Color|Red|So, is Wikiversity the right place for it?}} [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 09:15, 17 August 2024 (UTC)

:Thanks for asking. To be clear, it ''is'' acceptable to make [[:s:en:Category:Wikisource annotations|annotated editions]] of texts at Wikisource and Wikibooks does host at least one [[:b:en:Annotations of The Complete Peanuts|annotated guide to a copyright-protected work]]. So if what you're looking to do is to include inline annotations to a public domain text, you certainly can put that on Wikisource. If you have a textbook or guidebook that is a companion, that would go at Wikibooks. If you have some other kind of learning resources (like maintaining a list of relevant links, organizing a book reading group, etc.), that could go here. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:26, 17 August 2024 (UTC)
::Thank you for your quick answer. Actually, ''Wikibooks'' was my first thought. However, this project is not merely an annotated edition. Although at first it ''will'' be a faithful copy of the original text, I want the project to be &quot;open&quot;, i. e. adding articles should be possible. And the project should enable to do a lot more than mere inline annotation. See section [[w:User:CalRis25/Temp-RICH-Prospectus#Improving_RICH|Improving Rich]] in the project description a my user-page (en.Wikipedia). No ''Mediawiki''-project (Wikisource, Wikibooks, Wikipedia, Wiktionary) seemed to be a sufficiently applicable &quot;fit&quot; for the project, so I thought of Wikiversity as a last resort, because it is supposed to be home to all sorts of &quot;learning resources&quot;. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 09:57, 17 August 2024 (UTC)
:::The scope of Wikiversity ''is'' pretty catch-all and would allow for a pretty flexible place to host most learning resources that don't fit elsewhere.
:::Also, as nitpick, &quot;MediaWiki&quot; is the software that is the basis of these wikis (wikis being collections of interlinked documents that can be edited) and &quot;Wikimedia Foundation&quot; is the non-profit who owns the trademarks and hosts these projects like Wiktionary and Wikivoyage. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 10:06, 17 August 2024 (UTC)
::::Hello Justin, thank you for the reply. '''I think that settles it. I will create this project at ''Wikiversity''.''' Just for additional clarification, why I do so. Let's imagine a full transcription of the original 1849-edition of the ''Illustrated Companion'' by Anthony Rich and call it ''RICH-1849''. We shall call my project, for brevity sake, RICH-2K. And now, let's have a look at the article about the Roman toga (a piece of attire). In ''RICH-1849'' we can can call it ''RICH-1849/Toga'', and it contains ''exactly'' the content of the 1849-book. Now, let's look at the article ''RICH-2K/Toga''. At the beginning its only content would be the article ''RICH-1849/Toga''. Does that make ''RICH-2K/Toga'' and ''RICH-1849/Toga'' the same? Not at all, because in truth ''RICH-2K/Toga'' is a &quot;container&quot; which initially contains only the article ''RICH-1849/Toga'' but later on may include more stuff: images, external links, article text which builds on or extends ''RICH-1849/Toga'' and information from other sources of information (Wikipedia, specialized books). By the way, this added article information would not be a mere copy of the text at en.Wikipedia, because the information needs to looked at through the eyes of someone reading the original text (more citations with direct links to these etc.). [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 11:39, 17 August 2024 (UTC)

== Coming soon: A new sub-referencing feature – try it! ==

&lt;section begin=&quot;Sub-referencing&quot;/&gt;
[[File:Sub-referencing reuse visual.png|{{#ifeq:{{#dir}}|ltr|right|left}}|400px]]
Hello. For many years, community members have requested an easy way to re-use references with different details. Now, a MediaWiki solution is coming: The new sub-referencing feature will work for wikitext and Visual Editor and will enhance the existing reference system. You can continue to use different ways of referencing, but you will probably encounter sub-references in articles written by other users. More information on [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing|the project page]].

'''We want your feedback''' to make sure this feature works well for you:

* [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing#Test|Please try]] the current state of development on beta wiki and [[m:Talk:WMDE Technical Wishes/Sub-referencing|let us know what you think]]. 
* [[m:WMDE Technical Wishes/Sub-referencing/Sign-up|Sign up here]] to get updates and/or invites to participate in user research activities.

[[m:Special:MyLanguage/Wikimedia Deutschland|Wikimedia Deutschland]]’s [[m:Special:MyLanguage/WMDE Technical Wishes|Technical Wishes]] team is planning to bring this feature to Wikimedia wikis later this year. We will reach out to creators/maintainers of tools and templates related to references beforehand.

Please help us spread the message. --[[m:User:Johannes Richter (WMDE)|Johannes Richter (WMDE)]] ([[m:User talk:Johannes Richter (WMDE)|talk]]) 10:36, 19 August 2024 (UTC)
&lt;section end=&quot;Sub-referencing&quot;/&gt;
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== New [[Template:Form]] ==

Hi! Today I was bold and created [[Template:Form]] (which calls [[Module:WikiForm]] and [[MediaWiki:Gadget-WikiForm.js]]). The template allows to create user-friendly forms that can create pages or add content to existing pages. My motivation and first use case was [[Wikidebate/New|this form]] to create new [[wikidebates]], but I suspect the template can be useful elsewhere on Wikiversity. Let me know if you notice any issues or have any requests or concerns. Kind regards, [[User:Sophivorus|Sophivorus]] ([[User talk:Sophivorus|discuss]] • [[Special:Contributions/Sophivorus|contribs]]) 15:21, 21 August 2024 (UTC)

== Sign up for the language community meeting on August 30th, 15:00 UTC ==

Hi all,

The next language community meeting is scheduled in a few weeks—on August 30th at 15:00 UTC. If you're interested in joining, you can [https://www.mediawiki.org/wiki/Wikimedia_Language_and_Product_Localization/Community_meetings#30_August_2024 sign up on this wiki page].

This participant-driven meeting will focus on sharing language-specific updates related to various projects, discussing technical issues related to language wikis, and working together to find possible solutions. For example, in the last meeting, topics included the Language Converter, the state of language research, updates on the Incubator conversations, and technical challenges around external links not working with special characters on Bengali sites.

Do you have any ideas for topics to share technical updates or discuss challenges? Please add agenda items to the document [https://etherpad.wikimedia.org/p/language-community-meeting-aug-2024 here] and reach out to ssethi(__AT__)wikimedia.org. We look forward to your participation!

[[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 23:20, 22 August 2024 (UTC)
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== Template consolidation: User talk page block notice ==
Wondering if someone who likes templates could have a go at consolidating or helping decide between use of:
* [[Template:Block]]
* [[Template:Blocked]]

Unless I'm missing something, it seems like we don't need both?

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 07:16, 23 August 2024 (UTC)
: I tried to figure out a Wikidata item with most links to projects. I found this: [[Wikidata:Q6379131]], which is Template:Uw-block. There is even a corresponding Wikiversity template, [[Template:Uw-block1]] (not used anywhere).
: My impression is that of the three templates, we only need one. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:43, 13 September 2024 (UTC)

== Announcing the Universal Code of Conduct Coordinating Committee ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
:''[https://lists.wikimedia.org/hyperkitty/list/board-elections@lists.wikimedia.org/thread/OKCCN2CANIH2K7DXJOL2GPVDFWL27R7C/ Original message at wikimedia-l]. [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement - results|You can find this message translated into additional languages on Meta-wiki.]] [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-{{urlencode:Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Announcement - results}}&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]''

Hello all,

The scrutineers have finished reviewing the vote and the [[m:Special:MyLanguage/Elections Committee|Elections Committee]] have certified the [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election/Results|results]] for the [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee/Election/2024 Special Election|Universal Code of Conduct Coordinating Committee (U4C) special election]].

I am pleased to announce the following individual as regional members of the U4C, who will fulfill a term until 15 June 2026:

* North America (USA and Canada)
** Ajraddatz

The following seats were not filled during this special election:

* Latin America and Caribbean
* Central and East Europe (CEE)
* Sub-Saharan Africa
* South Asia
* The four remaining Community-At-Large seats

Thank you again to everyone who participated in this process and much appreciation to the candidates for your leadership and dedication to the Wikimedia movement and community.

Over the next few weeks, the U4C will begin meeting and planning the 2024-25 year in supporting the implementation and review of the UCoC and Enforcement Guidelines. You can follow their work on [[m:Special:MyLanguage/Universal Code of Conduct/Coordinating Committee|Meta-Wiki]].

On behalf of the U4C and the Elections Committee,&lt;section end=&quot;announcement-content&quot; /&gt;

[[m:User:RamzyM (WMF)|RamzyM (WMF)]] 14:07, 2 September 2024 (UTC)
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== Re: The Vector 2022 skin as the default in two weeks? ==

[[File:Vector 2022 video-en.webm|thumb|A two minute-long video about Vector 2022]]
Hello everyone, I'm reaching out on behalf of the [[mediawikiwiki:Reading/Web|Wikimedia Foundation Web team]] responsible for the MediaWiki skins. I'd like to revisit the topic of making Vector 2022 the default here on English Wikiversity. I [[Wikiversity:Colloquium/archives/September 2022#The Vector 2022 skin as the default in two weeks?|did post a message about this almost two years ago]] (where you can find all the details about the skin), but we didn't finalize it back then.

What happened in the meantime? We built [[mw:Reading/Web/Accessibility for reading|dark mode and different options for font sizes]], and made Vector 2022 the default on most wikis, including all other Wikiversities. With the not-so-new V22 skin being the default, existing and coming features, like dark mode and [[mw:Trust and Safety Product/Temporary Accounts|temporary accounts]] respectively, will become available for logged-out users here.

So, if no large concerns are raised, we will deploy Vector 2022 here in two weeks, in the week of September 16. Do let me know if you have any questions. Thank you! [[User:SGrabarczuk (WMF)|SGrabarczuk (WMF)]] ([[User talk:SGrabarczuk (WMF)|discuss]] • [[Special:Contributions/SGrabarczuk (WMF)|contribs]]) 21:48, 2 September 2024 (UTC)

:Sounds good, Szymon - we look forward to the upcoming change of skin {{smile}} Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 07:35, 13 September 2024 (UTC)
* I for one oppose a switch to Vector 2022. I do not find it preferable. Here is a staggering evidence of user refusal of Vector 2022 once it was deployed: [[W:en:Wikipedia:Requests for comment/Rollback of Vector 2022]], Junuary 2023. 355 voters supported rollback to Vector 2010 whereas 64 opposed, yielding 84.7% support, as clear a supermajority as one may wish. These people opposing Vector 2022 feel the same way as I do. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 10:48, 13 September 2024 (UTC)
*:Hey @[[User:Dan Polansky|Dan Polansky]]. Thanks for your comment. I'm open to discussion about problems with our software, and I hope we can maintain a respectful tone.
*:I understand that there are users who prefer Vector legacy or other skins, just as there are people who still stick to Monobook. Such people are active across many wikis. They can keep Vector legacy, although non-default skins don't have the support the default ones do. We are rolling out for technical reasons, as I mentioned above, with benefit to not logged-in users.
*:Regarding the rollback RfC on Wikipedia, two neutral users stated that there was no consensus for rollback, RfC is not a vote, and the numbers were different (355:226:24). I believe this all is pretty easy to verify.
*:So to sum up, Vector 2022 needs to become the default, tons and tons of comments were made about the skin and related stuff, and we have taken many ideas into account, and it's totally OK if you stick to Vector legacy.
*:Thanks! [[User:SGrabarczuk (WMF)|SGrabarczuk (WMF)]] ([[User talk:SGrabarczuk (WMF)|discuss]] • [[Special:Contributions/SGrabarczuk (WMF)|contribs]]) 19:30, 16 September 2024 (UTC)
*:: Today, I visited Wikiversity and found it switched to Vector 2022. I changed my preference settings to Vector 2010. From what I understand, non-registered visitors are now defaulted to Vector 2022 despite its unpopularity in [[W:en:Wikipedia:Requests for comment/Rollback of Vector 2022]]. I have not seen any evidence that users prefer Vector 2022, and given the evidence in the linked RfC, I tentatively conclude that the decision to switch has made the site experience worse for the majority of users. The logic of &quot;you can switch&quot; surely applies to Vector 2022 as well: those who prefer it can switch to it. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 05:08, 17 September 2024 (UTC)

== Have your say: Vote for the 2024 Board of Trustees! ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
Hello all,

The voting period for the [[m:Special:MyLanguage/Wikimedia Foundation elections/2024|2024 Board of Trustees election]] is now open. There are twelve (12) candidates running for four (4) seats on the Board.

Learn more about the candidates by [[m:Special:MyLanguage/Wikimedia Foundation elections/2024/Candidates|reading their statements]] and their [[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Questions_for_candidates|answers to community questions]].

When you are ready, go to the [[Special:SecurePoll/vote/400|SecurePoll]] voting page to vote. '''The vote is open from September 3rd at 00:00 UTC to September 17th at 23:59 UTC'''.

To check your voter eligibility, please visit the [[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Voter_eligibility_guidelines|voter eligibility page]].

Best regards,

The Elections Committee and Board Selection Working Group&lt;section end=&quot;announcement-content&quot; /&gt;

[[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 12:15, 3 September 2024 (UTC)
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== Separate page for hyperbola. ==

Good morning,

I notice that a search for &quot;hyperbola&quot; redirects to &quot;Conic sections&quot;.

At present there is a separate page for &quot;ellipse&quot;. Therefore a separate page for &quot;hyperbola&quot; seems to be justified.

Could this redirection be changed so that search for &quot;hyperbola&quot; goes to a separate page for &quot;hyperbola&quot;?

Many thanks,

[[User:ThaniosAkro|ThaniosAkro]] ([[User talk:ThaniosAkro|discuss]] • [[Special:Contributions/ThaniosAkro|contribs]]) 12:04, 15 September 2024 (UTC)

:It is true that ellipses are covered at [[Conic sections]] (along with hyperbolas, parabolas, etc.) and there is a separate page for [[ellipse]]s that elaborates. We certainly ''could'' have a page about [[hyperbola]]s that is separate, but no one has written sufficient content to spin it off yet. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 12:17, 15 September 2024 (UTC)

== I hereby request for your Unblocking IP address and just reviewed and received a reverted rec ==

Hi there. {{unsigned|Ishmael Raphasha}}
:No one has any clue what you're talking about. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 16:53, 18 September 2024 (UTC)

== RICH-2K: New project with some initial questions ==

Hello! I'm creating a new learning resource on ''Wikiversity''. The respective project is based on my transcription of a classical dictionary from 1849 by Anthony Rich. For more information about the project see its [[User:CalRis25/RICH: Description|description page]] (see also that page for why not ''Wikisource'' or ''Wikibooks''). The project's scope is fairly big: 3205 article-pages plus 304 REDIRECT-pages. The images (scanned by myself from an original copy) have been uploaded to ''Commons''. I have some initial technical questions (more of these and more detailed ones will follow):

* '''Upload''': Due to the large number of pages it is not realistic to create these manually. Is it possible to bulk-upload these in some way (the Wikitext of the pages is created using a Python-script with one file per article/page)? Is it possible to upload these to a test-environment first where any problems (hopefully none) can be identified and dealt with more easily than on the production-version of ''Wikiversity''?
* '''(Technical) Structure''': I am planning to set up this project at ''&lt;nowiki&gt;https://en.wikiversity.org/wiki/RICH-2K&lt;/nowiki&gt;'' as the main page and anything else as subpages: ''RICH-2K/Subpage_1 ... RICH-2K/Subpage_n''. However, these subpages fall into two categories: 1. Article-pages (content) and 2. Meta/Administrative pages. This project requires search capability restricted to the ''RICH-2K''-namespace. The ''Mediawiki''-software seems to supply a ''Search''-input field with the possibility to restrict the search to some namespace. I would like, however, to restrict the search further to the first group of pages, namely the articles. Is that possible, perhaps by use of (hidden) categories?
* '''External links''': This project will need many external links, and yes, I have read the relevant ''Wikiversity''-pages, but this specific project needs them. The ''Recommended Editions''-page (used for recommended online editions, to which to link when citing texts) alone probably will require several hundred external links. However, only relatively few [[w:Second-level domain|second-level domains]] will be involved, and most of these should be trustworthy (Perseus Digital library, digital collections of universities etc., in some cases, however, also ''Archive.org''). Perhaps there is a list of web-sites, for which external links are generally allowed? And who is allowed to create external links on ''Wikiversity''-pages (I haven't found the relevant policy)?
* '''Categories''': This project requires quite a few of its own categories, which belong to two large groups: 1. Categories (2 levels) of the ''Classed Index'' (about 170 categories), a thematic index of some (but not all) of the articles. 2. Administrative categories. Is there a recommended way to distinguish between different classes of categories within a project (category name or other method)? What about naming conventions for project-specific categories?

I am looking forward to your input. If you think that it's preferable we can move the discussions to the [[User_talk:CalRis25/RICH:_Description|Talk-page]] of the project's description. Thank you in advance. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 05:29, 20 September 2024 (UTC)

:*Admins have access to [[Special:Import]] and can bulk import XML pages. You can create pages in your sandbox if you'd like and make an indefinite amount of them at pages like [[User:CalRis25/sandbox]]. What can and cannot be hosted in user namespace is very loose, but still has to follow in principle Wikiversity's scope.
:*Using subpages is in principle a good way to organize these various resources. Please do not name them after a user name or something obscure. I personally think that &quot;RICH-2K&quot; is a not optimal name. I may recommend something like [[Anthony Rich Dictionary Project]] or [[21st-Century Anthony Rich Dictionary]] or something more obviously intelligible. While we have very few actual policies and guidelines, see [[Wikiversity:Naming conventions]] for a rough consensus of what is probably best practice for naming pages.
:*External linking generally does not use an allowed list (a.k.a. whitelist model), but a disallow (a.k.a. blacklist) model. See [[MediaWiki:Spam-blacklist]] and [[Special:BlockedExternalDomains]] (which is currently empty but is another method of listing blocked domains). It's perfectly fine to aggregate external links in learning resources.
:*I'm not 100% sure what the distinction is that you're drawing, but you can freely arrange categories underneath a main category that has the same name as your larger project. So, following the suggestions I gave, you could have a category like [[:Category:Anthony Rich Dictionary Project]] and then create any number of subcategories that logically help users navigate all these pages. Please make sure the main category you create is itself categorized under some relevant category(ies). If you need help, please ask.
:I think this answers your questions, please let me know if I'm unclear or you have more. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 06:11, 20 September 2024 (UTC)
::Hello Justin!
::* '''Upload:''' Creating the project in sandbox pages of my User-namespace defeats the purpose, as this is an ''open'' project. Also that would not solve, as such, the problem of having to manually create thousands of pages. I wonder, does ''Wikiversity'' support creation of pages using its API. ''Mediawiki's'' [[mw:API:Main_page|API-description]] seems to imply that it ought to be possible. If that's the case, I should be able to create a Python-script which automatically creates the pages (of course, a few trial pages first).
::* '''(Technical) Structure''': You may be right, here. RICH-2K is, for now, merely a technical name to make a clear but not too verbose distinction between the original text and the current project. I'll give this more thought.
::* '''External links''': I brought this up mainly because when I first edited my ''Wikiversity''-page, I got a message that I was not allowed to create external links. However, I just now tested creating an external link on my user-page and got no error, so this problem seems to be solved.
::* '''Categories''': I think I know what you mean. I'll create a category structure and maybe ask some specific questions once I am ready to do so.
::Thank you for your quick help. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 18:51, 20 September 2024 (UTC)
:::re: upload, I'm just suggesting your sandbox(es) as you asked about &quot;a test-environment&quot;. Anyone can edit someone else's sandboxes, but you typically defer to other users to control what's in their own subpages as a collegial thing. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:39, 23 September 2024 (UTC)

Hello! I have two further questions:
# I created a category-structure for the project. Could you (or someone else) have a look at it ([[User:CalRis25/RICH: Categories]]) and answer the questions in the section [[User:CalRis25/RICH:_Categories#Questions|Questions]]? I gave it some thought and believe that this would work fine for the project.
# ''Project boxes'' (see [[Help:Tour of project boxes]]): It is unclear to me, whether these belong only on the main page of the project (that makes the most sense to me), or on every single subpage.
Thanks in advance for your help. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 17:51, 24 September 2024 (UTC)

:To answer your questions here:
:*No, you are not contravening any policies we have.
:*A leading &quot;The&quot; is acceptable, but if you want it to sort alphabetically, you will have to use &lt;nowiki&gt;{{DEFAULTSORT:}}&lt;/nowiki&gt;. E.g. to get Category:The Best Stuff to sort under &quot;B&quot;, insert &quot;&lt;nowiki&gt;{{DEFAULTSORT:Best Stuff, The}}&lt;/nowiki&gt;.
:*Trailing &quot;etc.&quot; is acceptable.
:*An accent in a category title is acceptable.
:I'll also note that it looks like you have in mind some tracking categories that are redundant. Pages such as [[Special:LonelyPages]] and [[Special:DeadendPages]] already do automatically what you're proposing to do manually.
:As for project boxes, it's typically the case that the subjects are only placed on the main resource, but as you may imagine, [[Help:Tour of project boxes/1|status completion ones]] may vary from subpage to subpage. As with most things at Wikiversity, there are very few actual rules, so it's pretty much the wild west, even tho this project has been around for almost 20&amp;nbsp;years. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:18, 25 September 2024 (UTC)
::Hello Justin, thanks for the DEFAULTSORT-hint for categories beginning with ''The''. I will restrict the project boxes to the main page. As for the the orphaned/dead-end-categories, I prefer these to be project-specific. Once the project is up and running, putting articles &quot;on the map&quot; (making them accessible from other articles and creating links to other articles) is one of the first tasks to be dealt with. I already know which articles are involved and will add these categories to these articles. [[User:CalRis25|CalRis25]] ([[User talk:CalRis25|discuss]] • [[Special:Contributions/CalRis25|contribs]]) 16:51, 25 September 2024 (UTC)

== Your wiki will be in read-only soon ==

&lt;section begin=&quot;server-switch&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;

[[:m:Special:MyLanguage/Tech/Server switch|Read this message in another language]] • [https://meta.wikimedia.org/w/index.php?title=Special:Translate&amp;group=page-Tech%2FServer+switch&amp;language=&amp;action=page&amp;filter= {{int:please-translate}}]

The [[foundation:|Wikimedia Foundation]] will switch the traffic between its data centers. This will make sure that Wikipedia and the other Wikimedia wikis can stay online even after a disaster.

All traffic will switch on '''{{#time:j xg|2024-09-25|en}}'''. The switch will start at '''[https://zonestamp.toolforge.org/{{#time:U|2024-09-25T15:00|en}} {{#time:H:i e|2024-09-25T15:00}}]'''.

Unfortunately, because of some limitations in [[mw:Special:MyLanguage/Manual:What is MediaWiki?|MediaWiki]], all editing must stop while the switch is made. We apologize for this disruption, and we are working to minimize it in the future. 

A banner will be displayed on all wikis 30 minutes before this operation happens. This banner will remain visible until the end of the operation. 

'''You will be able to read, but not edit, all wikis for a short period of time.'''

*You will not be able to edit for up to an hour on {{#time:l j xg Y|2024-09-25|en}}.
*If you try to edit or save during these times, you will see an error message. We hope that no edits will be lost during these minutes, but we can't guarantee it. If you see the error message, then please wait until everything is back to normal. Then you should be able to save your edit. But, we recommend that you make a copy of your changes first, just in case.

''Other effects'':

*Background jobs will be slower and some may be dropped. Red links might not be updated as quickly as normal. If you create an article that is already linked somewhere else, the link will stay red longer than usual. Some long-running scripts will have to be stopped.
* We expect the code deployments to happen as any other week. However, some case-by-case code freezes could punctually happen if the operation require them afterwards.
* [[mw:Special:MyLanguage/GitLab|GitLab]] will be unavailable for about 90 minutes.

This project may be postponed if necessary. You can [[wikitech:Switch_Datacenter|read the schedule at wikitech.wikimedia.org]]. Any changes will be announced in the schedule. 

'''Please share this information with your community.'''&lt;/div&gt;&lt;section end=&quot;server-switch&quot;/&gt;

[[User:Trizek_(WMF)|Trizek_(WMF)]], 09:37, 20 September 2024 (UTC)
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== 'Wikidata item' link is moving. Find out where... ==

&lt;div lang=&quot;en&quot; dir=&quot;ltr&quot; class=&quot;mw-content-ltr&quot;&gt;&lt;i&gt;Apologies for cross-posting in English. Please consider translating this message.&lt;/i&gt;{{tracked|T66315}}

Hello everyone, a small change will soon be coming to the user-interface of your Wikimedia project.
The [[d:Q16222597|Wikidata item]] [[w:|sitelink]] currently found under the &lt;span style=&quot;color: #54595d;&quot;&gt;&lt;u&gt;''General''&lt;/u&gt;&lt;/span&gt; section of the '''Tools''' sidebar menu will move into the &lt;span style=&quot;color: #54595d;&quot;&gt;&lt;u&gt;''In Other Projects''&lt;/u&gt;&lt;/span&gt; section.

We would like the Wiki communities feedback so please let us know or ask questions on the   [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Discussion page]] before we enable the change which can take place October 4 2024, circa 15:00 UTC+2.
More information can be found on [[m:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|the project page]].&lt;br&gt;&lt;br&gt;We welcome your feedback and questions.&lt;br&gt; [[User:MediaWiki message delivery|MediaWiki message delivery]] ([[User talk:MediaWiki message delivery|discuss]] • [[Special:Contributions/MediaWiki message delivery|contribs]]) 18:56, 27 September 2024 (UTC)
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==Download as PDF==
[[Phabricator:T376438]]: &quot;Download to PDF&quot; on en.wv is returning error: &quot;{&quot;name&quot;:&quot;HTTPError&quot;,&quot;message&quot;:&quot;500&quot;,&quot;status&quot;:500,&quot;detail&quot;:&quot;Internal Server Error&quot;}&quot;

-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:36, 3 October 2024 (UTC)

:I just downloaded this page as a PDF and it worked just fine. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 23:04, 3 October 2024 (UTC)

== Protected template bug for Pp ==

It seems that templates derivative of {{tlx|Pp}} (compiled in {{tlx|Protection templates}}) are being sorted into protection categories using the name 'Wikipedia' instead of 'Wikiversity' (e.g., [[:Category:Wikipedia pages with incorrect protection templates]]). From what I can tell, it is not in the publicly accessible source code of any of the templates. The only other impacted pages are modules which call {{tlx|pp}}-derivatives (e.g., [[Module:Navbar/styles.css]]).

This does not seem to affect any other pages in [[:Category:Wikiversity protected templates]]. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 18:59, 4 October 2024 (UTC)

:The problem is that &quot;Wikipedia&quot; is [https://en.wikiversity.org/w/index.php?title=Special%3ASearch&amp;limit=500&amp;offset=0&amp;ns828=1&amp;search=Wikipedia&amp;searchToken=9svkpqlxxoquoq7bnkt55ugts mentioned in several modules that were copied over from en.wp]; many of these are legit and many of them need to be replaced with &quot;Wikiversity&quot; ([https://en.wikiversity.org/w/index.php?title=Module%3APp-move-indef&amp;diff=2662815&amp;oldid=1944984 e.g.]) This particular change ''may'' fix all of these issues...? But 1.) it will take time to propagate across the site and 2.) there are still many more &quot;Wikipedia&quot;s that need to be changed, so I'll go thru a few more, but if you want to give me an assist, if you can just check this one week from now and ping me if the problem persists, that would be nice. Sometimes, I make calendar reminders to follow up on these, but I'm not a perfect person. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 04:55, 5 October 2024 (UTC)

== Invitation to Participate in Wiki Loves Ramadan Community Engagement Survey ==

Dear all,


We are excited to announce the upcoming [[m:Wiki Loves Ramadan|Wiki Loves Ramadan]] event, a global initiative aimed at celebrating Ramadan by enriching Wikipedia and its sister projects with content related to this significant time of year. As we plan to organize this event globally, your insights and experiences are crucial in shaping the best possible participation experience for the community.

To ensure that Wiki Loves Ramadan is engaging, inclusive, and impactful, we kindly invite you to participate in our community engagement survey. Your feedback will help us understand the needs of the community, set the event's focus, and guide our strategies for organizing this global event.

Survey link: https://forms.gle/f66MuzjcPpwzVymu5

Please take a few minutes to share your thoughts. Your input will make a difference!

Thank you for being a part of our journey to make Wiki Loves Ramadan a success.


Warm regards,

User:ZI Jony 03:19, 6 October 2024 (UTC)

Wiki Loves Ramadan Organizing Team
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== 'Edit to my talk page' notification bug? ==

This may belong at the bug tracker, but does anyone else have an issue disabling ''email'' notifications upon  an 'Edit to my talk page' in [[Special:GlobalPreferences]]? Oddly I ''am'' able to disable the global preference on Wikipedia, MediaWiki, etc, but not here. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 09:23, 7 October 2024 (UTC)

:I have not experienced this, but to be clear, do you also have the option to get emails when items on your talk page are edited turned on? —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:39, 7 October 2024 (UTC)
::The only (non-grayed out) options I have enabled for email are 'Failed login attempts' and 'Login from an unfamiliar device'. 'Edit to my talk page' re-checks after every save. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 09:54, 7 October 2024 (UTC)
:::That does sound like a [[phab:]] issue, with the caveat that I don't 100% recall how global preferences work and if they override local ones, etc. If you have parsed that and still have this issue, you'll probably need to file a ticket. Maybe someone else has this issue. Wish I could help. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 09:57, 7 October 2024 (UTC)
::::[[phab:T376601|Off 'n away]] 🫡 [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 10:35, 7 October 2024 (UTC)

== [[Portal:Computer Science]] ➝ [[Portal:Information sciences]] ==

Seeking consensus to complete the merge into the broader portal. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 06:28, 8 October 2024 (UTC)

:Why should it be merged? Computer Science seems well-enough designed. What is the incentive to collapse it into a broader field of study? —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:18, 8 October 2024 (UTC)
::Portals as top level organizations allowing for content to be best centralized. Also note that I did not start the merge, just offering to finish it. Perhaps a {{tlx|prod}} instead? [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 07:20, 8 October 2024 (UTC)
:::I have no objections, personally. If it gets done, please use a redirect and should someone want to come along to resurrect it later, it will be easier. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 07:21, 8 October 2024 (UTC)
: Is computer science really a branch of information sciences? I would not think so, but what do I know. Do we have some external resources/links confirm this idea? [[W:Information science]] currently says: &quot;Information science, documentology[1] or informatology[2][3] is an academic field which is primarily concerned with analysis, collection, classification, manipulation, storage, retrieval, movement, dissemination, and protection of information.&quot; --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 14:49, 11 October 2024 (UTC)
::Looking through [https://stackoverflow.com/q/1047014/22673230] [https://businessdegrees.uab.edu/mis-degree-bachelors/resources/computer-information-systems-vs-computer-science/] [https://www.si.umich.edu/student-experience/what-information-science] a few top (not necessarily RS) searches I'm inclined to agree. I am more familiar with the grafted [[:w:Information and computer science|information ''and'' computer science]] which makes an effort to merge the disciplines, but it does not seem like reaching to say that IS is presented as more applications-concerned (certainly with no lack of theoretical abstraction), whereas CS can be more freely associated with any and all 'science related to computers'. It is easy to reason about the connection between the fields, but I think it is clear academia maintains this taxonomy for a good reason.
::With these considerations, I think I will ''stop'' the process of merging in favor of expanding the existing [[School:Library and Information Science]]. 

::Let me know if there is not consensus to redirect [[Portal:Information sciences]] to [[School:Library and Information Science]] (with enough expansion it can generalize away from just library sciences). [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 16:16, 11 October 2024 (UTC)
::: I do not see that a merge of a ''portal'' to a ''school'' is a good thing. Do you have a clear idea of the concepts of school and portal and how they relate to each other? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 16:34, 11 October 2024 (UTC)
::::Found [[:Category:Information sciences]]; there are enough existing resources in there to make my other proposed merge excessive. I will simply continue developing the existing [[Portal:Information sciences]] instead. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 17:05, 11 October 2024 (UTC)
::::: Frankly, I would ideally see [[Portal:Information sciences]] deleted: I don't see what it does that a category would not do well enough. There does not seem to be any material specific to &quot;Information sciences&quot; (whatever that is) in that portal at all. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 17:11, 11 October 2024 (UTC)
::::::Tacked a {{tlx|prod}} for an eventual deletion, but I may still try to develop it as proof of concept at some point. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 17:33, 11 October 2024 (UTC)

== [[:Category:Occupational Epidemiology]] ==

I propose moving the pages in this category (without leaving redirects) to their equivalent under the parent resource [[Occupational Health Risk Surveillance]]. Also due to the number of subpages, it seems &lt;code&gt;|filing=deep&lt;/code&gt; would be a justified. (Also [[Special:PrefixIndex/Occupational_Epidemiology|there are quite a few]] untagged subpages.) [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 05:11, 9 October 2024 (UTC)
: I above all think that the content should be ''moved out of the mainspace'': I do not see readers learning anything from e.g. [[Occupational Epidemiology/Research tools/Reading of scientific articles for learning epidemiology and biostatstics]] or [[Occupational Epidemiology/Research tools/Ongoing projects/Risk Communication in Seafaring/Writing the article guideline IMRAD]]. Wikiversity can be kind enough to host that material in, say, subspace of [[User:Saltrabook]], but more should not be asked, I think. Let us recall that per [[WV:Deletions]], &quot;Resources may be eligible for proposed deletion when education objectives and learning outcomes are scarce, and objections to deletion are unlikely&quot;; I do not see how learning outcomes can be anything but scarce. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 15:04, 11 October 2024 (UTC)
::thank you, agree @ [[User:Saltrabook|Saltrabook]] ([[User talk:Saltrabook|discuss]] • [[Special:Contributions/Saltrabook|contribs]]) 21:03, 13 November 2024 (UTC)

== Active editors ==

It is interesting to observe the stats on [https://stats.wikimedia.org/#/en.wikiversity.org/contributing/active-editors/normal|line|all|(page_type)~content*non-content|monthly active editors] through the project's history. October is our month! [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 20:44, 8 October 2024 (UTC)

:Odd. Maybe related to the school year? —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 02:10, 9 October 2024 (UTC)
::I wonder how many are [[User:Jtneill|Jtneill]]'s crowd... the number is in the hundreds though, so that is one chunky cohort —[[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:16, 9 October 2024 (UTC)
:::Yes, [[Motivation and emotion/Book]] involves ~100-150 students editing most intensely during October each year. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:20, 9 October 2024 (UTC)
::::Neat, that still leaves around ~50-100 other students from other avenues each year since 2021. I also wonder which projects were involved in the COVID enrollment spike. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 02:26, 9 October 2024 (UTC)
:::::Personally I can admit that my editing is much more active during the school season vs. the summer break, so I'm in the same boat as Jtneill's students. —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 21:24, 13 November 2024 (UTC)
:@[[User:Tule-hog|Tule-hog]] This is an interesting topic, but it is not clear to me as an outsider what you and other participants in this discussion find interesting. I  find this graph not very meaningful because it does not tell me if the number of  Active editors has gone up or down during the period covered, which I think was 2000-now. 
:I can see a big jump between 2000 and 2007, but what happened since then? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 15:45, 16 December 2024 (UTC)

== Intentionally incorrect resource ==

There is a [[Special:Diff/2583464|disclaimer inserted onto a resource]] (by not the original author) that:
&lt;blockquote&gt;I am merely [making this page false] to show you (The viewer) that Wikipedia and this page 'Wikiversity' is bull sh*t and it will not give you the reliability you need when writing an academic piece of writing.&lt;/blockquote&gt;
However, that IP has [[Special:Contributions/86.22.73.151|not made any other edits]], so unless they vandalized via a sock, the intent went un-realized and only that portion need be removed. Bumping here in case there is some obvious jumbo in that essay that someone else can catch. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 16:58, 9 October 2024 (UTC)

:Removed that portion, which was obviously vandalism. No perspective on the rest of the essay. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 18:38, 9 October 2024 (UTC)

== [[:Category:Proposed guidelines]] ==

Noting for future editors that WV has collapsed all proposals into [[:Category:Proposed policies|proposed policies]]. Seeking consensus to further collapse [[:Category:Wikiversity proposals]] into the former, or to restore [[:Category:Proposed guidelines]]. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 19:19, 9 October 2024 (UTC)

== [[Around Wikiversity in 80 Seconds|Broken 80-second tour]] ==

Bumping a [[Talk:Around_Wikiversity_in_80_Seconds|comment]] on the ''Wikiversity in 80 seconds'' tour. Appears wikisuite is not working with the Vector 2022 appearance. 

Also see [[:w:Wikipedia:Miscellany_for_deletion/Wikiversuite_pages|this thread]] on the Wikiversal package - may not be relevant to Wikiversity, but FYC. [[User:Tule-hog|Tule-hog]] ([[User talk:Tule-hog|discuss]] • [[Special:Contributions/Tule-hog|contribs]]) 00:26, 10 October 2024 (UTC)
: I would just delete the material; I do not see value in it. If others agree, I would try to articulate why I think it should be deleted (or move to author user space). --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 06:57, 13 October 2024 (UTC)
::Just mark as {{tl|historical}}. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 17:39, 13 October 2024 (UTC)
::: This thing was created by [[User:Planotse]]. His creations are now being discussed in Wikibooks for deletion: [[B:Wikibooks:Requests for deletion#Wikiversal generated pages]]. It seems he used some kind of tool that is no longer available (the above mentioned &quot;Wikiversal&quot; package) to create this kind of slideshow-like material (believing the Wikibooks discussion). I do not see value of this in the mainspace, not even as historical (I am okay with userspace, but maybe even that is not the best option?). A look at the source code of [[Around Wikiversity in 80 Seconds/Introduction]] confirms the words of Omphalographer, namely that &quot;the HTML-heavy markup generated by Wikiversal makes them [the pages] unreasonably difficult to edit.&quot;
::: I went ahead and marked the page for proposed deletion. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:35, 14 October 2024 (UTC)

== Preliminary results of the 2024 Wikimedia Foundation Board of Trustees elections ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
Hello all,

Thank you to everyone who participated in the [[m:Special:MyLanguage/Wikimedia Foundation elections/2024|2024 Wikimedia Foundation Board of Trustees election]]. Close to 6000 community members from more than 180 wiki projects have voted.

The following four candidates were the most voted:

# [[User:Kritzolina|Christel Steigenberger]]
# [[User:Nadzik|Maciej Artur Nadzikiewicz]]
# [[User:Victoria|Victoria Doronina]]
# [[User:Laurentius|Lorenzo Losa]]

While these candidates have been ranked through the vote, they still need to be appointed to the Board of Trustees. They need to pass a successful background check and meet the qualifications outlined in the Bylaws. New trustees will be appointed at the next Board meeting in December 2024.

[[m:Special:MyLanguage/Wikimedia_Foundation_elections/2024/Results|Learn more about the results on Meta-Wiki.]]

Best regards,

The Elections Committee and Board Selection Working Group

&lt;section end=&quot;announcement-content&quot; /&gt;

[[User:MPossoupe_(WMF)|MPossoupe_(WMF)]] 08:26, 14 October 2024 (UTC)
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== Seeking volunteers to join several of the movement’s committees ==

&lt;section begin=&quot;announcement-content&quot; /&gt;
Each year, typically from October through December, several of the movement’s committees seek new volunteers. 

Read more about the committees on their Meta-wiki pages:
* [[m:Special:MyLanguage/Affiliations_Committee|Affiliations Committee (AffCom)]]
* [[m:Special:MyLanguage/Ombuds_commission|Ombuds commission (OC)]]
* [[m:Special:MyLanguage/Wikimedia Foundation/Legal/Community Resilience and Sustainability/Trust and Safety/Case Review Committee|Case Review Committee (CRC)]]

Applications for the committees open on 16 October 2024. Applications for the Affiliations Committee close on 18 November 2024, and applications for the Ombuds commission and the Case Review Committee close on 2 December 2024. Learn how to apply by [[m:Special:MyLanguage/Wikimedia_Foundation/Legal/Committee_appointments|visiting the appointment page on Meta-wiki]]. Post to the talk page or email [mailto:cst@wikimedia.org cst@wikimedia.org] with any questions you may have.

For the Committee Support team,
&lt;section end=&quot;announcement-content&quot; /&gt;

-- [[m:User:Keegan (WMF)|Keegan (WMF)]] ([[m:User talk:Keegan (WMF)|talk]]) 23:09, 16 October 2024 (UTC)
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== Interactive elements ==
Can we use interactive elements on Wikiversity? I'd like to add JavaScript to a page. If it's not possible now, where can I suggest this feature? I have a safe integration idea.
[[User:Отец Никифор|Отец Никифор]] ([[User talk:Отец Никифор|discuss]] • [[Special:Contributions/Отец Никифор|contribs]]) 12:10, 17 October 2024 (UTC)
: This is beyond my technical knowledge, but have you checked out:
:* https://www.mediawiki.org/wiki/Manual:Interface/JavaScript?
:* [[Wikipedia:WikiProject JavaScript]]
:* [[MediaWiki:Common.js]]

:What sort of interactive elements are you thinking about?

: Sincerely, James -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 03:39, 18 October 2024 (UTC)
::I was thinking about adding something like a graph with adjustable controls, where users can interact with it and see how different changes affect the outcome. It seems like this could be a useful feature. There might already be discussions about enhancing Wikiversity or similar platforms—perhaps on a relevant talk page or in a Discord group. Do you know where such discussions might be happening? [[User:Отец Никифор|Отец Никифор]] ([[User talk:Отец Никифор|discuss]] • [[Special:Contributions/Отец Никифор|contribs]]) 19:47, 18 October 2024 (UTC)
:::From a quick look, maybe check out:
:::* [[mw:Extension:Graph]]
:::* [[phab:tag/graphs]]
:::-- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 22:40, 18 October 2024 (UTC)
:::: mw:Extension:Graph is currently disabled on Wikipedia etc. wikis, for security reasons, and seems unlikely to be enabled again. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 09:30, 19 October 2024 (UTC)

== An unexplained spurt of Wikiversity page views ==
The [https://pageviews.wmcloud.org/siteviews/?platform=all-access&amp;source=pageviews&amp;agent=user&amp;start=2024-06-01&amp;end=2024-10-18&amp;sites=en.wikiversity.org|en.wikibooks.org|en.wikiquote.org|en.wikisource.org page view report] shows an unexplained spurt of Wikiversity page views, reaching over 4 times the baseline and then falling back again. Does anyone have any idea what is going on? --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 08:01, 19 October 2024 (UTC)

:Interesting. I wonder why only the English wikiquote and wikiversity and not Wikisource or wikibooks? How reliable do you think those stats are? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 15:44, 8 December 2024 (UTC)
:I guess the mention in mass media might be a cause. Someone metions it and then thousands go and look. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:02, 17 December 2024 (UTC)

== Center tempate failed on a contributors phone... ==

See the edit comment here - https://en.wikiversity.org/w/index.php?title=Wikiphilosophers&amp;diff=prev&amp;oldid=2673962. I'm puzzled as this is the first failure of this, I've noted recently. [[User:ShakespeareFan00|ShakespeareFan00]] ([[User talk:ShakespeareFan00|discuss]] • [[Special:Contributions/ShakespeareFan00|contribs]]) 08:45, 19 October 2024 (UTC)

== Essay-like page in user space that makes little sense and seems incoherent ==
The page [[User:TheoYalur/Illusions]] seems to match the description, at least by my assessment. My understanding is that since the page is only in user space and not in the mainspace, it can stay there even if it has those disqualifying qualities. But if I am wrong and the page belongs deleted, please correct me and let me know. I do not know which policy or guideline, if any, guides the case. --[[User:Dan Polansky|Dan Polansky]] ([[User talk:Dan Polansky|discuss]] • [[Special:Contributions/Dan Polansky|contribs]]) 12:30, 21 October 2024 (UTC)

== 'Wikidata item' link is moving, finally. ==

Hello everyone, I previously wrote on the 27th September to advise that the ''Wikidata item'' sitelink will change places in the sidebar menu, moving from the '''General''' section into the '''In Other Projects''' section. The scheduled rollout date of 04.10.2024 was delayed due to a necessary request for Mobile/MinervaNeue skin. I am happy to inform that the global rollout can now proceed and will occur later today, 22.10.2024 at 15:00 UTC-2. [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Please let us know]] if you notice any problems or bugs after this change. There should be no need for null-edits or purging cache for the changes to occur. Kind regards, -[[m:User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] 11:28, 22 October 2024 (UTC)
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:Hi @[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]]: I Just noticed your post above, and it is timely.
:I have been participating in the English WikiUniversity for a few years, much less often recently. I seems like something in the way the site displays is different, but I cannot put my finger on it. Your posting gave me a clue. Can you please tell me where  the link to wikidata items has moved to? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:23, 11 December 2024 (UTC)
::Hello @[[User:Ottawahitech|Ottawahitech]], sure, I would be happy to. The button/sitelink name didn't change, just its position. You should find it in the sidebar-menu under the section '''In other projects''' (where the links to all other Wikimedia Projects are displayed). If you do not see it, please reach out to us on the [[m:Talk:Wikidata_For_Wikimedia_Projects/Projects/Move_Wikidata_item_link|Move Wikidata item - Discussion page]]. Thank you, -[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] ([[User talk:Danny Benjafield (WMDE)|discuss]] • [[Special:Contributions/Danny Benjafield (WMDE)|contribs]]) 09:24, 12 December 2024 (UTC)
:::@[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]], thank you for responding. I intend to followup on the ''Move Wikidata item - Discussion page'' as per your post above by putting it on my ever growing todo list.
:::I don't know about others on this wiki, as I said I have not been visiting here frequently, but for me the constant changes are a big distraction. I have been around wikimedia projects since 2007, so why do I have to spend so much time learning and re-learning how to find what I came here for? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:41, 12 December 2024 (UTC)
::::Hi @[[User:Ottawahitech|Ottawahitech]], thanks for you thoughts. Your input whether positive or critical helps us understand the impacts to editors so we welcome your further thoughts when you reach us in your To Do List :)
::::I can't speak about the other changes you've experienced here but I do hope they are made with a spirit of improvement for the community as a whole. -[[User:Danny Benjafield (WMDE)|Danny Benjafield (WMDE)]] ([[User talk:Danny Benjafield (WMDE)|discuss]] • [[Special:Contributions/Danny Benjafield (WMDE)|contribs]]) 10:43, 16 December 2024 (UTC)

== Final Reminder: Join us in Making Wiki Loves Ramadan Success ==

Dear all,

We’re thrilled to announce the Wiki Loves Ramadan event, a global initiative to celebrate Ramadan by enhancing Wikipedia and its sister projects with valuable content related to this special time of year. As we organize this event globally, we need your valuable input to make it a memorable experience for the community.

Last Call to Participate in Our Survey: To ensure that Wiki Loves Ramadan is inclusive and impactful, we kindly request you to complete our community engagement survey. Your feedback will shape the event’s focus and guide our organizing strategies to better meet community needs.

* Survey Link: [https://docs.google.com/forms/d/e/1FAIpQLSffN4prPtR5DRSq9nH-t1z8hG3jZFBbySrv32YoxV8KbTwxig/viewform?usp=sf_link Complete the Survey]
* Deadline: November 10, 2024
Please take a few minutes to share your thoughts. Your input will truly make a difference!

'''Volunteer Opportunity''': Join the Wiki Loves Ramadan Team! We’re seeking dedicated volunteers for key team roles essential to the success of this initiative. If you’re interested in volunteer roles, we invite you to apply.
* Application Link: [https://docs.google.com/forms/d/e/1FAIpQLSfXiox_eEDH4yJ0gxVBgtL7jPe41TINAWYtpNp1JHSk8zhdgw/viewform?usp=sf_link Apply Here]
* Application Deadline: October 31, 2024
Explore Open Positions: For a detailed list of roles and their responsibilities, please refer to the position descriptions here: [https://docs.google.com/document/d/1oy0_tilC6kow5GGf6cEuFvdFpekcubCqJlaxkxh-jT4/ Position Descriptions]

Thank you for being part of this journey. We look forward to working together to make Wiki Loves Ramadan a success!


Warm regards,&lt;br&gt;
The Wiki Loves Ramadan Organizing Team 05:11, 29 October 2024 (UTC)
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== Android app for Wikiversity ==

Hi, is there an Android app for Wikiversity? How does it work? I have been advised that there is no infrastructure for push notifications for Android apps for sister wikis and I would be interested to know more. Related: [[:phab:T378545]]. Thanks! [[User:Gryllida|Gryllida]] 23:15, 29 October 2024 (UTC)

:Thanks for suggesting this - I agree that it would be useful. -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 00:56, 31 October 2024 (UTC)
:@[[User:Gryllida|Gryllida]]: Would you explain your terminology  for those of us not in the know. What does ''push notifications'' mean? I use [https://www.mediawiki.org/wiki/Help:Notifications notifications] when I am communicating on wikimedia projects, but have never heard this term before. [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:13, 11 December 2024 (UTC)
:I dont think there is an app.  [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 10:01, 17 December 2024 (UTC)

à

== Import Resource From Wikibooks? ==

Hello! [[wikibooks:Character_List_for_Baxter&amp;Sagart|Character List for Baxter&amp;Sagart]] and related titles [[wikibooks:Wikibooks:Requests_for_deletion#Character_List_for_Baxter&amp;Sagart|are up for deletion at Wikibooks]] because WB policy does not allow dictionaries like them. However, because they are useful as learning tools, I am wondering if they might have a home here at Wikiversity. Pinging @[[User:Tibetologist|Tibetologist]] here to link them in to this discussion, since they are the affected user. Thank you! —[[User:Kittycataclysm|Kittycataclysm]] ([[User talk:Kittycataclysm|discuss]] • [[Special:Contributions/Kittycataclysm|contribs]]) 18:18, 1 November 2024 (UTC)

:Sure, I can do it. That said, as mentioned there, it does seem like something like this is ideally suited for Wiktionary in the Appendix namespace, but I'm not very familiar with CJK characters and languages. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:23, 3 November 2024 (UTC)
::Oh man, these pages are too big to import and while I've already tried a half-dozen times, it will constantly fail. Strictly speaking, we don't have to use the import feature for licensing purposes. We can just copy and paste the contents and list the usernames or on the talk page. I think that's the solution. {{Ping|Tibetologist}}, are you interested in doing that? If you just copied and pasted these pages and then added [[:Category:Chinese]] and maybe include a couple of links to the pages, that would probably be ideal. —[[User:Koavf|Justin (&lt;span style=&quot;color:grey&quot;&gt;ko'''a'''vf&lt;/span&gt;)]]&lt;span style=&quot;color:red&quot;&gt;❤[[User talk:Koavf|T]]☮[[Special:Contributions/Koavf|C]]☺[[Special:Emailuser/Koavf|M]]☯&lt;/span&gt; 22:31, 3 November 2024 (UTC)

== Language translation requests? ==

Is there anywhere on Wikiversity to request translation, for example, requesting Latin or French translation? I would be asking from the context as a student, so I would be interested in translation explanation as well. [[User:Indexcard88|Indexcard88]] ([[User talk:Indexcard88|discuss]] • [[Special:Contributions/Indexcard88|contribs]]) 04:56, 20 November 2024 (UTC)

== Sign up for the language community meeting on November 29th, 16:00 UTC ==

Hello everyone,

The next language community meeting is coming up next week, on November 29th, at 16:00 UTC (Zonestamp! For your timezone &lt;https://zonestamp.toolforge.org/1732896000&gt;). If you're interested in joining, you can sign up on this wiki page: &lt;https://www.mediawiki.org/wiki/Wikimedia_Language_and_Product_Localization/Community_meetings#29_November_2024&gt;.

This participant-driven meeting will be organized by the Wikimedia Foundation’s Language Product Localization team and the Language Diversity Hub. There will be presentations on topics like developing language keyboards, the creation of the Moore Wikipedia, and the language support track at Wiki Indaba. We will also have members from the Wayuunaiki community joining us to share their experiences with the Incubator and as a new community within our movement. This meeting will have a Spanish interpretation.

Looking forward to seeing you at the language community meeting! Cheers, [[User:SSethi (WMF)|Srishti]] 19:55, 21 November 2024 (UTC)
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== Events on Wikiversity ==

Since Wikipedia and Wikivoyage are having their &quot;Asian Month&quot; editathon, I was thinking if we could start up a Wikiversity version of that. This would be an &quot;Asian Month&quot; as well, but it would be about creating resources based on Asia and its culture. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 17:57, 6 December 2024 (UTC)

:Not immediately opposed, but the question is, do we have an active enough community to facilitate this? —[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:31, 6 December 2024 (UTC)
::I'm not too sure. As long as we get enough traffic, this could happen. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 08:45, 7 December 2024 (UTC)
:::This is to increase traffic on Wikiversity, which is promoted amongst other communities. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 10:47, 7 December 2024 (UTC)
:Hi @[[User:RockTransport|RockTransport]], This is a good idea, but will it also involve users who are not &quot;professors and scientists&quot;. Just curious. cheers, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:30, 9 December 2024 (UTC)
::Yes, considering the fact that Wikiversity is for everyone, and not just for specific users. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 09:09, 10 December 2024 (UTC)
:::because I'm personally not a &quot;professor&quot; or a &quot;scientist&quot; and because '''anyone''' can create resources on Wikiversity. We want to make Wikiversity open for everyone, and not just for certain users. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 09:10, 10 December 2024 (UTC)
::::I am also not a professor or a scientist, but it seems to me that as result I am viewed here as a visitor rather than someone who can contribute. Just my $.02. [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:05, 12 December 2024 (UTC)
:I am affraid, that creation of educational resources on certain topic is way harder then wikipedia. Secondly while wikipedia stub does not matter, education resource stub is uselless completly. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 09:59, 17 December 2024 (UTC)

== Wikiversity - Newsletters ==

Hello All,

I wanted to create a newsletter on Wikiversity, which would highlight what is going on in certain months and events on Wikiversity; which would bolster engagement by many people. This would be on the website and would have its dedicated 'Newsletter' tab.

I hope you acknowledge this idea. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 21:05, 8 December 2024 (UTC)

:@[[User:RockTransport|RockTransport]], What sort of things do you plan to include in your newsletter? Will they be different than what is currently in [[Main Page/News]]? Just curious.
:I am also wondering about your motive which I think is: to bolster engagement by many people. I am asking because I wonder if others who are currently active here also think this I is desirable? Have you asked them? [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 17:34, 11 December 2024 (UTC)
::Not yet, which was why I was asking this on the colloquium. I plan to include things that many people have created on Wikiversity over the month, as it is a monthly newsletter. It would be somewhere on the website here. It will be more frequent that the ones seen on [[Main Page/News]]. We will include people's resources to essentially promote them. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 06:50, 12 December 2024 (UTC)
:::@[[User:RockTransport|RockTransport]], I Think what you are saying is that ''Main Page/News'' does not update frequently enough? 
:::If this is the reason, why not start small by simply increasing the frequency of posting news on the main page, instead of trying to start a newsletter?
:::If there is more, can you articulate what else is missing. Thanks in advance, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 16:51, 12 December 2024 (UTC)
::::I meant going to detail into topics covered in that month, rather than just giving a few points. [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 16:53, 12 December 2024 (UTC)
:::::What sort of details did you have in mind? You can pick one of the links provided in [[Main Page/News]] to illustrate. cheers, [[User:Ottawahitech|Ottawahitech]] ([[User talk:Ottawahitech|discuss]] • [[Special:Contributions/Ottawahitech|contribs]]) 15:29, 16 December 2024 (UTC)
::::::I'm thinking of the community entering their projects, and discussing those in the newsletter. It depends on what they want, though. There would be a dedicated page for giving the information about their projects [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 17:24, 16 December 2024 (UTC)</text>
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  <page>
    <title>Wikiversity:Sandbox</title>
    <ns>4</ns>
    <id>1558</id>
    <revision>
      <id>2692178</id>
      <parentid>2692033</parentid>
      <timestamp>2024-12-16T12:16:37Z</timestamp>
      <contributor>
        <username>MathXplore</username>
        <id>2888076</id>
      </contributor>
      <comment>reset sandbox</comment>
      <origin>2692178</origin>
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      <text bytes="50" sha1="phlij3i0lq7l17sacctmpzowd8epftu" xml:space="preserve">{{Please leave this line alone (sandbox heading)}}</text>
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      <id>2692196</id>
      <parentid>2692178</parentid>
      <timestamp>2024-12-16T15:36:18Z</timestamp>
      <contributor>
        <ip>82.219.7.140</ip>
      </contributor>
      <origin>2692196</origin>
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      <text bytes="131" sha1="nq3jmn414na7znkvxgttj7wvevm31b8" xml:space="preserve">{{Please leave this line alone (sandbox heading)}}
guys stanley harold-salvage said that he likes stalking people... what the sigma</text>
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    <revision>
      <id>2692197</id>
      <parentid>2692196</parentid>
      <timestamp>2024-12-16T15:37:01Z</timestamp>
      <contributor>
        <ip>82.219.7.140</ip>
      </contributor>
      <origin>2692197</origin>
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Guys this platform  is very nice.</text>
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    <revision>
      <id>2692198</id>
      <parentid>2692197</parentid>
      <timestamp>2024-12-16T15:40:07Z</timestamp>
      <contributor>
        <ip>82.219.7.140</ip>
      </contributor>
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Guys this platform  is very nice. 
WHAT THE SIGMA, SKIBIDI, SLICERS, OHIO, GYATT, RIZZLER, (IBNSERT OTHER BRAINMROT) THGIS IS BRAINRROT AEAEAEAEAEAEAEAE,</text>
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    <revision>
      <id>2692199</id>
      <parentid>2692198</parentid>
      <timestamp>2024-12-16T15:42:50Z</timestamp>
      <contributor>
        <ip>82.219.7.140</ip>
      </contributor>
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Wikiversity, thanks for this opportunity. Learning is an ongoing concern.</text>
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    <revision>
      <id>2692207</id>
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      <timestamp>2024-12-16T18:36:26Z</timestamp>
      <contributor>
        <username>RockTransport</username>
        <id>2992610</id>
      </contributor>
      <comment>Resetting sandbox</comment>
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  <page>
    <title>User:Brendia</title>
    <ns>2</ns>
    <id>38409</id>
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      <id>2692274</id>
      <parentid>143254</parentid>
      <timestamp>2024-12-17T11:11:08Z</timestamp>
      <contributor>
        <username>Gpkp</username>
        <id>1243490</id>
      </contributor>
      <minor />
      <comment>([[c:GR|GR]]) [[c:COM:FR|File renamed]]: [[File:Sparkle.jpg]] → [[File:Fire cracker stick sparkles (India, 2015).jpg]] C2</comment>
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      <text bytes="79" sha1="0drcnlhszyxcl7guz9xee62i8ygo2nb" xml:space="preserve">im very cool
== yeah ==
[[Image:Fire cracker stick sparkles (India, 2015).jpg]]</text>
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  <page>
    <title>Applied linear operators and spectral methods/Lecture 1</title>
    <ns>0</ns>
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      <parentid>2388699</parentid>
      <timestamp>2024-12-16T22:21:01Z</timestamp>
      <contributor>
        <username>Reinforcing</username>
        <id>2995105</id>
      </contributor>
      <minor />
      <comment>/* Addition and scalar multiplication */ Correct the tense of &quot;defined&quot; to &quot;define&quot;</comment>
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      <text bytes="10080" sha1="bv9nmjrc1xf9bmvutmftxqkvja6jlr6" xml:space="preserve">Linear operators can be thought of as infinite dimensional matrices.  Hence we
can use well known results from matrix theory when dealing with linear operators.
However, we have to be careful.  A finite dimensional matrix has an inverse if
none of its eigenvalues are zero.  For an infinite dimensional matrix, even 
though all the eigenvectors may be nonzero, we might have a sequence of 
eigenvalues that tend to zero.  There are several other subtleties that we will
discuss in the course of this series of lectures.

Let us start off with the basics, i.e., linear vector spaces.

==Linear Vector Spaces (''S'')==
Let &lt;math&gt;\mathcal{S}&lt;/math&gt; be a linear vector space.  

===Addition and scalar multiplication===
Let us first define addition and scalar
multiplication in this space.  The addition operation acts completely in &lt;math&gt;\mathcal{S}&lt;/math&gt; 
while the scalar multiplication operation may involve multiplication either
by a real (in &lt;math&gt;\mathbb{R}&lt;/math&gt;) or by a complex number (in &lt;math&gt;\mathbb{C}&lt;/math&gt;).  These
operations must have the following closure properties:

#If &lt;math&gt;\mathbf{x}, \mathbf{y} \in \mathcal{S}&lt;/math&gt; then &lt;math&gt;\mathbf{x} + \mathbf{y} \in \mathcal{S}&lt;/math&gt;.
#If &lt;math&gt;\alpha \in \mathbb{R}&lt;/math&gt; (or &lt;math&gt;\mathbb{C}&lt;/math&gt;) and &lt;math&gt;\mathbf{x} \in \mathcal{S}&lt;/math&gt; then &lt;math&gt;\alpha~\mathbf{x} \in \mathcal{S}&lt;/math&gt;.

And the following laws must hold for addition

#&lt;math&gt;\mathbf{x} + \mathbf{y}&lt;/math&gt; = &lt;math&gt;\mathbf{y} + \mathbf{x} \qquad&lt;/math&gt; Commutative law.
#&lt;math&gt;\mathbf{x} + (\mathbf{y} + \mathbf{z})&lt;/math&gt; = &lt;math&gt;(\mathbf{x} + \mathbf{y}) + \mathbf{z} \qquad&lt;/math&gt; Associative law.
#&lt;math&gt;\exists \mathbf{0} \in \mathcal{S}&lt;/math&gt; such that &lt;math&gt;\mathbf{0} + \mathbf{x} = \mathbf{x} \quad \forall \mathbf{x} \in \mathcal{S} \qquad &lt;/math&gt; Additive identity.
#&lt;math&gt;\forall \mathbf{x} \in \mathcal{S} \quad \exists -\mathbf{x} \in \mathcal{S}&lt;/math&gt; such that &lt;math&gt;-\mathbf{x} + \mathbf{x} = \mathbf{0} \qquad &lt;/math&gt; Additive inverse.

For scalar multiplication we have the properties

#&lt;math&gt;\alpha~(\beta~\mathbf{x}) = (\alpha~\beta)~\mathbf{x}&lt;/math&gt;.
#&lt;math&gt;(\alpha + \beta)~\mathbf{x} = \alpha~\mathbf{x} + \beta~\mathbf{x}&lt;/math&gt;.
#&lt;math&gt;\alpha~(\mathbf{x}+\mathbf{y}) = \alpha~\mathbf{x} + \alpha~\mathbf{y}&lt;/math&gt;.
#&lt;math&gt;\mathbf{1}~\mathbf{x} = \mathbf{x}&lt;/math&gt;.
#&lt;math&gt;\mathbf{0}~\mathbf{x} = \mathbf{0}&lt;/math&gt;.

====Example 1: ''n'' tuples====
The &lt;math&gt;n&lt;/math&gt; tuples &lt;math&gt;(x_1, x_2, \dots, x_n)&lt;/math&gt; with
:&lt;math&gt;
  \begin{align}
  (x_1, x_2, \dots, x_n) + (y_1, y_2, \dots, y_n)  &amp; = 
  (x_1 + y_1, x_2 + y_2, \dots, x_n + y_n) \\
  \alpha~(x_1, x_2, \dots, x_n)  &amp; = 
  (\alpha~x_1, \alpha~x_2, \dots, \alpha~x_n) 
  \end{align}
&lt;/math&gt;
form a linear vector space.

====Example 2: Matrices====
Another example of a linear vector space is the set of &lt;math&gt;2 \times 2&lt;/math&gt; matrices 
with addition as usual and scalar multiplication, or more generally &lt;math&gt;n\times m&lt;/math&gt;
matrices.
:&lt;math&gt;
  \alpha\begin{bmatrix} x_{11} &amp; x_{12} \\ x_{21} &amp; x_{22} \end{bmatrix} = 
  \begin{bmatrix} \alpha~x_{11} &amp; \alpha~x_{12} \\ \alpha~x_{21} &amp; \alpha~x_{22} 
  \end{bmatrix}
&lt;/math&gt;

====Example 3: Polynomials====
The space of &lt;math&gt;n&lt;/math&gt;-th order polynomials forms a linear vector space.
:&lt;math&gt;
  p_n = \sum_{j=1}^n \alpha_j~x^j
&lt;/math&gt;

====Example 4: Continuous functions====
The space of continuous functions, say in &lt;math&gt;[0, 1]&lt;/math&gt;, also forms a linear vector 
space with addition and scalar multiplication defined as usual.

===Linear Dependence===
A set of vectors &lt;math&gt;\mathbf{x}_1, \mathbf{x}_2, \dots, \mathbf{x}_n \in \mathcal{S}&lt;/math&gt; are said to be linearly
dependent if &lt;math&gt;\exists~ \alpha_1, \alpha_2, \dots, \alpha_n&lt;/math&gt; not all zero such that
:&lt;math&gt;
  \alpha~\mathbf{x}_1 + \alpha~\mathbf{x}_2 + \dots + \alpha~\mathbf{x}_n = \mathbf{0}
&lt;/math&gt;
If  such a set of constants &lt;math&gt;\alpha_1, \alpha_2, \dots, \alpha_n&lt;/math&gt; do not exists 
then the vectors are said to be linearly independent.

====Example====
Consider the matrices
:&lt;math&gt;
  \boldsymbol{M}_1 = \begin{bmatrix} 1 &amp; 0 \\ 0 &amp; 2 \end{bmatrix}, 
  \boldsymbol{M}_2 = \begin{bmatrix} 1 &amp; 0 \\ 0 &amp; 0 \end{bmatrix}, 
  \boldsymbol{M}_3 = \begin{bmatrix} 0 &amp; 0 \\ 0 &amp; -1 \end{bmatrix} 
&lt;/math&gt;
These are linearly dependent since &lt;math&gt;\boldsymbol{M}_1 - \boldsymbol{M}_2 + 2~\boldsymbol{M}_3 = \mathbf{0}&lt;/math&gt;.

===Span===
The span of a set of vectors &lt;math&gt;(\boldsymbol{T})&lt;/math&gt; is the set of all vectors that are
linear combinations of the vectors &lt;math&gt;\mathbf{x}_i&lt;/math&gt;.  Thus
:&lt;math&gt;
  \text{span}(\boldsymbol{T}) = \{\boldsymbol{T}_1, \boldsymbol{T}_2, \dots, \boldsymbol{T}_n\} 
&lt;/math&gt;
where
:&lt;math&gt;
  \boldsymbol{T}_i = \alpha_1~\mathbf{x}_1 + \alpha_2~\mathbf{x}_2 + \dots + \alpha_n~\mathbf{x}_n
&lt;/math&gt;
as &lt;math&gt;\alpha_1, \alpha_2, \dots, \alpha_n&lt;/math&gt; vary.

====Spanning set====
If the span = &lt;math&gt;\mathcal{S}&lt;/math&gt; then &lt;math&gt;\boldsymbol{T}&lt;/math&gt; is said to be a spanning set.

===Basis===
If &lt;math&gt;\boldsymbol{T}&lt;/math&gt; is a spanning set and its elements are linearly independent then we call 
it a basis for &lt;math&gt;\mathcal{S}&lt;/math&gt;.  A vector in &lt;math&gt;\mathcal{S}&lt;/math&gt; has a unique representation as a 
linear combination of the basis elements. '' why is it unqiue?''

====Dimension====
The dimension of a space &lt;math&gt;\mathcal{S}&lt;/math&gt; is the number of elements in the basis.  This is 
independent of actual elements that form the basis and is a property of &lt;math&gt;\mathcal{S}&lt;/math&gt;.

====Example 1: Vectors in ''R&lt;sup&gt;2&lt;/sup&gt;''====
Any two non-collinear vectors &lt;math&gt;\mathbb{R}^2&lt;/math&gt; is a basis for &lt;math&gt;\mathbb{R}^2&lt;/math&gt; 
because any other vector in &lt;math&gt;\mathbb{R}^2&lt;/math&gt; can be expressed as a linear 
combination of the two vectors.

====Example 2: Matrices====
A basis for the linear space of &lt;math&gt;2 \times 2&lt;/math&gt; matrices is
:&lt;math&gt;
  \begin{bmatrix} 1 &amp; 0 \\ 0 &amp; 0 \end{bmatrix}, 
  \begin{bmatrix} 1 &amp; 1 \\ 0 &amp; 0 \end{bmatrix}, 
  \begin{bmatrix} 1 &amp; 1 \\ 0 &amp; 1 \end{bmatrix}, 
  \begin{bmatrix} 1 &amp; 3 \\ 1 &amp; 1 \end{bmatrix} 
&lt;/math&gt;
Note that there is a lot of nonuniqueness in the choice of bases.  One 
important skill that you should develop is to choose the right basis to solve
a particular problem.

====Example 3: Polynomials====
The set &lt;math&gt;\{1, x, x^2, \dots, x^n\}&lt;/math&gt; is a basis for polynomials of degree &lt;math&gt;n&lt;/math&gt;.
  
====Example 4: The natural basis====
A natural basis is the set &lt;math&gt;\{ \mathbf{e}_1, \mathbf{e}_2, \dots, \mathbf{e}_n\}&lt;/math&gt; where the &lt;math&gt;j&lt;/math&gt;th 
entry of &lt;math&gt;\mathbf{e}_k&lt;/math&gt; is
:&lt;math&gt;
  \delta_{jk} = \begin{cases} 1 &amp; \mbox{for} ~j = k \\
      0 &amp; \mbox{for}~ j \ne k \end{cases}
&lt;/math&gt;
The quantity &lt;math&gt;\delta_{jk}&lt;/math&gt; is also called the Kronecker delta.

===Inner Product Spaces===
To give more structure to the idea of a vector space we need concepts such as
magnitude and angle.  The inner product provides that structure.

The inner product generalizes the concept of an angle and is defined as a 
function
:&lt;math&gt;
  \langle\bullet,~\bullet\rangle : \mathcal{S}\times\mathcal{S} \rightarrow \mathbb{R}
  \quad (\text{or}~\mathbb{C}~\text{for a complex vector space})
&lt;/math&gt;
with the properties

#&lt;math&gt;\langle\mathbf{x},~\mathbf{y}\rangle = \overline{\langle\mathbf{y},~\mathbf{x}\rangle} \qquad&lt;/math&gt; overbar indicates complex conjugation.
#&lt;math&gt;\langle\alpha~\mathbf{x},~\mathbf{y}\rangle = \alpha~\langle\mathbf{x},~\mathbf{y}\rangle \quad&lt;/math&gt; Linear with respect to scalar multiplication.
#&lt;math&gt;\langle\mathbf{x}+\mathbf{y},~\mathbf{z}\rangle = \langle\mathbf{x},~\mathbf{z}\rangle + \langle\mathbf{y},~\mathbf{z}\rangle \quad&lt;/math&gt; Linearity with respect to addition.
#&lt;math&gt;\langle\mathbf{x},~\mathbf{x}\rangle &gt; \mathbf{0}&lt;/math&gt; if &lt;math&gt;\mathbf{x} \ne 0&lt;/math&gt; and &lt;math&gt;\langle\mathbf{x},~\mathbf{x}\rangle = \mathbf{0}&lt;/math&gt; if and only if &lt;math&gt;\mathbf{x} = \mathbf{0}&lt;/math&gt;.

A vector space with an inner product is called an inner product space.

====Example 1:====
:&lt;math&gt;
  \langle\mathbf{x},~\beta~\mathbf{y}\rangle = \overline{\langle\beta~\mathbf{y},~\mathbf{x}\rangle} 
    = \overline{\beta}~\overline{\langle\mathbf{y},~\mathbf{x}\rangle} 
    = \overline{\beta}\langle\mathbf{x},~\mathbf{y}\rangle
&lt;/math&gt;

====Example 2: Discrete vectors====
In &lt;math&gt;\mathbb{R}^n&lt;/math&gt; with &lt;math&gt;\mathbf{x} = \{x_1, x_2, \dots, x_n \}&lt;/math&gt; and 
&lt;math&gt;\mathbf{y} = \{y_1, y_2, \dots, y_n \}&lt;/math&gt; the Eulidean norm is given by
:&lt;math&gt;
  \langle\mathbf{x},~\mathbf{y}\rangle = \sum_n x_n~y_n 
&lt;/math&gt;
With &lt;math&gt;\mathbf{x}, \mathbf{y} \in \mathbb{C}^n&lt;/math&gt; the standard norm is
:&lt;math&gt;
  \langle\mathbf{x},~\mathbf{y}\rangle = \sum_k x_k~\overline{y_k}
&lt;/math&gt;

====Example 3: Continuous functions====
For two complex valued continuous functions &lt;math&gt;f(x)&lt;/math&gt; and &lt;math&gt;g(x)&lt;/math&gt; in &lt;math&gt;[0, 1]&lt;/math&gt;
we could approximately represent them by their function values at 
'' equally spaced points.''

Approximate &lt;math&gt;f(x)&lt;/math&gt; and &lt;math&gt;g(x)&lt;/math&gt; by
:&lt;math&gt;
  \begin{align}
  F &amp; = \{f(x_1), f(x_2), \dots, f(x_n)\} \qquad \text{with} ~x_k = \cfrac{k}{n}\\
  G &amp; = \{g(x_1), g(x_2), \dots, g(x_n)\} \qquad \text{with} ~x_k = \cfrac{k}{n}
  \end{align}
&lt;/math&gt;
With that approximation, a natural norm is
:&lt;math&gt;
  \langle F,~G\rangle = \cfrac{1}{n}~\sum_{k=1}^n f(x_k)~\overline{g(x_k)}
&lt;/math&gt;
Taking the limit as &lt;math&gt;n \rightarrow \infty&lt;/math&gt; (show this)
:&lt;math&gt;
  \langle f,~g\rangle = \int_0^1 f(x)~\overline{g(x)}~dx 
&lt;/math&gt;

If we took '' non-equally spaced'' yet smoothly distributed points we would get
:&lt;math&gt;
  \langle f,~g\rangle = \int_0^1 f(x)~\overline{g(x)}~w(x)~dx 
&lt;/math&gt;
where &lt;math&gt;w(x) &gt; 0&lt;/math&gt; is a smooth weighting function (show this).

There are many other inner products possible.  For functions that are not only
continuous but also differentiable, a useful norm is
:&lt;math&gt;
  \langle f,~g\rangle = \int_0^1 \left[f(x)~\overline{g(x)} + 
      f^{'}(x)~\overline{g^{'}(x)}\right]~dx
&lt;/math&gt;

We will continue further explorations into linear vector spaces in the next 
lecture.

[[Category:Functional analysis]]
{{lecture}}</text>
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  <page>
    <title>User:Chocoman</title>
    <ns>2</ns>
    <id>40813</id>
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      <parentid>2692147</parentid>
      <timestamp>2024-12-16T12:15:51Z</timestamp>
      <contributor>
        <username>MathXplore</username>
        <id>2888076</id>
      </contributor>
      <minor />
      <comment>Reverted edits by [[Special:Contributions/62.254.28.193|62.254.28.193]] ([[User_talk:62.254.28.193|talk]]) to last version by [[User:Dionysios|Dionysios]] using [[Wikiversity:Rollback|rollback]]</comment>
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      <text bytes="414" sha1="colgjjrwseuuwncjd1r2mm97w1n60xy" xml:space="preserve">==CHOCOMAN==
Welcome to my page!! I am glad to be a wikiversitian,and very new here.
I heard that this site is the best school on the internet,I'm in college.
No [[Wikipedia:Administrators]] or anyone else from wikipedia can go to my page!!
That's why I have an account in Wikipedia,so Wikipedians can cooment.Only Wikiversitians
can comment me.
Wikipedia username:[[w:User:S495|S495]]
The page is to be continued.</text>
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  <page>
    <title>Aymaran</title>
    <ns>0</ns>
    <id>48124</id>
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      <timestamp>2024-12-16T22:30:46Z</timestamp>
      <contributor>
        <username>CommonsDelinker</username>
        <id>9184</id>
      </contributor>
      <comment>Replacing Banner_of_the_Qulla_Suyu.svg with [[File:Banner_of_the_Qulla_Suyu_(1979).svg]] (by [[:c:User:CommonsDelinker|CommonsDelinker]] because: [[:c:COM:FR|File renamed]]: [[:c:COM:FR#FR4|Criterion 4]] (harmonizing names of file set)).</comment>
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      <text bytes="2582" sha1="7bpbl7jhac32pqa9fhuxs93oc41pxc7" xml:space="preserve">{| cellpadding=&quot;10&quot; cellspacing=&quot;5&quot; style=&quot;width: 99%; background-color: inherit; margin-left: auto; margin-right: auto&quot;
| style=&quot;background-color: #bbd2e1; border: 1px solid #777777; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;&quot; colspan=&quot;1&quot; |&lt;br&gt;
[[File:Banner of the Qulla Suyu (1979).svg|200px]]&lt;big&gt;&amp;nbsp;'''Introduction to Aymaran'''&lt;/big&gt;
[[File:Sillustani, Perú, 2015-08-01, DD 108-112.JPG|250px|Perou]]
|-
| style=&quot;width: 60%; background-color: #fffff0; border: 1px solid #777777; vertical-align: top; -moz-border-radius-topleft: 8px; -moz-border-radius-bottomleft: 8px; -moz-border-radius-topright: 8px; -moz-border-radius-bottomright: 8px;&quot; &lt;!--rowspan=&quot;2&quot;--&gt; |
{{center top}}&lt;big&gt;&lt;b&gt;Aymara&lt;/b&gt;&lt;/big&gt;{{center bottom}}
{{languages}}
{{testing}}
[[File:Aymara-language-domain-en-001.svg|thumb|Geographic distribution of the Aymara language.]]

'''Aymaran''' or '''Aymara''' is a South American language.

== Geographical distribution ==
There are roughly 1.6 million Bolivian speakers, 420,000 Peruvian speakers and 15,000 Chilean speakers.&lt;ref&gt;Cerron-Palomino, 2000, pp 68-70.&lt;/ref&gt; At the time of the Spanish conquest, in the sixteenth century, Aymara was the dominant language over a much larger area than today, including most of highland Peru south of Cuzco. Over the centuries Aymara has gradually lost speakers both to Spanish and to Quechua; today, many Peruvian and Bolivian communities which were once Aymara-speaking speak Quechua.&lt;ref&gt;Xavier Albó, &quot;Andean People in the Twentieth Century,&quot; in ''The Cambridge History of the Native Peoples of the Americas. Vol. III: South America'', ed. Frank Salomon and Stuart B. Schwartz (New York: Cambridge University Press, 1999), 765-871.&lt;/ref&gt;

==References==
{{reflist}}

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==Topics==
* First text&lt;nowiki&gt;: &lt;/nowiki&gt;'''[[/About/]]'''
* Second text&lt;nowiki&gt;: &lt;/nowiki&gt;'''[[/Pronunciation/]]'''
* Third text&lt;nowiki&gt;: &lt;/nowiki&gt;'''[[/Bases/]]'''
* Fourth text&lt;nowiki&gt;: &lt;/nowiki&gt;'''[[/Dialects/]]'''
* Fifth text&lt;nowiki&gt;: &lt;/nowiki&gt;'''[[/Phonology/]]'''
|}

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{{CourseCat}}
[[Category:Humanities courses]]
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    <title>Help talk:Resources by type</title>
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      <text bytes="7773" sha1="ksf5gc9qntf28resimlvm2qdew2s0h1" xml:space="preserve">==Available project boxes==
How might we make this section into a sortable table? -- [[User:Jtneill|Jtneill]] - &lt;small&gt;[[User talk:Jtneill|Talk]] - [[Special:Contributions/Jtneill|c]]&lt;/small&gt; 02:41, 20 August 2008 (UTC)

== table ==

The table on this page may create a horizontal scroll for some users. [[User:Emesee|Emesee]] 17:05, 29 August 2008 (UTC)

== Sub-categories? ==

So… these project boxes force pages to be in generic categories!  What this means is that any page with a History userbox immediately gets thrown into the [[:Category:History|History]] supercategory.  With hundreds of history pages, this could quickly make the History category completely unmanageable!  Can you please make it so the Category: inclusion here is a subst: inclusion, instead, so that conscientious categorizers can leave the templates up but move them to relevant subcategories?  Please?  [[User:Jade Knight|The Jade Knight]] 10:22, 10 September 2008 (UTC)
:Better yet would be to create a subst: category that puts them in [[:Category:Unsorted Pages in History]] or something like that, so the rest of us know and can then go and sort them.  [[User:Jade Knight|The Jade Knight]] 11:00, 10 September 2008 (UTC)
:: Give me a moment - I'm thinking! :) --[[User:McCormack|McCormack]] 11:00, 10 September 2008 (UTC)
:: Yes - I thought about this one. There are a number of ways to deal with this. It depends on the complexity with which an editor is editing, and your editing requirements are higher than the average ;-) It should be possible, if I have done these templates right, and have them on the latest version, to override the category with something more specific. --[[User:McCormack|McCormack]] 11:02, 10 September 2008 (UTC)
:: OK - I've checked the history template, and no, I haven't updated that one yet to the latest project box version. So you can't yet override the category. I can update on request. --[[User:McCormack|McCormack]] 11:04, 10 September 2008 (UTC)
=== Explanation ===
==== The bigger picture ====

The main thing about using templates for categorisation is that '''''they can be retrospectively updated''''' and the updates then spread to all the pages where the templates are used. Suppose that on day 1, WV has 2 history pages. We categorise them as &quot;history&quot;. On day 1000, WV has 2000 history pages (let's hope), and then we find it is no longer good to categorise them just as history. So we change the category to [[:Category:Unsorted Pages in History]] (or something like that), insert a parser function into the template which divides them automatically depending on various criteria such as date, size, title keywords, etc, which then helps us to create a better category system for them. The problem with categories is that as the whole system evolves, the categories must get more complex, and try as you might, you can never predict the most sensible category system. Using templates rather than analogue manual text for categories allows us to get a step deeper into efficient category management because of the possibility of retrospective updating. Am I making sense? --[[User:McCormack|McCormack]] 11:09, 10 September 2008 (UTC)

==== Other approaches ====

Suppose an expert historian comes to WV and wishes to create a better history category system. The historian is free to clone many derivatives of the history template - e.g. for history by country, continent, century, topic, etc. The historian can then apply these templates to pages instead. Put it in other words, my initial attempt at subject-related project boxes was not intended to be exhaustive - quite the contrary, just a start. --[[User:McCormack|McCormack]] 11:12, 10 September 2008 (UTC)

==== Using subst: ====

Using ''subst:'' would defeat the main point, which is to enable retrospective changes to the category system to be applied site-wide. But discuss... --[[User:McCormack|McCormack]] 11:14, 10 September 2008 (UTC)

=== My dilemma ===

I think I understand what you're saying, but I don't see a solution to my problem.  Let me be more specific:  I'm going around totally restructuring the History category system right now to a) make it more user-friendly, and b) make it more powerful (functional).  We currently have literally hundreds of History categories right now (I think), and I'm going around and trying to put pages where they go in these categories.  Sometimes pages go in only one, sometimes they go in several categories.  When I see a page with a Project Box: History on it, I see only one category.  I want to change this category, but I do not know how without removing the project box, which someone may not like.  What should I do?  [[User:Jade Knight|The Jade Knight]] 11:37, 10 September 2008 (UTC)

: Hi. Here are some options about what to do. (1) Replace the box with a &quot;bare category&quot; (i.e. manual, not template) - which is not a preferred solution. (2) Create 100 history templates each with a different built-in category - possibly not an ideal solution - little too much work and organisation there! (3) Update history template with a &quot;cat&quot; parameter for overriding the built-in category. --[[User:McCormack|McCormack]] 04:11, 11 September 2008 (UTC)
::Frankly, I'd really prefer making the ''cat'' element a subst: within the template itself (though the template itself wouldn't be subst:ed in, is it possible to have a normal template with a piece which is subst:?  Or is that impossible with the software?)  Next to that, my preference would be option 3; least work for greatest utility… though ''I'm'' not opposed to option 1, it would kind of kick History out of the resource template club, and some editors might not like that.  [[User:Jade Knight|The Jade Knight]] 05:42, 11 September 2008 (UTC)

One more issue:  If the project box vanishes, so does the category.  So if someone doesn't end up liking the box, then the page has to be recategorized all over again (assuming they even notice the category is gone).  [[User:Jade Knight|The Jade Knight]] 11:37, 10 September 2008 (UTC)

: Yes, you are correct. Two points: (1) People tend not to remove these boxes unless they know what they're doing, so the problem may not be that large. (2) If worried, surround the template with &lt;nowiki&gt;&lt;!-- Don't remove this code! --&gt;&lt;/nowiki&gt; messages in the source. --[[User:McCormack|McCormack]] 04:11, 11 September 2008 (UTC)

::An infrequent minor headache is still a minor headache, even if it's an infrequent minor one.  [[User:Jade Knight|The Jade Knight]] 05:42, 11 September 2008 (UTC)

==Further Ideas==

* &lt;nowiki&gt;{{reflection}}&lt;/nowiki&gt; - pause for reflection ....{{unsigned|Ktucker}}

== Difference between an article and a paper? ==

Can someone clarify the difference between {{tl|article}} and {{tl|paper}} for me?   I'm assuming the latter is more formal and traditionally academic?  &amp;mdash; [[User:Samwilson|Sam Wilson]] ( &lt;span style=&quot;font-size:0.9em&quot;&gt;[[User_talk:Samwilson|Talk]] &amp;bull; [[Special:Contributions/Samwilson|Contribs]]&lt;/span&gt; ) &amp;hellip; 08:02, 7 July 2011 (UTC)

:@[[User:Samwilson|Samwilson]] Yup, I would understand it that way. Article could be less formal, paper would be more academic. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:44, 7 September 2024 (UTC)

== Learning by research ==

What type of learning material would be that attendees learn by research? Literautre or practical. [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 08:43, 7 September 2024 (UTC)

== No template for research ==

So there is not a box for research projects? [[User:Juandev|Juandev]] ([[User talk:Juandev|discuss]] • [[Special:Contributions/Juandev|contribs]]) 07:17, 17 December 2024 (UTC)</text>
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      <comment>Links in the &quot;See also&quot; section led to an advertisement template or were a dead link. I deleted the &quot;See also&quot; section. It could be reinstated after updating the links.</comment>
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      <text bytes="1072" sha1="ehc2ubfs3j66u3m0o8toj56xc9utkbn" xml:space="preserve">* [http://webh01.ua.ac.be/biomag/ Bioimaging Lab], University of Antwerp
* [http://www.brunel.ac.uk/about/acad/sss/research/centres/cfcan Centre for Cognition and Neuroimaging], Brunel University 
* [http://www.wbic.cam.ac.uk/ Wolfson Brain Imaging Centre, University of Cambridge]
* [http://www.wbic.cam.ac.uk/ Cardiff University Brain and Repair Imaging Centre], Cardiff
* [http://www.dcn.ed.ac.uk/bic/ SFC Brain Imaging Research Centre], University of Edinburgh
* [http://medisip.elis.ugent.be Medical Image and Signal Processing], Ghent University
* [http://www.bcn-nic.nl/ BCN Neuroimaging Center], University Medical Center Groningen, University of Groningen
* [http://www.kuleuven.ac.be/radiology/Research MR Research Centre], KUleuven
* [http://w3.umh.ac.be/~nmrlab/index.html NMR and Molecular Imaging Laboratory], Université de Mons-Hainaut
* [http://www.fmrib.ox.ac.uk/ FMRIB Centre, University of Oxford]
* [http://www.fil.ion.ucl.ac.uk Wellcome Trust Centre for Neuroimaging], University College London

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      <text bytes="34456" sha1="to42vjvmwp34zfn3tfe3gf9klvmxx6j" xml:space="preserve">==Strong (nuclear) gravitation ==
In Astronomy the only one available characteristic empirical physical constant is the gravitational constant. Without completing the charge-mass unification or final unification: one cannot say, whether it is an ‘input to the unification’ or ‘output of unification’. The same idea can be applied to the atomic physical constants also. Sitting in a grand unified roof one cannot make an  ‘absolute measurement’ but can make an ‘absolute finding’. Up till now, no atomic model has implemented the gravitational constant in the atomic or nuclear physics. Then, whatever may be its magnitude, measuring its value from existing atomic principles is impossible. Its value has been measured in the lab only within a range of 1 cm to a few metres, whereas the observed nuclear size is 1.2 fermi. Until one measures the value of the gravitational constant in microscopic physics, the debate of strong (nuclear) gravitation can be considered positively. The idea of strong gravitation originally referred specifically to mathematical approach of Abdus Salam of unification of gravitation and quantum chromodynamics, but is now often used for any particle level gravitation approach. Now many persons are working on this subject. A main advantage of this subject is: it couples black hole physics and particle physics.

==Strong gravitational constant==
The '''strong gravitational constant''', denoted &lt;math&gt;~\;\; \Gamma &lt;/math&gt; or &lt;math&gt;~G_s &lt;/math&gt;, is a grand unified physical constant of strong gravitation, involved in calculation of gravitational attraction at the level of elementary particles and atoms.

According to [[w:Isaac Newton| Newton]]'s law of universal gravitation, the force of gravitational attraction between two massive points with masses &lt;math&gt; ~ m_1 &lt;/math&gt; and &lt;math&gt; ~ m_2 &lt;/math&gt;, located at a distance &lt;math&gt; ~ R &lt;/math&gt; between them, is:

: &lt;math&gt;F=G \frac{m_1 m_2}{R^2}.&lt;/math&gt;

The coefficient of proportionality &lt;math&gt; ~ G &lt;/math&gt; in this expression is called [[w:gravitational constant |gravitational constant]]. It is assumed, that in contrast to the usual force of gravity, at the level of elementary particles acts [[Physics/Essays/Fedosin/Strong gravitation |strong gravitation]]. In order to describe it &lt;math&gt; ~ G &lt;/math&gt; in the formula for gravitational force must be replaced on &lt;math&gt; ~ \Gamma &lt;/math&gt;:

: &lt;math&gt;F_{sg}=\Gamma \frac{m_1 m_2}{R^2}.&lt;/math&gt;

==Dimensions and magnitude==
The dimensions assigned to the strong gravitational constant may be found from the equation above — length cubed, divided by [[mass]] and by time squared (in SI units, metres cubed per kilogram per second squared).

There are several ways to assess the value of &lt;math&gt; ~ \Gamma &lt;/math&gt;. J. Dufour, under the assumption that the strong gravitational constant depends on the type of objects, from the interaction of two deuterium nuclei determined, &lt;ref&gt; J. Dufour. [http://www.iscmns.org/CMNS/CMNS.htm &quot;Very sizeable increase of gravity at pico-meter distance: a novel working hypothesis to explain anomalous heat effects and apparent transmutations in certain metal hydrogen systems&quot;]. J. of condensed matter nuclear science, Vol. 1, pp. 47-61 (2007). &lt;/ref&gt; that  &lt;math&gt; G' = 2.06 \times 10^{25} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

Based on the analogy between hadrons and Kerr-Newman [[black hole]]s &lt;ref&gt;Strong Interactions, Gravitation and Cosmology. Abdus Salam Publ. in: NATO Advanced Study Institute, Erice, June16-July 6, 1972 ; in: High Energy Astrophysics and its Relation to Elementary Particle Physics, 441-452 MIT Press, Cambridge (1974). &lt;/ref&gt; Sivaram, C. and Sinha, K.P, &lt;ref&gt; Sivaram, C. and Sinha, K.P. Strong gravity, black holes, and hadrons. Physical Review D, Vol. 16, Issue 6, pp. 1975-1978 (1977). &lt;/ref&gt;  &lt;ref&gt;Salam A. and Sivaram C. Strong Gravity Approach to QCD and Confinement. Mod. Phys. Lett., v. A8(4), pp. 321-326 (1993). &lt;/ref&gt; and Raut, Usha and Shina, KP &lt;ref&gt;Raut, Usha and Shina, KP (1983) [http://eprints.iisc.ernet.in/13571/  Strong gravity and the fine structure constant.] In: Proceedings of the Indian Academy of Sciences Part A: Physical Sciences, 49 (2). pp. 352-358. &lt;/ref&gt; accepted the value &lt;math&gt; \Gamma = 6.7 \times 10^{27} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

This value of the strong gravitational constant allowed estimating the strong spin-torsion interaction between spinning protons. &lt;ref&gt;V. de Sabbata, C. Sivaram. [http://prints.iiap.res.in/bitstream/2248/4394/3/Strong%20spin-torsion%20 Strong Spin-Torsion Interaction between Spinning Protons.] Il Nuovo Cimento, Vol. 101A, N. 2, pp. 273-283 (1989). &lt;/ref&gt;

In paper of Mongan &lt;ref&gt;T. R.  Mongan. [http://th1.ihep.su/archive/gr-qc/070622.html Cold dark matter from &quot;strong gravity&quot;.] General Relativity &amp; Quantum Cosmology, 20 Jun 2007; [http://ru.arxiv.org/abs/0706.3050v2 arXiv:0706.3050v2.] &lt;/ref&gt; strong gravitational constant is &lt;math&gt; G_s = 1.1 \times 10^{28} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

According to [https://en.everybodywiki.com/Robert_Oldershaw Robert L. Oldershaw] &lt;ref&gt; Oldershaw R.L. [http://arxiv.org/abs/physics/0701132v3 Discrete Scale Relativity.] Astrophysics and Space Science, Vol. 311, N. 4, pp. 431-433 (2007). DOI: 10.107/s10509-007-9557-x. &lt;/ref&gt; value of the strong gravitational constant is &lt;math&gt; G_s = 2.18 \times 10^{28} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

As in Oldershaw’ paper, strong gravitational constant could be related &lt;ref&gt;Stone R.A. Quark Confinement and Force Unification. Progress in Physics, April 2010, Vol. 2, P. 19–20. &lt;/ref&gt; with the proton radius &lt;math&gt; ~ R_p &lt;/math&gt;, the proton mass &lt;math&gt; ~ m_p &lt;/math&gt; and the speed of light &lt;math&gt;~c &lt;/math&gt;: 
: &lt;math&gt;sG_p= \frac{R_p c^2}{2 m_p }= 2.4 \times 10^{28} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

According to Tennakone who identified the electron and the proton as black holes in the strong gravitational field, strong gravitational constant is: &lt;ref&gt;K. Tennakone. [http://prd.aps.org/abstract/PRD/v10/i6/p1722_1 Electron, muon, proton, and strong gravity.]  Phys. Rev. D, Volume 10, Issue 6, pp.1722-1725 (1974). &lt;/ref&gt;
: &lt;math&gt;\Gamma =  3.9 \times 10^{28} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

Zane Andrea Quintili finds a strong gravitational constant based on the similarity between the Planck mass and radius, and accordingly the mass and radius of the proton: &lt;ref&gt; Zane Andrea Quintili. [http://vixra.org/abs/1904.0540 Gravitational Field and Proton Radius]. vixra.org. (2019). &lt;/ref&gt;
: &lt;math&gt; G_q = \frac {8\hbar c }{m^2_p} =9.04 \times 10^{28} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

Recami et al &lt;ref&gt;Recami, E.; Ammiraju, P.; Hernandez, H.E.; Kretly, L.C.; Rodrigues, W.A., Jr. Elementary particles as micro-universes: a geometric approach to &quot;strong gravity&quot;. Apeiron, January 01, 1997. &lt;/ref&gt; &lt;ref&gt;Recami E. and Tonin-Zanchin V. The strong coupling constant: its theoretical derivation from a geometric approach to hadron structure. Found. Phys. Lett., v, 7(1), pp. 85-92 (1994). &lt;/ref&gt; define strong gravitational constant through the mass of the pion &lt;math&gt; ~ m_{\pi} &lt;/math&gt; as follows:

: &lt;math&gt;N\approx \frac{h c}{ m^2_{\pi} }= 3.2 \times 10^{30} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;,

where &lt;math&gt; ~ h &lt;/math&gt; – [[w:Planck constant |Planck constant]].

From this they derive [[w:coupling constant | constant of strong interaction]] of two nucleons in the following form: &lt;ref&gt;Erasmo Recami, Tonin-Zanchin, Antonino Del Popolo, Mario Gambera. [http://arxiv.org/abs/physics/0105080 The strong coupling constant,] Heavy Ion Physics, Vol. 10, pp. 345-349 (1999). &lt;/ref&gt;

:&lt;math&gt; \frac{ N g^2}{\hbar c } \approx 14&lt;/math&gt; , where &lt;math&gt;~g &lt;/math&gt; indicates a strong charge, &lt;math&gt; ~ \hbar &lt;/math&gt; is reduced Planck constant.

Stanislav Fisenko et all found &lt;ref&gt;Stanislav Fisenko &amp; Igor Fisenko. [http://www.ccsenet.org/journal/index.php/apr/article/download/8060/6060 The Conception of Thermonuclear Reactor on the Principle of Gravitational Confinement of Dense High-temperature Plasma.] Applied Physics Research, Vol. 2, No. 2, pp. 71-79 (2010). &lt;/ref&gt; &lt;ref&gt;S. I. Fisenko, M. M. Beilinson and B. G. Umanov. Some notes on the concept of “strong” gravitation and possibilities of its experimental investigation. Physics Letters A, Volume 148, Issues 8-9, pp. 405-407 (1990).&lt;/ref&gt; a spectrum of steady states of the electron in proper gravitational field (0.511 MeV …0.681 MeV) on the base of strong coupling constant
: &lt;math&gt;N= 5.1 \times 10^{31} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

U. V. S. Seshavatharam and S. Lakshminarayana &lt;ref&gt; U. V. S. Seshavatharam and S. Lakshminarayana. Strong nuclear gravitational constant and the origin of nuclear planck scale. Progress in Physics, vol. 3, pp. 31-38 (2010). [http://fs.gallup.unm.edu/PP-03-2010.pdf] &lt;/ref&gt; in determining &lt;math&gt; ~ G_s &lt;/math&gt; repelled from the Fermi constant, which led them to the value &lt;math&gt; G_s = 6.94 \times 10^{31} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

In the paper &lt;ref&gt;Perng J. J. [https://springerlink3.metapress.com/content/1503066144130716/resource-secured/?target=fulltext.pdf&amp;sid=qwdy0a45odlh3a55v4k45u45&amp;sh=www.springerlink.com Strong gravitation and elementary particles.] Nuovo Cimento, Lettere, Serie 2, vol. 23, N. 15, pp. 552-554 (1978). &lt;/ref&gt; strong gravitational constant equal to &lt;math&gt;\Gamma =2.77 \times 10^{32} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

[[User:Fedosin | Sergey Fedosin]] entered the strong gravitational constant in 1999 on the basis of equality between the Coulomb electric force and gravitational force in the hydrogen atom on the [[w:Bohr radius |Bohr radius]]. This leads to the following expression for the value of the strong gravitational constant: &lt;ref name=&quot;fed&quot;&gt; Fedosin S.G. [http://lccn.loc.gov/2009457349 Fizika i filosofiia podobiia: ot preonov do metagalaktik,] Perm, (1999-06-09) 544 pp. {{ISBN|5-8131-0012-1}}. &lt;/ref&gt;

: &lt;math&gt;\Gamma= \frac{e^2}{4 \pi \varepsilon_{0} m_p m_e }=1.514 \times 10^{29} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;,

where &lt;math&gt; ~ e &lt;/math&gt; – [[w:elementary charge |elementary charge]], &lt;math&gt; ~ \pi &lt;/math&gt; – [[w:pi (number) | pi]], &lt;math&gt; ~ \varepsilon_{0} &lt;/math&gt; – [[electric constant]], &lt;math&gt; ~ m_p &lt;/math&gt; – the mass of [[w:proton |proton]], &lt;math&gt; ~ m_e &lt;/math&gt; – the mass of [[electron]]. 

It is assumed that strong gravitation, as a universal force, acts on the matter of nucleons, hadrons, electrons and elementary particles, regardless of the type of these particles. In contrast, the standard approach considers that strong interaction does not affect electrons and other leptons.

The small mass and large charge of matter do not allow the electron to be entirely in some small volume near the nucleus, and it gets disklike axisymmetric shape, which is limited by size of atom. In the hydrogen atom electrical forces between the nucleus and matter of the electron are attractive, but they are compensated by the repulsion of the intrinsic charge of the electron. There are the centripetal force of rotation of the electron around the nucleus, and the gravitational attraction between massive nucleus and matter of the electron. All these forces are equal in magnitude. From here follows that the action of strong gravitation between the masses of nucleus and electron on the one hand, and the electric force between charges of the nucleus and the electron, on the other hand, allows to estimate the value of &lt;math&gt; ~ \Gamma &lt;/math&gt;.
If &lt;math&gt;~ R_B = \frac {\hbar }{ m_e \alpha c } &lt;/math&gt; is the Bohr radius, then the equality of forces gives:

: &lt;math&gt; \frac {\Gamma m_p m_e }{R^2_B} = \frac{e^2}{4 \pi \varepsilon_{0} R^2_B } .&lt;/math&gt;

[[Physics/Essays/Fedosin/Fine structure constant | Fine structure constant]] is
:&lt;math&gt; \alpha = \frac{e^2}{4 \pi \varepsilon_0 \hbar c}, &lt;/math&gt;

So that
: &lt;math&gt; \Gamma= \frac{\alpha \hbar c }{m_p m_e }, \qquad \qquad \hbar = \frac{\Gamma m_p m_e }{ \alpha c }.&lt;/math&gt;

Bohr radius becomes equal
:&lt;math&gt;~ R_B = \frac{\Gamma m_p }{ \alpha^2 c^2 } = \frac{\Gamma m_p }{ V^2_B },&lt;/math&gt;

where &lt;math&gt;~ V_B = \alpha c &lt;/math&gt; is the orbital speed of the electron cloud at the first energy level.

Hence &lt;math&gt;~ V^2_B = \frac{\Gamma m_p }{ R_B }&lt;/math&gt;, and the kinetic energy of the electron, taking into account determination of strong gravitational constant, is equal to:
:&lt;math&gt;~ K = \frac{m_e V^2_B }{ 2 } = \frac{\Gamma m_p m_e }{ 2 R_B }=\frac { e^2}{8 \pi \varepsilon_0 R_B } = - \frac {W}{2} ,&lt;/math&gt;

where &lt;math&gt;~ W &lt;/math&gt;  is the potential energy of electron in the electric field of the nucleus of a hydrogen atom.

It turns out the virial theorem in the form &lt;math&gt;~ K = - \frac {W}{2} &lt;/math&gt;. The total electron energy is also found at the first energy level:

:&lt;math&gt;~ E = K+W = \frac {W}{2} = -K = -13.6 &lt;/math&gt; eV.

With the help of the constant &lt;math&gt; ~ \Gamma &lt;/math&gt; the [[w:Invariant mass#Rest energy | rest energy]] of proton in the form of a ball is equal to half of its potential energy of strong gravitational field in accordance with [[w:virial theorem |virial theorem]], &lt;ref&gt; [[User:Fedosin | Sergey Fedosin]], [https://www.lap-publishing.com/catalog/details//store/gb/book/978-3-659-57301-9/the-physical-theories-and-infinite-hierarchical-nesting-of-matter-volume-1 The physical theories and infinite hierarchical nesting of matter], Volume 1, LAP LAMBERT Academic Publishing, pages: 580, {{ISBN|978-3-659-57301-9}}. &lt;/ref&gt; if we assume that the binding energy &lt;math&gt; ~ E_b &lt;/math&gt; for the proton up to a sign is equal to the total energy of proton, and &lt;math&gt; ~ E_b &lt;/math&gt; becomes very close to relativistic energy in the form of rest energy:

: &lt;math&gt;~ m_p c^2 \approx E_b = -\frac {W_p}{2} = \frac{ k \Gamma m^2_p }{ 2R_p},&lt;/math&gt;

where &lt;math&gt; ~ R_p =8.73 \times 10^{-16} &lt;/math&gt; m  is the proton radius,
&lt;math&gt; ~ k=0.62 &lt;/math&gt; (in the hypothetical case of a uniform mass density of the proton there must be &lt;math&gt; ~ k = 0.6 &lt;/math&gt;). This implies that the mass of nucleons is determined by the energy of the strong gravitation according to the principle of [[w:mass–energy equivalence |mass–energy equivalence]].

If we assume that the magnetic moment of the proton is created by the maximum rotation of its positive charge distributed over the volume of the proton in the form of a ball, when the centripetal acceleration at the equator becomes equal to acceleration of strong gravitation, the formula for the magnetic moment is as follows:
: &lt;math&gt; ~ P_m = \delta e \sqrt {\Gamma m_p R_p}, &lt;/math&gt;

where &lt;math&gt; ~ P_m = 1.41 \times 10^{-26} &lt;/math&gt; J / T  is the magnetic moment of the proton,
&lt;math&gt; ~ \delta = 0.1875 &lt;/math&gt; (in the case of uniform density and charge should be &lt;math&gt; ~ \delta = 0.2 &lt;/math&gt;).

From the formulas for the energy and the magnetic moment the radius of the proton is determined in the self-consistent model. &lt;ref&gt; Fedosin S.G. [http://vixra.org/abs/1208.0006 The radius of the proton in the self-consistent model.] Hadronic Journal, Vol. 35, No. 4, pp. 349-363 (2012). &lt;/ref&gt;

The strong gravitational constant is also included in the formula describing the [[w:nuclear force |nuclear force]] through strong gravitation and [[gravitational torsion field]] of rotating particles. &lt;ref&gt;[http://sergf.ru/com.htm Comments to the book]: Fedosin S.G. Fizicheskie teorii i beskonechnaia vlozhennost’ materii. – Perm, 2009, 844 pages, Tabl. 21, Pic. 41, Ref. 289. {{ISBN|978-5-9901951-1-0}}. (in Russian). &lt;/ref&gt; A feature of the [[gravitational induction]] is that if two bodies rotate along one axis and come close by the force of gravitation, then these bodies will increase the angular velocity of its rotation. In this regard, it is assumed that the nucleons in atomic nuclei rotate at maximum speed. This may explain the equilibrium of the nucleons in atomic nuclei as a balance between the attractive force of strong gravitation and the strong force of the torsion field (of gravitomagnetic forces in [[Physics/Essays/Fedosin/Gravitoelectromagnetism|gravitoelectromagnetism]]). In particular, the [[coupling constant]] is

:&lt;math&gt;\alpha_{pp}=  \frac{\beta \Gamma m^2_p }{\hbar c }=13.4 \beta &lt;/math&gt;,
where &lt;math&gt; ~ \beta &lt;/math&gt; is equal to 0.26 for the interaction of two nucleons, and tending to 1 for bodies with a lower mass density.

The constant &lt;math&gt;~\alpha_{pp}&lt;/math&gt; is close to [[w:coupling constant |coupling constant]] of [[Charges/Interactions/Strong|strong interaction]] of two nucleons in [[w:Standard Model |Standard Model]] 
:&lt;math&gt;\alpha_s=  \frac{ g^2_{N \pi}}{4\pi\hbar c } \approx 14.6&lt;/math&gt; , where &lt;math&gt;~g_{N \pi} &lt;/math&gt; is the constant of the pseudoscalar nucleon-pionic interaction.

[[Physics/Essays/Fedosin/Fine structure constant | Fine structure constant]] is coupling constant of electromagnetic interaction and may be written so:

:&lt;math &gt; ~ \alpha = \frac { \Gamma m_p m_e }{\hbar c }\approx \frac {1}{137.036}.&lt;/math&gt;

==Role of squared Avogadro number ==
Considering Avogadro number &lt;math&gt;N&lt;/math&gt; as a scaling factor, U. V. S. Seshavatharam and S. Lakshminarayana finally arrived at a value of &lt;ref&gt; U. V. S. Seshavatharam and S. Lakshminarayana. Unified model of universe and the atom. Book. {{ISBN|9783843393966}}, LAP LAMBERT Academic Publishing GmbH &amp; Co. KG, Germany, 2011 March 30.  &lt;/ref&gt; &lt;ref&gt; U. V. S. Seshavatharam and S. Lakshminarayana. Role of Avogadro number in grand unification. Hadronic Journal. Vol. 33, No 5, p. 513 (2010).  &lt;/ref&gt; &lt;ref&gt; U. V. S. Seshavatharam and S. Lakshminarayana. Atomic gravitational constant and the origin of elementary magnetic moments. To be published.  &lt;/ref&gt; 
&lt;math&gt; G_s = N^2G = 2.42 \times 10^{37} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.
It is noticed that in Hydrogen atom, ratio of total energy of electron and nuclear potential is equal to the electromagnetic and gravitational force ratio of electron where the operating gravitational constant is nothing but the atomic gravitational having a value N&lt;sup&gt;2&lt;/sup&gt;G. This is a direct confirmation of the existence of the atomic or nuclear gravitational constant in nuclear physics. Therefore, this subject can now be considered as part of the mainstream research in quantum gravity.

The central idea is: for mole number of particles, strength of gravity is &lt;math&gt;N.G&lt;/math&gt; and force required to bind &lt;math&gt;N&lt;/math&gt; particles is &lt;math&gt;\frac{c^4}{N.G}.&lt;/math&gt; Force required to bind one particle is &lt;math&gt;\frac{c^4}{N^2.G}.&lt;/math&gt; By considering this force magnitude as the characteristic weak force magnitude, it is observed that, &lt;math&gt;\ln \sqrt{\frac{e^2}{4 \pi \epsilon_0 G m_p^2}} \cong \sqrt{\frac{m_p}{m_e}-\ln\left(N^2\right)}&lt;/math&gt; where &lt;math&gt;m_p&lt;/math&gt; is the rest mass of proton and &lt;math&gt;m_e&lt;/math&gt; is the rest mass of electron. Obtained value of &lt;math&gt; G\cong \; 6.{\rm 6}66270{\rm 1}79\times {\rm 1}0^{-{\rm 1}1} {\rm \; m}^{{\rm 3}} {\rm Kg}^{{\rm -1}} {\rm sec}^{{\rm -2}.}&lt;/math&gt; Here the most important point to be emphasized is &lt;math&gt;\frac{c^4}{G}&lt;/math&gt; can be considered as the classical or upper limit of gravitational or electromagnetic force. It can be considered as the grand unified force. It is the origin of Planck scale and of the black hole astrophysics.

==Connection with usual gravitational constant==
With the help of [[Physics/Essays/Fedosin/Similarity of matter levels|similarity of matter levels]] and [[Physics/Essays/Fedosin/SPФ symmetry|SPФ symmetry]] in Theory of [[Physics/Essays/Fedosin/Infinite Hierarchical Nesting of Matter|Infinite Hierarchical Nesting of Matter]] the value of &lt;math&gt; ~ \Gamma &lt;/math&gt; can also be defined in terms of coefficients of similarity and the gravitational constant:

: &lt;math&gt;\Gamma = G \frac{ \Phi }{ P S^2},&lt;/math&gt;

where &lt;math&gt; ~ \Phi =1.62 \times 10^{57} &lt;/math&gt;, &lt;math&gt; ~ P= 1.4 \times 10^{19} &lt;/math&gt;, &lt;math&gt; ~ S= 0.23 &lt;/math&gt; are the coefficients of similarity in mass, size and speed, respectively, for the degenerate quantum objects at the atomic and stellar levels of matter.&lt;ref name=&quot;fed&quot;/&gt;  The powers of similarity coefficients in this equation correspond to dimension of gravitational constant according to [[w:Dimensional analysis |dimensional analysis]]. 

From the standpoint of Infinite Hierarchical Nesting of Matter and [[w:Le Sage's theory of gravitation |Le Sage's theory of gravitation]], the presence of two gravitational constants &lt;math&gt; ~ \Gamma &lt;/math&gt; and &lt;math&gt; ~ G &lt;/math&gt; reflects the difference between the properties of gravitons and properties of matter at different levels of matter. &lt;ref &gt; Fedosin S.G. [http://sergf.ru/mgen.htm  Model of Gravitational Interaction in the Concept of Gravitons]. Journal of Vectorial Relativity, Vol. 4, No. 1, pp. 1-24 (2009). &lt;/ref&gt; &lt;ref&gt; Fedosin S.G. [http://vixra.org/abs/1503.0127 The graviton field as the source of mass and gravitational force in the modernized Le Sage’s model.] Physical Science International Journal, ISSN: 2348-0130, Vol. 8, Issue 4, pp. 1-18 (2015). http://dx.doi.org/10.9734/PSIJ/2015/22197. &lt;/ref&gt;

In particular, for the strong gravitational constant and the ordinary gravitational constant it is possible to write similar relations, in which these constants are expressed in terms of the corresponding energy densities of gravitons’ fluxes in [[electrogravitational vacuum]] and the parameters of the densest object of the corresponding level of matter: &lt;ref name=&quot;cha&quot;&gt; Fedosin S.G. [https://www.ajol.info/index.php/jfas/article/view/168204  The charged component of the vacuum field as the source of electric force in the modernized Le Sage’s model.] Journal of Fundamental and Applied Sciences, Vol. 8, No. 3, pp. 971-1020 (2016). http://dx.doi.org/10.4314/jfas.v8i3.18, https://dx.doi.org/10.5281/zenodo.845357.&lt;/ref&gt;
:&lt;math&gt;~ \Gamma = \frac { \varepsilon_c \vartheta^2}{4 \pi M^2_n } , \qquad \qquad  G = \frac { \varepsilon_{cs} \vartheta^2_s}{4 \pi M^2_s } , &lt;/math&gt;

where &lt;math&gt;~ \varepsilon_c = 7.4 \cdot 10^{35}&lt;/math&gt; J/m³  is the energy density of the graviton fluxes for cubic distribution;
&lt;math&gt;~ \vartheta = 2.67 \cdot 10^{-30} &lt;/math&gt; m²  is the cross-section of interaction of the charged particles of the electrogravitational vacuum ([[Physics/Essays/Fedosin/Praon|praon]]s) with nucleons, which is very close in magnitude to the geometrical cross-section of the nucleon and is used to calculate the [[electric constant]]; &lt;math&gt;~ M_n &lt;/math&gt; is the mass of the nucleon;
&lt;math&gt; \varepsilon_{cs} = \varepsilon_c \frac {\Phi S^2}{ P^3} = 2.3 \cdot 10^{34}&lt;/math&gt; J/m³ is the energy density of the graviton fluxes at the stellar level for cubic distribution; &lt;math&gt;~ \vartheta_s = \vartheta P^2 = 5.2 \cdot 10^{8} &lt;/math&gt; m²  is the cross-section of interaction between the gravitons and a neutron star; &lt;math&gt;~ M_s = M_n \Phi = 2.7 \cdot 10^{30} &lt;/math&gt; kg is the mass of the neutron star.

At the matter level of [[Physics/Essays/Fedosin/Praon|praon]]s, its own strong gravitational constant &lt;math&gt;~G_{pr} &lt;/math&gt; must act. Considering that the coefficient of similarity in speed between the nucleon and praon levels of matter is &lt;math&gt;~S \approx 1 &lt;/math&gt;, we can write: 

: &lt;math&gt; G_{pr} = \Gamma \frac{ \Phi }{ P S^2} = \frac{ q^2_{pr}\beta}{4 \pi \varepsilon_{0} m^2_{pr} } =1.752 \cdot 10^{67}&lt;/math&gt; m&lt;sup&gt;3&lt;/sup&gt;•s&lt;sup&gt;–2&lt;/sup&gt;•kg&lt;sup&gt;–1&lt;/sup&gt;,

where &lt;math&gt;~ q_{pr} = 1.06 \cdot 10^{-57} &lt;/math&gt; C is the charge of the praon, &lt;math&gt;~ m_{pr} = 1 \cdot 10^{-84}&lt;/math&gt; kg is the mass of the praon, &lt;math&gt;~ \beta = \frac { m_p }{ m_e }= 1836.152&lt;/math&gt; is the proton to electron mass ratio.

==Connection with mass and unification of interaction==
The main object of unification is to understand the origin of elementary particles mass, (Dirac) magnetic moments and their forces. Right now and till today ‘string theory’ with 4 + 6 extra dimensions not in a position to explain the unification of gravitational and non-gravitational forces. More clearly speaking it is not in a position to bring down the planck scale to the nuclear size. Physicists say – if strength of strong interaction is unity, with reference to the strong interaction, strength of gravitation is 10&lt;sup&gt;−39&lt;/sup&gt;. The fundamental question to be answered is: is mass an inherent property of any elementary particle?

One can say: for any elementary particle mass is an induced property. This idea makes grand unification easy. Note that [[Theory of relativity/General relativity|general relativity]] does not throw any light on the ‘mass generation’ of charged particles. It only suggests that space-time is curved near the massive celestial objects. More over it couples the cosmic (dust) matter with geometry. But how matter is created? Why and how elementary particle possesses both charge and mass? Such types of questions are not discussed in the frame work of general relativity.

The first step in unification is to understand the origin of the [[w:Invariant mass |rest mass]] of a charged elementary particle. Second step is to understand the combined effects of its electromagnetic (or charged) and gravitational interactions. Third step is to understand its behavior with surroundings when it is created. Fourth step is to understand its behavior with cosmic space-time or other particles. Right from its birth to death, in all these steps the underlying fact is that whether it is a strongly interacting particle or weakly interacting particle, it is having some rest mass. To understand the first two steps somehow one can implement the gravitational constant in sub atomic physics.

To bring down the [[Physics/Essays/Fedosin/Planck mass|Planck mass]] scale to the observed elementary particles mass scale a large scale factor is required. Just like [[w:relative permeability |relative permeability]] and [[w:relative permittivity |relative permittivity]] by any suitable reason in atomic space if one is able to increase the value of classical gravitational constant, it helps in four ways. Observed elementary particles mass can be generated and grand unification can be achieved. Third important application is characteristic building block of the cosmological [[dark matter]] can be quantified in terms of fundamental physical constants. Fourth important application is – no [[w:extra dimensions |extra dimensions]] are required. Finally nuclear physics and quantum mechanics can be studied in the view of [[strong gravitation |strong nuclear gravity]] where nuclear charge and atomic gravitational constant play a crucial role in the nuclear space-time curvature, [[w:quantum chromodynamics |quantum chromodynamics]] and [[w:Color confinement |quark confinement]]. Not only that cosmology and particle physics can be studied in a unified way. In this connection it is suggested that square root of ratio of atomic gravitational constant and classical gravitational constant is equal to the Avogadro number. &lt;ref&gt; [http://www.journal-of-nuclear-physics.com/?p=316 AGNI – Avogadro's gravity for nuclear interactions.] Nuclear experiments blog: Journal of Nuclear Physics, Nov. 2010. &lt;/ref&gt; The Avogadro constant expresses the number of elementary entities per mole of substance and it has the value mol&lt;sup&gt;–1&lt;/sup&gt;. Avogadro's constant is a scaling factor between macroscopic and microscopic (atomic scale) observations of nature. It is an observed fact. The very unfortunate thing is that even though it is a large number it is neither implemented in cosmology nor implemented in grand unification. 

Modern physics is having hardly 100 years of ‘strong nuclear’ back ground. By Einstein’s time very little information was available on nuclear strong and weak forces. Avogadro hypothesis was proposed in 1811. Compared to modern nuclear physics,  Avogadro number is having 100 years of old history. Avogadro number may not be a fundamental physical constant but can be considered as a ‘scale factor’. But quantitatively it can be linked with the fundamental force ratios. Future thoughts and experiments may give some clue of it. Best present example is the ratio of planck mass and electron mass.  Considering this ratio automatically N&lt;sup&gt;2&lt;/sup&gt; comes into picture. It is noticed that in Hydrogen atom, ratio of total energy of electron and nuclear potential is equal to the electromagnetic and gravitational force ratio of electron where the operating gravitational constant is nothing but the atomic gravitational having a value N&lt;sup&gt;2&lt;/sup&gt;G. This is a direct confirmation of the existence of the atomoc or nuclear gravitational constant.  

Here the very important question to be answered is – which is more fundamental either &lt;math&gt;  G &lt;/math&gt; or &lt;math&gt; G_s &lt;/math&gt; ? It is proposed that both can be considered as the 'head' and 'tail' of matter coin. It can also be suggested that classical &lt;math&gt;  G &lt;/math&gt; is a consequence of the existence of atomic &lt;math&gt; G_s &lt;/math&gt;. It is known that there is a difference in between 'absolute findings' and 'absolute measurements'. Absolute findings can be understood where as 'absolute measurements' can not be made by nuclear experiments which are being conducted under the sky of universal gravity with unknown origin of elementary particles mass.

Till today there is no explanation for this fantastic and large difference between &lt;math&gt; G &lt;/math&gt; or &lt;math&gt; G_s &lt;/math&gt; or between gravitation and strong interaction, about 10&lt;sup&gt;−39&lt;/sup&gt;. It can be supposed that elementary particles construction is much more fundamental than the black hole's construction. If one wishes to unify electroweak, strong and gravitational interactions it is a must to implement the classical gravitational constant &lt;math&gt; G &lt;/math&gt; in the sub atomic physics. &lt;ref&gt; Seshavatharam, U. V. S.; Lakshminarayana, S. [http://adsabs.harvard.edu/abs/2010IJMPE..19..263S  Super symmetry in strong and weak interactions.] International journal of modern physics E, Issue 02, pp. 263-280, Feb.2010. &lt;/ref&gt; By any reason if one implements the Planck scale in elementary particle physics and nuclear physics automatically &lt;math&gt; G &lt;/math&gt; comes into subatomic physics. Then a large arbitrary number has to be considered as a proportionality constant. After that its physical significance has to be analyzed. Alternatively its equivalent 'strong atomic gravitational constant' can also be assumed. Some attempts have been done in physics history.

Whether it may be real or an equivalent if it is existing as a 'single constant' its physical significance can be understood. Nuclear size can be fitted with 'nuclear Schwarzschild radius'. Nucleus can be considered as 'strong nuclear black hole'. This idea requires a basic nuclear fermion! Nuclear binding energy constants can be generated directly. Proton-neutron stability can be studied. Origin of strong [[w:coupling constant |coupling constant]] and [[w:Fermi's interaction |Fermi's weak coupling constant]] can be understood. Charged lepton masses can be fitted. Such applications can be considered favorable for the proposed assumptions and further analysis can be carried out positively for understanding and developing this proposed 'Avogadro's strong nuclear gravity'.

Unification means: finding the similarities, finding the limiting physical constants, finding the key numbers, coupling the key physical constants, coupling the key physical concepts, coupling the key physical properties, minimizing the number of dimensions, minimizing the number of inputs and implementing the key physical constant or key number in different branches of physics. This is a very lengthy process. In all these cases observations, interpretations, experiments and imagination play a key role. The main difficulty is with interpretations and observations. As the interpretation changes physical concept changes, physical equation changes and finally the destiny changes. 

Note that human beings are part of this universal gravity. There are some natural restrictions to experiments. Seeing a black hole is highly speculative. But indirectly its significances can be well understood. In the similar way in nuclear and particle physics: any experimental setup which is being run under the influence of the proposed strong nuclear gravity, without knowing the probing particle’s massive origin, without knowing the massive origin of the nucleus: based on ‘grand unified scheme’ one may not be able to unearth the absolute findings. Note that observer, experimental setup and the probing particle all are under the same influence of universal gravity. When searching for an experimental proof in grand/final unification scheme or dark matter projects this fact may be considered positively for further analysis. 

To conclude it can be suggested that – existence of strong gravitational constant as Atomic gravitational constant is true and its consequences can be understood easily and can be implemented easily in grand unification program and dark matter projects.

== Notes ==
&lt;references/&gt;

== See also ==
* [[Strong gravitation]]
* [[Coupling constant]]
* [[Gravitational constant]]
* [[w:Gravitational coupling constant |Gravitational coupling constant]]
* [[Fine structure constant]]
* [[w:Dimensionless physical constant |Dimensionless physical constant]]
* [[Physics/Essays/Fedosin/Infinite Hierarchical Nesting of Matter|Infinite Hierarchical Nesting of Matter]]
* [[Physics/Essays/Fedosin/Similarity of matter levels|Similarity of matter levels]]
* [[Physics/Essays/Fedosin/SPФ symmetry|SPФ symmetry]]
* [[Quantization of parameters of cosmic systems]]
* [[Hydrogen system]]
* [[Stellar constants]]
* [[Gravitational model of strong interaction]]
* [[Model of quark quasiparticles]]
* [[Physics/Essays/Fedosin/Substantial electron model|Substantial electron model]]
* [[Physics/Essays/Fedosin/Substantial neutron model|Substantial neutron model]]
* [[Physics/Essays/Fedosin/Substantial proton model|Substantial proton model]]
* [[Physics/Essays/Fedosin/Substantial photon model|Substantial photon model]]
* [[Electrogravitational vacuum]]

==External links==
* [http://www.wikiznanie.ru/ru-wz/index.php/%D0%9F%D0%BE%D1%81%D1%82%D0%BE%D1%8F%D0%BD%D0%BD%D0%B0%D1%8F_%D1%81%D0%B8%D0%BB%D1%8C%D0%BD%D0%BE%D0%B9_%D0%B3%D1%80%D0%B0%D0%B2%D0%B8%D1%82%D0%B0%D1%86%D0%B8%D0%B8 Strong gravitational constant (in Russian)]

[[Category:Gravitation]]
[[Category:Fundamental constants]]</text>
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      <text bytes="34618" sha1="a566mg9if3neimpcrz90axfnwocrh2f" xml:space="preserve">==Strong (nuclear) gravitation ==
In Astronomy the only one available characteristic empirical physical constant is the gravitational constant. Without completing the charge-mass unification or final unification: one cannot say, whether it is an ‘input to the unification’ or ‘output of unification’. The same idea can be applied to the atomic physical constants also. Sitting in a grand unified roof one cannot make an  ‘absolute measurement’ but can make an ‘absolute finding’. Up till now, no atomic model has implemented the gravitational constant in the atomic or nuclear physics. Then, whatever may be its magnitude, measuring its value from existing atomic principles is impossible. Its value has been measured in the lab only within a range of 1 cm to a few metres, whereas the observed nuclear size is 1.2 fermi. Until one measures the value of the gravitational constant in microscopic physics, the debate of strong (nuclear) gravitation can be considered positively. The idea of strong gravitation originally referred specifically to mathematical approach of Abdus Salam of unification of gravitation and quantum chromodynamics, but is now often used for any particle level gravitation approach. Now many persons are working on this subject. A main advantage of this subject is: it couples black hole physics and particle physics.

==Strong gravitational constant==
The '''strong gravitational constant''', denoted &lt;math&gt;~\;\; \Gamma &lt;/math&gt; or &lt;math&gt;~G_s &lt;/math&gt;, is a grand unified physical constant of strong gravitation, involved in calculation of gravitational attraction at the level of elementary particles and atoms.

According to [[w:Isaac Newton| Newton]]'s law of universal gravitation, the force of gravitational attraction between two massive points with masses &lt;math&gt; ~ m_1 &lt;/math&gt; and &lt;math&gt; ~ m_2 &lt;/math&gt;, located at a distance &lt;math&gt; ~ R &lt;/math&gt; between them, is:

: &lt;math&gt;F=G \frac{m_1 m_2}{R^2}.&lt;/math&gt;

The coefficient of proportionality &lt;math&gt; ~ G &lt;/math&gt; in this expression is called [[w:gravitational constant |gravitational constant]]. It is assumed, that in contrast to the usual force of gravity, at the level of elementary particles acts [[Physics/Essays/Fedosin/Strong gravitation |strong gravitation]]. In order to describe it &lt;math&gt; ~ G &lt;/math&gt; in the formula for gravitational force must be replaced on &lt;math&gt; ~ \Gamma &lt;/math&gt;:

: &lt;math&gt;F_{sg}=\Gamma \frac{m_1 m_2}{R^2}.&lt;/math&gt;

==Dimensions and magnitude==
The dimensions assigned to the strong gravitational constant may be found from the equation above — length cubed, divided by [[w:mass |mass]] and by time squared (in SI units, metres cubed per kilogram per second squared).

There are several ways to assess the value of &lt;math&gt; ~ \Gamma &lt;/math&gt;. J. Dufour, under the assumption that the strong gravitational constant depends on the type of objects, from the interaction of two deuterium nuclei determined, &lt;ref&gt; J. Dufour. [http://www.iscmns.org/CMNS/CMNS.htm &quot;Very sizeable increase of gravity at pico-meter distance: a novel working hypothesis to explain anomalous heat effects and apparent transmutations in certain metal hydrogen systems&quot;]. J. of condensed matter nuclear science, Vol. 1, pp. 47-61 (2007). &lt;/ref&gt; that  &lt;math&gt; G' = 2.06 \times 10^{25} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

Based on the analogy between hadrons and Kerr-Newman [[black hole]]s &lt;ref&gt;Strong Interactions, Gravitation and Cosmology. Abdus Salam Publ. in: NATO Advanced Study Institute, Erice, June16-July 6, 1972 ; in: High Energy Astrophysics and its Relation to Elementary Particle Physics, 441-452 MIT Press, Cambridge (1974). &lt;/ref&gt; Sivaram, C. and Sinha, K.P, &lt;ref&gt; Sivaram, C. and Sinha, K.P. Strong gravity, black holes, and hadrons. Physical Review D, Vol. 16, Issue 6, pp. 1975-1978 (1977). &lt;/ref&gt;  &lt;ref&gt;Salam A. and Sivaram C. Strong Gravity Approach to QCD and Confinement. Mod. Phys. Lett., v. A8(4), pp. 321-326 (1993). &lt;/ref&gt; and Raut, Usha and Shina, KP &lt;ref&gt;Raut, Usha and Shina, KP (1983) [http://eprints.iisc.ernet.in/13571/  Strong gravity and the fine structure constant.] In: Proceedings of the Indian Academy of Sciences Part A: Physical Sciences, 49 (2). pp. 352-358. &lt;/ref&gt; accepted the value &lt;math&gt; \Gamma = 6.7 \times 10^{27} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

This value of the strong gravitational constant allowed estimating the strong spin-torsion interaction between spinning protons. &lt;ref&gt;V. de Sabbata, C. Sivaram. [http://prints.iiap.res.in/bitstream/2248/4394/3/Strong%20spin-torsion%20 Strong Spin-Torsion Interaction between Spinning Protons.] Il Nuovo Cimento, Vol. 101A, N. 2, pp. 273-283 (1989). &lt;/ref&gt;

In paper of Mongan &lt;ref&gt;T. R.  Mongan. [http://th1.ihep.su/archive/gr-qc/070622.html Cold dark matter from &quot;strong gravity&quot;.] General Relativity &amp; Quantum Cosmology, 20 Jun 2007; [http://ru.arxiv.org/abs/0706.3050v2 arXiv:0706.3050v2.] &lt;/ref&gt; strong gravitational constant is &lt;math&gt; G_s = 1.1 \times 10^{28} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

According to [https://en.everybodywiki.com/Robert_Oldershaw Robert L. Oldershaw] &lt;ref&gt; Oldershaw R.L. [http://arxiv.org/abs/physics/0701132v3 Discrete Scale Relativity.] Astrophysics and Space Science, Vol. 311, N. 4, pp. 431-433 (2007). DOI: 10.107/s10509-007-9557-x. &lt;/ref&gt; value of the strong gravitational constant is &lt;math&gt; G_s = 2.18 \times 10^{28} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

As in Oldershaw’ paper, strong gravitational constant could be related &lt;ref&gt;Stone R.A. Quark Confinement and Force Unification. Progress in Physics, April 2010, Vol. 2, P. 19–20. &lt;/ref&gt; with the proton radius &lt;math&gt; ~ R_p &lt;/math&gt;, the proton mass &lt;math&gt; ~ m_p &lt;/math&gt; and the speed of light &lt;math&gt;~c &lt;/math&gt;: 
: &lt;math&gt;sG_p= \frac{R_p c^2}{2 m_p }= 2.4 \times 10^{28} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

According to Tennakone who identified the electron and the proton as black holes in the strong gravitational field, strong gravitational constant is: &lt;ref&gt;K. Tennakone. [http://prd.aps.org/abstract/PRD/v10/i6/p1722_1 Electron, muon, proton, and strong gravity.]  Phys. Rev. D, Volume 10, Issue 6, pp.1722-1725 (1974). &lt;/ref&gt;
: &lt;math&gt;\Gamma =  3.9 \times 10^{28} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

Zane Andrea Quintili finds a strong gravitational constant based on the similarity between the Planck mass and radius, and accordingly the mass and radius of the proton: &lt;ref&gt; Zane Andrea Quintili. [http://vixra.org/abs/1904.0540 Gravitational Field and Proton Radius]. vixra.org. (2019). &lt;/ref&gt;
: &lt;math&gt; G_q = \frac {8\hbar c }{m^2_p} =9.04 \times 10^{28} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

Recami et al &lt;ref&gt;Recami, E.; Ammiraju, P.; Hernandez, H.E.; Kretly, L.C.; Rodrigues, W.A., Jr. Elementary particles as micro-universes: a geometric approach to &quot;strong gravity&quot;. Apeiron, January 01, 1997. &lt;/ref&gt; &lt;ref&gt;Recami E. and Tonin-Zanchin V. The strong coupling constant: its theoretical derivation from a geometric approach to hadron structure. Found. Phys. Lett., v, 7(1), pp. 85-92 (1994). &lt;/ref&gt; define strong gravitational constant through the mass of the pion &lt;math&gt; ~ m_{\pi} &lt;/math&gt; as follows:

: &lt;math&gt;N\approx \frac{h c}{ m^2_{\pi} }= 3.2 \times 10^{30} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;,

where &lt;math&gt; ~ h &lt;/math&gt; – [[w:Planck constant |Planck constant]].

From this they derive [[w:coupling constant | constant of strong interaction]] of two nucleons in the following form: &lt;ref&gt;Erasmo Recami, Tonin-Zanchin, Antonino Del Popolo, Mario Gambera. [http://arxiv.org/abs/physics/0105080 The strong coupling constant,] Heavy Ion Physics, Vol. 10, pp. 345-349 (1999). &lt;/ref&gt;

:&lt;math&gt; \frac{ N g^2}{\hbar c } \approx 14&lt;/math&gt; , where &lt;math&gt;~g &lt;/math&gt; indicates a strong charge, &lt;math&gt; ~ \hbar &lt;/math&gt; is reduced Planck constant.

Stanislav Fisenko et all found &lt;ref&gt;Stanislav Fisenko &amp; Igor Fisenko. [http://www.ccsenet.org/journal/index.php/apr/article/download/8060/6060 The Conception of Thermonuclear Reactor on the Principle of Gravitational Confinement of Dense High-temperature Plasma.] Applied Physics Research, Vol. 2, No. 2, pp. 71-79 (2010). &lt;/ref&gt; &lt;ref&gt;S. I. Fisenko, M. M. Beilinson and B. G. Umanov. Some notes on the concept of “strong” gravitation and possibilities of its experimental investigation. Physics Letters A, Volume 148, Issues 8-9, pp. 405-407 (1990).&lt;/ref&gt; a spectrum of steady states of the electron in proper gravitational field (0.511 MeV …0.681 MeV) on the base of strong coupling constant
: &lt;math&gt;N= 5.1 \times 10^{31} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

U. V. S. Seshavatharam and S. Lakshminarayana &lt;ref&gt; U. V. S. Seshavatharam and S. Lakshminarayana. Strong nuclear gravitational constant and the origin of nuclear planck scale. Progress in Physics, vol. 3, pp. 31-38 (2010). [http://fs.gallup.unm.edu/PP-03-2010.pdf] &lt;/ref&gt; in determining &lt;math&gt; ~ G_s &lt;/math&gt; repelled from the Fermi constant, which led them to the value &lt;math&gt; G_s = 6.94 \times 10^{31} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

In the paper &lt;ref&gt;Perng J. J. [https://springerlink3.metapress.com/content/1503066144130716/resource-secured/?target=fulltext.pdf&amp;sid=qwdy0a45odlh3a55v4k45u45&amp;sh=www.springerlink.com Strong gravitation and elementary particles.] Nuovo Cimento, Lettere, Serie 2, vol. 23, N. 15, pp. 552-554 (1978). &lt;/ref&gt; strong gravitational constant equal to &lt;math&gt;\Gamma =2.77 \times 10^{32} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

[[User:Fedosin | Sergey Fedosin]] entered the strong gravitational constant in 1999 on the basis of equality between the Coulomb electric force and gravitational force in the hydrogen atom on the [[w:Bohr radius |Bohr radius]]. This leads to the following expression for the value of the strong gravitational constant: &lt;ref name=&quot;fed&quot;&gt; Fedosin S.G. [http://lccn.loc.gov/2009457349 Fizika i filosofiia podobiia: ot preonov do metagalaktik,] Perm, (1999-06-09) 544 pp. {{ISBN|5-8131-0012-1}}. &lt;/ref&gt;

: &lt;math&gt;\Gamma= \frac{e^2}{4 \pi \varepsilon_{0} m_p m_e }=1.514 \times 10^{29} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;,

where &lt;math&gt; ~ e &lt;/math&gt; – [[w:elementary charge |elementary charge]], &lt;math&gt; ~ \pi &lt;/math&gt; – [[w:pi (number) | pi]], &lt;math&gt; ~ \varepsilon_{0} &lt;/math&gt; – [[electric constant]], &lt;math&gt; ~ m_p &lt;/math&gt; – the mass of [[w:proton |proton]], &lt;math&gt; ~ m_e &lt;/math&gt; – the mass of [[w:electron| electron]]. 

It is assumed that strong gravitation, as a universal force, acts on the matter of nucleons, hadrons, electrons and elementary particles, regardless of the type of these particles. In contrast, the standard approach considers that strong interaction does not affect electrons and other leptons.

The small mass and large charge of matter do not allow the electron to be entirely in some small volume near the nucleus, and it gets disklike axisymmetric shape, which is limited by size of atom. In the hydrogen atom electrical forces between the nucleus and matter of the electron are attractive, but they are compensated by the repulsion of the intrinsic charge of the electron. There are the centripetal force of rotation of the electron around the nucleus, and the gravitational attraction between massive nucleus and matter of the electron. All these forces are equal in magnitude. From here follows that the action of strong gravitation between the masses of nucleus and electron on the one hand, and the electric force between charges of the nucleus and the electron, on the other hand, allows to estimate the value of &lt;math&gt; ~ \Gamma &lt;/math&gt;.
If &lt;math&gt;~ R_B = \frac {\hbar }{ m_e \alpha c } &lt;/math&gt; is the Bohr radius, then the equality of forces gives:

: &lt;math&gt; \frac {\Gamma m_p m_e }{R^2_B} = \frac{e^2}{4 \pi \varepsilon_{0} R^2_B } .&lt;/math&gt;

[[Physics/Essays/Fedosin/Fine structure constant | Fine structure constant]] is
:&lt;math&gt; \alpha = \frac{e^2}{4 \pi \varepsilon_0 \hbar c}, &lt;/math&gt;

So that
: &lt;math&gt; \Gamma= \frac{\alpha \hbar c }{m_p m_e }, \qquad \qquad \hbar = \frac{\Gamma m_p m_e }{ \alpha c }.&lt;/math&gt;

Bohr radius becomes equal
:&lt;math&gt;~ R_B = \frac{\Gamma m_p }{ \alpha^2 c^2 } = \frac{\Gamma m_p }{ V^2_B },&lt;/math&gt;

where &lt;math&gt;~ V_B = \alpha c &lt;/math&gt; is the orbital speed of the electron cloud at the first energy level.

Hence &lt;math&gt;~ V^2_B = \frac{\Gamma m_p }{ R_B }&lt;/math&gt;, and the kinetic energy of the electron, taking into account determination of strong gravitational constant, is equal to:
:&lt;math&gt;~ K = \frac{m_e V^2_B }{ 2 } = \frac{\Gamma m_p m_e }{ 2 R_B }=\frac { e^2}{8 \pi \varepsilon_0 R_B } = - \frac {W}{2} ,&lt;/math&gt;

where &lt;math&gt;~ W &lt;/math&gt;  is the potential energy of electron in the electric field of the nucleus of a hydrogen atom.

It turns out the virial theorem in the form &lt;math&gt;~ K = - \frac {W}{2} &lt;/math&gt;. The total electron energy is also found at the first energy level:

:&lt;math&gt;~ E = K+W = \frac {W}{2} = -K = -13.6 &lt;/math&gt; eV.

With the help of the constant &lt;math&gt; ~ \Gamma &lt;/math&gt; the [[w:Invariant mass#Rest energy | rest energy]] of proton in the form of a ball is equal to half of its potential energy of strong gravitational field in accordance with [[w:virial theorem |virial theorem]], &lt;ref&gt; [[User:Fedosin | Sergey Fedosin]], [https://www.lap-publishing.com/catalog/details//store/gb/book/978-3-659-57301-9/the-physical-theories-and-infinite-hierarchical-nesting-of-matter-volume-1 The physical theories and infinite hierarchical nesting of matter], Volume 1, LAP LAMBERT Academic Publishing, pages: 580, {{ISBN|978-3-659-57301-9}}. &lt;/ref&gt; if we assume that the binding energy &lt;math&gt; ~ E_b &lt;/math&gt; for the proton up to a sign is equal to the total energy of proton, and &lt;math&gt; ~ E_b &lt;/math&gt; becomes very close to relativistic energy in the form of rest energy:

: &lt;math&gt;~ m_p c^2 \approx E_b = -\frac {W_p}{2} = \frac{ k \Gamma m^2_p }{ 2R_p},&lt;/math&gt;

where &lt;math&gt; ~ R_p =8.73 \times 10^{-16} &lt;/math&gt; m  is the proton radius,
&lt;math&gt; ~ k=0.62 &lt;/math&gt; (in the hypothetical case of a uniform mass density of the proton there must be &lt;math&gt; ~ k = 0.6 &lt;/math&gt;). This implies that the mass of nucleons is determined by the energy of the strong gravitation according to the principle of [[w:mass–energy equivalence |mass–energy equivalence]].

If we assume that the magnetic moment of the proton is created by the maximum rotation of its positive charge distributed over the volume of the proton in the form of a ball, when the centripetal acceleration at the equator becomes equal to acceleration of strong gravitation, the formula for the magnetic moment is as follows:
: &lt;math&gt; ~ P_m = \delta e \sqrt {\Gamma m_p R_p}, &lt;/math&gt;

where &lt;math&gt; ~ P_m = 1.41 \times 10^{-26} &lt;/math&gt; J / T  is the magnetic moment of the proton,
&lt;math&gt; ~ \delta = 0.1875 &lt;/math&gt; (in the case of uniform density and charge should be &lt;math&gt; ~ \delta = 0.2 &lt;/math&gt;).

From the formulas for the energy and the magnetic moment the radius of the proton is determined in the self-consistent model. &lt;ref&gt; Fedosin S.G. [http://vixra.org/abs/1208.0006 The radius of the proton in the self-consistent model.] Hadronic Journal, Vol. 35, No. 4, pp. 349-363 (2012). &lt;/ref&gt;

The strong gravitational constant is also included in the formula describing the [[w:nuclear force |nuclear force]] through strong gravitation and [[Physics/Essays/Fedosin/Gravitational torsion field |gravitational torsion field]] of rotating particles. &lt;ref&gt;[http://sergf.ru/com.htm Comments to the book]: Fedosin S.G. Fizicheskie teorii i beskonechnaia vlozhennost’ materii. – Perm, 2009, 844 pages, Tabl. 21, Pic. 41, Ref. 289. {{ISBN|978-5-9901951-1-0}}. (in Russian). &lt;/ref&gt; A feature of the [[Physics/Essays/Fedosin/Gravitational induction |gravitational induction]] is that if two bodies rotate along one axis and come close by the force of gravitation, then these bodies will increase the angular velocity of its rotation. In this regard, it is assumed that the nucleons in atomic nuclei rotate at maximum speed. This may explain the equilibrium of the nucleons in atomic nuclei as a balance between the attractive force of strong gravitation and the strong force of the torsion field (of gravitomagnetic forces in [[Physics/Essays/Fedosin/Gravitoelectromagnetism|gravitoelectromagnetism]]). In particular, the [[Physics/Essays/Fedosin/Coupling constant |coupling constant]] is

:&lt;math&gt;\alpha_{pp}=  \frac{\beta \Gamma m^2_p }{\hbar c }=13.4 \beta &lt;/math&gt;,
where &lt;math&gt; ~ \beta &lt;/math&gt; is equal to 0.26 for the interaction of two nucleons, and tending to 1 for bodies with a lower mass density.

The constant &lt;math&gt;~\alpha_{pp}&lt;/math&gt; is close to [[w:coupling constant |coupling constant]] of [[Charges/Interactions/Strong|strong interaction]] of two nucleons in [[w:Standard Model |Standard Model]] 
:&lt;math&gt;\alpha_s=  \frac{ g^2_{N \pi}}{4\pi\hbar c } \approx 14.6&lt;/math&gt; , where &lt;math&gt;~g_{N \pi} &lt;/math&gt; is the constant of the pseudoscalar nucleon-pionic interaction.

[[Physics/Essays/Fedosin/Fine structure constant | Fine structure constant]] is coupling constant of electromagnetic interaction and may be written so:

:&lt;math &gt; ~ \alpha = \frac { \Gamma m_p m_e }{\hbar c }\approx \frac {1}{137.036}.&lt;/math&gt;

==Role of squared Avogadro number ==
Considering Avogadro number &lt;math&gt;N&lt;/math&gt; as a scaling factor, U. V. S. Seshavatharam and S. Lakshminarayana finally arrived at a value of &lt;ref&gt; U. V. S. Seshavatharam and S. Lakshminarayana. Unified model of universe and the atom. Book. {{ISBN|9783843393966}}, LAP LAMBERT Academic Publishing GmbH &amp; Co. KG, Germany, 2011 March 30.  &lt;/ref&gt; &lt;ref&gt; U. V. S. Seshavatharam and S. Lakshminarayana. Role of Avogadro number in grand unification. Hadronic Journal. Vol. 33, No 5, p. 513 (2010).  &lt;/ref&gt; &lt;ref&gt; U. V. S. Seshavatharam and S. Lakshminarayana. Atomic gravitational constant and the origin of elementary magnetic moments. To be published.  &lt;/ref&gt; 
&lt;math&gt; G_s = N^2G = 2.42 \times 10^{37} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.
It is noticed that in Hydrogen atom, ratio of total energy of electron and nuclear potential is equal to the electromagnetic and gravitational force ratio of electron where the operating gravitational constant is nothing but the atomic gravitational having a value N&lt;sup&gt;2&lt;/sup&gt;G. This is a direct confirmation of the existence of the atomic or nuclear gravitational constant in nuclear physics. Therefore, this subject can now be considered as part of the mainstream research in quantum gravity.

The central idea is: for mole number of particles, strength of gravity is &lt;math&gt;N.G&lt;/math&gt; and force required to bind &lt;math&gt;N&lt;/math&gt; particles is &lt;math&gt;\frac{c^4}{N.G}.&lt;/math&gt; Force required to bind one particle is &lt;math&gt;\frac{c^4}{N^2.G}.&lt;/math&gt; By considering this force magnitude as the characteristic weak force magnitude, it is observed that, &lt;math&gt;\ln \sqrt{\frac{e^2}{4 \pi \epsilon_0 G m_p^2}} \cong \sqrt{\frac{m_p}{m_e}-\ln\left(N^2\right)}&lt;/math&gt; where &lt;math&gt;m_p&lt;/math&gt; is the rest mass of proton and &lt;math&gt;m_e&lt;/math&gt; is the rest mass of electron. Obtained value of &lt;math&gt; G\cong \; 6.{\rm 6}66270{\rm 1}79\times {\rm 1}0^{-{\rm 1}1} {\rm \; m}^{{\rm 3}} {\rm Kg}^{{\rm -1}} {\rm sec}^{{\rm -2}.}&lt;/math&gt; Here the most important point to be emphasized is &lt;math&gt;\frac{c^4}{G}&lt;/math&gt; can be considered as the classical or upper limit of gravitational or electromagnetic force. It can be considered as the grand unified force. It is the origin of Planck scale and of the black hole astrophysics.

==Connection with usual gravitational constant==
With the help of [[Physics/Essays/Fedosin/Similarity of matter levels|similarity of matter levels]] and [[Physics/Essays/Fedosin/SPФ symmetry|SPФ symmetry]] in Theory of [[Physics/Essays/Fedosin/Infinite Hierarchical Nesting of Matter|Infinite Hierarchical Nesting of Matter]] the value of &lt;math&gt; ~ \Gamma &lt;/math&gt; can also be defined in terms of coefficients of similarity and the gravitational constant:

: &lt;math&gt;\Gamma = G \frac{ \Phi }{ P S^2},&lt;/math&gt;

where &lt;math&gt; ~ \Phi =1.62 \times 10^{57} &lt;/math&gt;, &lt;math&gt; ~ P= 1.4 \times 10^{19} &lt;/math&gt;, &lt;math&gt; ~ S= 0.23 &lt;/math&gt; are the coefficients of similarity in mass, size and speed, respectively, for the degenerate quantum objects at the atomic and stellar levels of matter.&lt;ref name=&quot;fed&quot;/&gt;  The powers of similarity coefficients in this equation correspond to dimension of gravitational constant according to [[w:Dimensional analysis |dimensional analysis]]. 

From the standpoint of Infinite Hierarchical Nesting of Matter and [[w:Le Sage's theory of gravitation |Le Sage's theory of gravitation]], the presence of two gravitational constants &lt;math&gt; ~ \Gamma &lt;/math&gt; and &lt;math&gt; ~ G &lt;/math&gt; reflects the difference between the properties of gravitons and properties of matter at different levels of matter. &lt;ref &gt; Fedosin S.G. [http://sergf.ru/mgen.htm  Model of Gravitational Interaction in the Concept of Gravitons]. Journal of Vectorial Relativity, Vol. 4, No. 1, pp. 1-24 (2009). &lt;/ref&gt; &lt;ref&gt; Fedosin S.G. [http://vixra.org/abs/1503.0127 The graviton field as the source of mass and gravitational force in the modernized Le Sage’s model.] Physical Science International Journal, ISSN: 2348-0130, Vol. 8, Issue 4, pp. 1-18 (2015). http://dx.doi.org/10.9734/PSIJ/2015/22197. &lt;/ref&gt;

In particular, for the strong gravitational constant and the ordinary gravitational constant it is possible to write similar relations, in which these constants are expressed in terms of the corresponding energy densities of gravitons’ fluxes in [[electrogravitational vacuum]] and the parameters of the densest object of the corresponding level of matter: &lt;ref name=&quot;cha&quot;&gt; Fedosin S.G. [https://www.ajol.info/index.php/jfas/article/view/168204  The charged component of the vacuum field as the source of electric force in the modernized Le Sage’s model.] Journal of Fundamental and Applied Sciences, Vol. 8, No. 3, pp. 971-1020 (2016). http://dx.doi.org/10.4314/jfas.v8i3.18, https://dx.doi.org/10.5281/zenodo.845357.&lt;/ref&gt;
:&lt;math&gt;~ \Gamma = \frac { \varepsilon_c \vartheta^2}{4 \pi M^2_n } , \qquad \qquad  G = \frac { \varepsilon_{cs} \vartheta^2_s}{4 \pi M^2_s } , &lt;/math&gt;

where &lt;math&gt;~ \varepsilon_c = 7.4 \cdot 10^{35}&lt;/math&gt; J/m³  is the energy density of the graviton fluxes for cubic distribution;
&lt;math&gt;~ \vartheta = 2.67 \cdot 10^{-30} &lt;/math&gt; m²  is the cross-section of interaction of the charged particles of the electrogravitational vacuum ([[Physics/Essays/Fedosin/Praon|praon]]s) with nucleons, which is very close in magnitude to the geometrical cross-section of the nucleon and is used to calculate the [[electric constant]]; &lt;math&gt;~ M_n &lt;/math&gt; is the mass of the nucleon;
&lt;math&gt; \varepsilon_{cs} = \varepsilon_c \frac {\Phi S^2}{ P^3} = 2.3 \cdot 10^{34}&lt;/math&gt; J/m³ is the energy density of the graviton fluxes at the stellar level for cubic distribution; &lt;math&gt;~ \vartheta_s = \vartheta P^2 = 5.2 \cdot 10^{8} &lt;/math&gt; m²  is the cross-section of interaction between the gravitons and a neutron star; &lt;math&gt;~ M_s = M_n \Phi = 2.7 \cdot 10^{30} &lt;/math&gt; kg is the mass of the neutron star.

At the matter level of [[Physics/Essays/Fedosin/Praon|praon]]s, its own strong gravitational constant &lt;math&gt;~G_{pr} &lt;/math&gt; must act. Considering that the coefficient of similarity in speed between the nucleon and praon levels of matter is &lt;math&gt;~S \approx 1 &lt;/math&gt;, we can write: 

: &lt;math&gt; G_{pr} = \Gamma \frac{ \Phi }{ P S^2} = \frac{ q^2_{pr}\beta}{4 \pi \varepsilon_{0} m^2_{pr} } =1.752 \cdot 10^{67}&lt;/math&gt; m&lt;sup&gt;3&lt;/sup&gt;•s&lt;sup&gt;–2&lt;/sup&gt;•kg&lt;sup&gt;–1&lt;/sup&gt;,

where &lt;math&gt;~ q_{pr} = 1.06 \cdot 10^{-57} &lt;/math&gt; C is the charge of the praon, &lt;math&gt;~ m_{pr} = 1 \cdot 10^{-84}&lt;/math&gt; kg is the mass of the praon, &lt;math&gt;~ \beta = \frac { m_p }{ m_e }= 1836.152&lt;/math&gt; is the proton to electron mass ratio.

==Connection with mass and unification of interaction==
The main object of unification is to understand the origin of elementary particles mass, (Dirac) magnetic moments and their forces. Right now and till today ‘string theory’ with 4 + 6 extra dimensions not in a position to explain the unification of gravitational and non-gravitational forces. More clearly speaking it is not in a position to bring down the planck scale to the nuclear size. Physicists say – if strength of strong interaction is unity, with reference to the strong interaction, strength of gravitation is 10&lt;sup&gt;−39&lt;/sup&gt;. The fundamental question to be answered is: is mass an inherent property of any elementary particle?

One can say: for any elementary particle mass is an induced property. This idea makes grand unification easy. Note that [[Theory of relativity/General relativity|general relativity]] does not throw any light on the ‘mass generation’ of charged particles. It only suggests that space-time is curved near the massive celestial objects. More over it couples the cosmic (dust) matter with geometry. But how matter is created? Why and how elementary particle possesses both charge and mass? Such types of questions are not discussed in the frame work of general relativity.

The first step in unification is to understand the origin of the [[w:Invariant mass |rest mass]] of a charged elementary particle. Second step is to understand the combined effects of its electromagnetic (or charged) and gravitational interactions. Third step is to understand its behavior with surroundings when it is created. Fourth step is to understand its behavior with cosmic space-time or other particles. Right from its birth to death, in all these steps the underlying fact is that whether it is a strongly interacting particle or weakly interacting particle, it is having some rest mass. To understand the first two steps somehow one can implement the gravitational constant in sub atomic physics.

To bring down the [[Physics/Essays/Fedosin/Planck mass|Planck mass]] scale to the observed elementary particles mass scale a large scale factor is required. Just like [[w:relative permeability |relative permeability]] and [[w:relative permittivity |relative permittivity]] by any suitable reason in atomic space if one is able to increase the value of classical gravitational constant, it helps in four ways. Observed elementary particles mass can be generated and grand unification can be achieved. Third important application is characteristic building block of the cosmological [[dark matter]] can be quantified in terms of fundamental physical constants. Fourth important application is – no [[w:extra dimensions |extra dimensions]] are required. Finally nuclear physics and quantum mechanics can be studied in the view of [[strong gravitation |strong nuclear gravity]] where nuclear charge and atomic gravitational constant play a crucial role in the nuclear space-time curvature, [[w:quantum chromodynamics |quantum chromodynamics]] and [[w:Color confinement |quark confinement]]. Not only that cosmology and particle physics can be studied in a unified way. In this connection it is suggested that square root of ratio of atomic gravitational constant and classical gravitational constant is equal to the Avogadro number. &lt;ref&gt; [http://www.journal-of-nuclear-physics.com/?p=316 AGNI – Avogadro's gravity for nuclear interactions.] Nuclear experiments blog: Journal of Nuclear Physics, Nov. 2010. &lt;/ref&gt; The Avogadro constant expresses the number of elementary entities per mole of substance and it has the value mol&lt;sup&gt;–1&lt;/sup&gt;. Avogadro's constant is a scaling factor between macroscopic and microscopic (atomic scale) observations of nature. It is an observed fact. The very unfortunate thing is that even though it is a large number it is neither implemented in cosmology nor implemented in grand unification. 

Modern physics is having hardly 100 years of ‘strong nuclear’ back ground. By Einstein’s time very little information was available on nuclear strong and weak forces. Avogadro hypothesis was proposed in 1811. Compared to modern nuclear physics,  Avogadro number is having 100 years of old history. Avogadro number may not be a fundamental physical constant but can be considered as a ‘scale factor’. But quantitatively it can be linked with the fundamental force ratios. Future thoughts and experiments may give some clue of it. Best present example is the ratio of planck mass and electron mass.  Considering this ratio automatically N&lt;sup&gt;2&lt;/sup&gt; comes into picture. It is noticed that in Hydrogen atom, ratio of total energy of electron and nuclear potential is equal to the electromagnetic and gravitational force ratio of electron where the operating gravitational constant is nothing but the atomic gravitational having a value N&lt;sup&gt;2&lt;/sup&gt;G. This is a direct confirmation of the existence of the atomoc or nuclear gravitational constant.  

Here the very important question to be answered is – which is more fundamental either &lt;math&gt;  G &lt;/math&gt; or &lt;math&gt; G_s &lt;/math&gt; ? It is proposed that both can be considered as the 'head' and 'tail' of matter coin. It can also be suggested that classical &lt;math&gt;  G &lt;/math&gt; is a consequence of the existence of atomic &lt;math&gt; G_s &lt;/math&gt;. It is known that there is a difference in between 'absolute findings' and 'absolute measurements'. Absolute findings can be understood where as 'absolute measurements' can not be made by nuclear experiments which are being conducted under the sky of universal gravity with unknown origin of elementary particles mass.

Till today there is no explanation for this fantastic and large difference between &lt;math&gt; G &lt;/math&gt; or &lt;math&gt; G_s &lt;/math&gt; or between gravitation and strong interaction, about 10&lt;sup&gt;−39&lt;/sup&gt;. It can be supposed that elementary particles construction is much more fundamental than the black hole's construction. If one wishes to unify electroweak, strong and gravitational interactions it is a must to implement the classical gravitational constant &lt;math&gt; G &lt;/math&gt; in the sub atomic physics. &lt;ref&gt; Seshavatharam, U. V. S.; Lakshminarayana, S. [http://adsabs.harvard.edu/abs/2010IJMPE..19..263S  Super symmetry in strong and weak interactions.] International journal of modern physics E, Issue 02, pp. 263-280, Feb.2010. &lt;/ref&gt; By any reason if one implements the Planck scale in elementary particle physics and nuclear physics automatically &lt;math&gt; G &lt;/math&gt; comes into subatomic physics. Then a large arbitrary number has to be considered as a proportionality constant. After that its physical significance has to be analyzed. Alternatively its equivalent 'strong atomic gravitational constant' can also be assumed. Some attempts have been done in physics history.

Whether it may be real or an equivalent if it is existing as a 'single constant' its physical significance can be understood. Nuclear size can be fitted with 'nuclear Schwarzschild radius'. Nucleus can be considered as 'strong nuclear black hole'. This idea requires a basic nuclear fermion! Nuclear binding energy constants can be generated directly. Proton-neutron stability can be studied. Origin of strong [[w:coupling constant |coupling constant]] and [[w:Fermi's interaction |Fermi's weak coupling constant]] can be understood. Charged lepton masses can be fitted. Such applications can be considered favorable for the proposed assumptions and further analysis can be carried out positively for understanding and developing this proposed 'Avogadro's strong nuclear gravity'.

Unification means: finding the similarities, finding the limiting physical constants, finding the key numbers, coupling the key physical constants, coupling the key physical concepts, coupling the key physical properties, minimizing the number of dimensions, minimizing the number of inputs and implementing the key physical constant or key number in different branches of physics. This is a very lengthy process. In all these cases observations, interpretations, experiments and imagination play a key role. The main difficulty is with interpretations and observations. As the interpretation changes physical concept changes, physical equation changes and finally the destiny changes. 

Note that human beings are part of this universal gravity. There are some natural restrictions to experiments. Seeing a black hole is highly speculative. But indirectly its significances can be well understood. In the similar way in nuclear and particle physics: any experimental setup which is being run under the influence of the proposed strong nuclear gravity, without knowing the probing particle’s massive origin, without knowing the massive origin of the nucleus: based on ‘grand unified scheme’ one may not be able to unearth the absolute findings. Note that observer, experimental setup and the probing particle all are under the same influence of universal gravity. When searching for an experimental proof in grand/final unification scheme or dark matter projects this fact may be considered positively for further analysis. 

To conclude it can be suggested that – existence of strong gravitational constant as Atomic gravitational constant is true and its consequences can be understood easily and can be implemented easily in grand unification program and dark matter projects.

== Notes ==
&lt;references/&gt;

== See also ==
* [[Strong gravitation]]
* [[Coupling constant]]
* [[Gravitational constant]]
* [[w:Gravitational coupling constant |Gravitational coupling constant]]
* [[Fine structure constant]]
* [[w:Dimensionless physical constant |Dimensionless physical constant]]
* [[Physics/Essays/Fedosin/Infinite Hierarchical Nesting of Matter|Infinite Hierarchical Nesting of Matter]]
* [[Physics/Essays/Fedosin/Similarity of matter levels|Similarity of matter levels]]
* [[Physics/Essays/Fedosin/SPФ symmetry|SPФ symmetry]]
* [[Quantization of parameters of cosmic systems]]
* [[Hydrogen system]]
* [[Stellar constants]]
* [[Gravitational model of strong interaction]]
* [[Model of quark quasiparticles]]
* [[Physics/Essays/Fedosin/Substantial electron model|Substantial electron model]]
* [[Physics/Essays/Fedosin/Substantial neutron model|Substantial neutron model]]
* [[Physics/Essays/Fedosin/Substantial proton model|Substantial proton model]]
* [[Physics/Essays/Fedosin/Substantial photon model|Substantial photon model]]
* [[Electrogravitational vacuum]]

==External links==
* [http://www.wikiznanie.ru/ru-wz/index.php/%D0%9F%D0%BE%D1%81%D1%82%D0%BE%D1%8F%D0%BD%D0%BD%D0%B0%D1%8F_%D1%81%D0%B8%D0%BB%D1%8C%D0%BD%D0%BE%D0%B9_%D0%B3%D1%80%D0%B0%D0%B2%D0%B8%D1%82%D0%B0%D1%86%D0%B8%D0%B8 Strong gravitational constant (in Russian)]

[[Category:Gravitation]]
[[Category:Fundamental constants]]</text>
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      <comment>/* Connection with usual gravitational constant */</comment>
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      <text bytes="34670" sha1="9ltwgdbtqgrmtfe8wktg8wdiddqi7x6" xml:space="preserve">==Strong (nuclear) gravitation ==
In Astronomy the only one available characteristic empirical physical constant is the gravitational constant. Without completing the charge-mass unification or final unification: one cannot say, whether it is an ‘input to the unification’ or ‘output of unification’. The same idea can be applied to the atomic physical constants also. Sitting in a grand unified roof one cannot make an  ‘absolute measurement’ but can make an ‘absolute finding’. Up till now, no atomic model has implemented the gravitational constant in the atomic or nuclear physics. Then, whatever may be its magnitude, measuring its value from existing atomic principles is impossible. Its value has been measured in the lab only within a range of 1 cm to a few metres, whereas the observed nuclear size is 1.2 fermi. Until one measures the value of the gravitational constant in microscopic physics, the debate of strong (nuclear) gravitation can be considered positively. The idea of strong gravitation originally referred specifically to mathematical approach of Abdus Salam of unification of gravitation and quantum chromodynamics, but is now often used for any particle level gravitation approach. Now many persons are working on this subject. A main advantage of this subject is: it couples black hole physics and particle physics.

==Strong gravitational constant==
The '''strong gravitational constant''', denoted &lt;math&gt;~\;\; \Gamma &lt;/math&gt; or &lt;math&gt;~G_s &lt;/math&gt;, is a grand unified physical constant of strong gravitation, involved in calculation of gravitational attraction at the level of elementary particles and atoms.

According to [[w:Isaac Newton| Newton]]'s law of universal gravitation, the force of gravitational attraction between two massive points with masses &lt;math&gt; ~ m_1 &lt;/math&gt; and &lt;math&gt; ~ m_2 &lt;/math&gt;, located at a distance &lt;math&gt; ~ R &lt;/math&gt; between them, is:

: &lt;math&gt;F=G \frac{m_1 m_2}{R^2}.&lt;/math&gt;

The coefficient of proportionality &lt;math&gt; ~ G &lt;/math&gt; in this expression is called [[w:gravitational constant |gravitational constant]]. It is assumed, that in contrast to the usual force of gravity, at the level of elementary particles acts [[Physics/Essays/Fedosin/Strong gravitation |strong gravitation]]. In order to describe it &lt;math&gt; ~ G &lt;/math&gt; in the formula for gravitational force must be replaced on &lt;math&gt; ~ \Gamma &lt;/math&gt;:

: &lt;math&gt;F_{sg}=\Gamma \frac{m_1 m_2}{R^2}.&lt;/math&gt;

==Dimensions and magnitude==
The dimensions assigned to the strong gravitational constant may be found from the equation above — length cubed, divided by [[w:mass |mass]] and by time squared (in SI units, metres cubed per kilogram per second squared).

There are several ways to assess the value of &lt;math&gt; ~ \Gamma &lt;/math&gt;. J. Dufour, under the assumption that the strong gravitational constant depends on the type of objects, from the interaction of two deuterium nuclei determined, &lt;ref&gt; J. Dufour. [http://www.iscmns.org/CMNS/CMNS.htm &quot;Very sizeable increase of gravity at pico-meter distance: a novel working hypothesis to explain anomalous heat effects and apparent transmutations in certain metal hydrogen systems&quot;]. J. of condensed matter nuclear science, Vol. 1, pp. 47-61 (2007). &lt;/ref&gt; that  &lt;math&gt; G' = 2.06 \times 10^{25} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

Based on the analogy between hadrons and Kerr-Newman [[black hole]]s &lt;ref&gt;Strong Interactions, Gravitation and Cosmology. Abdus Salam Publ. in: NATO Advanced Study Institute, Erice, June16-July 6, 1972 ; in: High Energy Astrophysics and its Relation to Elementary Particle Physics, 441-452 MIT Press, Cambridge (1974). &lt;/ref&gt; Sivaram, C. and Sinha, K.P, &lt;ref&gt; Sivaram, C. and Sinha, K.P. Strong gravity, black holes, and hadrons. Physical Review D, Vol. 16, Issue 6, pp. 1975-1978 (1977). &lt;/ref&gt;  &lt;ref&gt;Salam A. and Sivaram C. Strong Gravity Approach to QCD and Confinement. Mod. Phys. Lett., v. A8(4), pp. 321-326 (1993). &lt;/ref&gt; and Raut, Usha and Shina, KP &lt;ref&gt;Raut, Usha and Shina, KP (1983) [http://eprints.iisc.ernet.in/13571/  Strong gravity and the fine structure constant.] In: Proceedings of the Indian Academy of Sciences Part A: Physical Sciences, 49 (2). pp. 352-358. &lt;/ref&gt; accepted the value &lt;math&gt; \Gamma = 6.7 \times 10^{27} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

This value of the strong gravitational constant allowed estimating the strong spin-torsion interaction between spinning protons. &lt;ref&gt;V. de Sabbata, C. Sivaram. [http://prints.iiap.res.in/bitstream/2248/4394/3/Strong%20spin-torsion%20 Strong Spin-Torsion Interaction between Spinning Protons.] Il Nuovo Cimento, Vol. 101A, N. 2, pp. 273-283 (1989). &lt;/ref&gt;

In paper of Mongan &lt;ref&gt;T. R.  Mongan. [http://th1.ihep.su/archive/gr-qc/070622.html Cold dark matter from &quot;strong gravity&quot;.] General Relativity &amp; Quantum Cosmology, 20 Jun 2007; [http://ru.arxiv.org/abs/0706.3050v2 arXiv:0706.3050v2.] &lt;/ref&gt; strong gravitational constant is &lt;math&gt; G_s = 1.1 \times 10^{28} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

According to [https://en.everybodywiki.com/Robert_Oldershaw Robert L. Oldershaw] &lt;ref&gt; Oldershaw R.L. [http://arxiv.org/abs/physics/0701132v3 Discrete Scale Relativity.] Astrophysics and Space Science, Vol. 311, N. 4, pp. 431-433 (2007). DOI: 10.107/s10509-007-9557-x. &lt;/ref&gt; value of the strong gravitational constant is &lt;math&gt; G_s = 2.18 \times 10^{28} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

As in Oldershaw’ paper, strong gravitational constant could be related &lt;ref&gt;Stone R.A. Quark Confinement and Force Unification. Progress in Physics, April 2010, Vol. 2, P. 19–20. &lt;/ref&gt; with the proton radius &lt;math&gt; ~ R_p &lt;/math&gt;, the proton mass &lt;math&gt; ~ m_p &lt;/math&gt; and the speed of light &lt;math&gt;~c &lt;/math&gt;: 
: &lt;math&gt;sG_p= \frac{R_p c^2}{2 m_p }= 2.4 \times 10^{28} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

According to Tennakone who identified the electron and the proton as black holes in the strong gravitational field, strong gravitational constant is: &lt;ref&gt;K. Tennakone. [http://prd.aps.org/abstract/PRD/v10/i6/p1722_1 Electron, muon, proton, and strong gravity.]  Phys. Rev. D, Volume 10, Issue 6, pp.1722-1725 (1974). &lt;/ref&gt;
: &lt;math&gt;\Gamma =  3.9 \times 10^{28} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

Zane Andrea Quintili finds a strong gravitational constant based on the similarity between the Planck mass and radius, and accordingly the mass and radius of the proton: &lt;ref&gt; Zane Andrea Quintili. [http://vixra.org/abs/1904.0540 Gravitational Field and Proton Radius]. vixra.org. (2019). &lt;/ref&gt;
: &lt;math&gt; G_q = \frac {8\hbar c }{m^2_p} =9.04 \times 10^{28} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

Recami et al &lt;ref&gt;Recami, E.; Ammiraju, P.; Hernandez, H.E.; Kretly, L.C.; Rodrigues, W.A., Jr. Elementary particles as micro-universes: a geometric approach to &quot;strong gravity&quot;. Apeiron, January 01, 1997. &lt;/ref&gt; &lt;ref&gt;Recami E. and Tonin-Zanchin V. The strong coupling constant: its theoretical derivation from a geometric approach to hadron structure. Found. Phys. Lett., v, 7(1), pp. 85-92 (1994). &lt;/ref&gt; define strong gravitational constant through the mass of the pion &lt;math&gt; ~ m_{\pi} &lt;/math&gt; as follows:

: &lt;math&gt;N\approx \frac{h c}{ m^2_{\pi} }= 3.2 \times 10^{30} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;,

where &lt;math&gt; ~ h &lt;/math&gt; – [[w:Planck constant |Planck constant]].

From this they derive [[w:coupling constant | constant of strong interaction]] of two nucleons in the following form: &lt;ref&gt;Erasmo Recami, Tonin-Zanchin, Antonino Del Popolo, Mario Gambera. [http://arxiv.org/abs/physics/0105080 The strong coupling constant,] Heavy Ion Physics, Vol. 10, pp. 345-349 (1999). &lt;/ref&gt;

:&lt;math&gt; \frac{ N g^2}{\hbar c } \approx 14&lt;/math&gt; , where &lt;math&gt;~g &lt;/math&gt; indicates a strong charge, &lt;math&gt; ~ \hbar &lt;/math&gt; is reduced Planck constant.

Stanislav Fisenko et all found &lt;ref&gt;Stanislav Fisenko &amp; Igor Fisenko. [http://www.ccsenet.org/journal/index.php/apr/article/download/8060/6060 The Conception of Thermonuclear Reactor on the Principle of Gravitational Confinement of Dense High-temperature Plasma.] Applied Physics Research, Vol. 2, No. 2, pp. 71-79 (2010). &lt;/ref&gt; &lt;ref&gt;S. I. Fisenko, M. M. Beilinson and B. G. Umanov. Some notes on the concept of “strong” gravitation and possibilities of its experimental investigation. Physics Letters A, Volume 148, Issues 8-9, pp. 405-407 (1990).&lt;/ref&gt; a spectrum of steady states of the electron in proper gravitational field (0.511 MeV …0.681 MeV) on the base of strong coupling constant
: &lt;math&gt;N= 5.1 \times 10^{31} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

U. V. S. Seshavatharam and S. Lakshminarayana &lt;ref&gt; U. V. S. Seshavatharam and S. Lakshminarayana. Strong nuclear gravitational constant and the origin of nuclear planck scale. Progress in Physics, vol. 3, pp. 31-38 (2010). [http://fs.gallup.unm.edu/PP-03-2010.pdf] &lt;/ref&gt; in determining &lt;math&gt; ~ G_s &lt;/math&gt; repelled from the Fermi constant, which led them to the value &lt;math&gt; G_s = 6.94 \times 10^{31} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

In the paper &lt;ref&gt;Perng J. J. [https://springerlink3.metapress.com/content/1503066144130716/resource-secured/?target=fulltext.pdf&amp;sid=qwdy0a45odlh3a55v4k45u45&amp;sh=www.springerlink.com Strong gravitation and elementary particles.] Nuovo Cimento, Lettere, Serie 2, vol. 23, N. 15, pp. 552-554 (1978). &lt;/ref&gt; strong gravitational constant equal to &lt;math&gt;\Gamma =2.77 \times 10^{32} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

[[User:Fedosin | Sergey Fedosin]] entered the strong gravitational constant in 1999 on the basis of equality between the Coulomb electric force and gravitational force in the hydrogen atom on the [[w:Bohr radius |Bohr radius]]. This leads to the following expression for the value of the strong gravitational constant: &lt;ref name=&quot;fed&quot;&gt; Fedosin S.G. [http://lccn.loc.gov/2009457349 Fizika i filosofiia podobiia: ot preonov do metagalaktik,] Perm, (1999-06-09) 544 pp. {{ISBN|5-8131-0012-1}}. &lt;/ref&gt;

: &lt;math&gt;\Gamma= \frac{e^2}{4 \pi \varepsilon_{0} m_p m_e }=1.514 \times 10^{29} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;,

where &lt;math&gt; ~ e &lt;/math&gt; – [[w:elementary charge |elementary charge]], &lt;math&gt; ~ \pi &lt;/math&gt; – [[w:pi (number) | pi]], &lt;math&gt; ~ \varepsilon_{0} &lt;/math&gt; – [[electric constant]], &lt;math&gt; ~ m_p &lt;/math&gt; – the mass of [[w:proton |proton]], &lt;math&gt; ~ m_e &lt;/math&gt; – the mass of [[w:electron| electron]]. 

It is assumed that strong gravitation, as a universal force, acts on the matter of nucleons, hadrons, electrons and elementary particles, regardless of the type of these particles. In contrast, the standard approach considers that strong interaction does not affect electrons and other leptons.

The small mass and large charge of matter do not allow the electron to be entirely in some small volume near the nucleus, and it gets disklike axisymmetric shape, which is limited by size of atom. In the hydrogen atom electrical forces between the nucleus and matter of the electron are attractive, but they are compensated by the repulsion of the intrinsic charge of the electron. There are the centripetal force of rotation of the electron around the nucleus, and the gravitational attraction between massive nucleus and matter of the electron. All these forces are equal in magnitude. From here follows that the action of strong gravitation between the masses of nucleus and electron on the one hand, and the electric force between charges of the nucleus and the electron, on the other hand, allows to estimate the value of &lt;math&gt; ~ \Gamma &lt;/math&gt;.
If &lt;math&gt;~ R_B = \frac {\hbar }{ m_e \alpha c } &lt;/math&gt; is the Bohr radius, then the equality of forces gives:

: &lt;math&gt; \frac {\Gamma m_p m_e }{R^2_B} = \frac{e^2}{4 \pi \varepsilon_{0} R^2_B } .&lt;/math&gt;

[[Physics/Essays/Fedosin/Fine structure constant | Fine structure constant]] is
:&lt;math&gt; \alpha = \frac{e^2}{4 \pi \varepsilon_0 \hbar c}, &lt;/math&gt;

So that
: &lt;math&gt; \Gamma= \frac{\alpha \hbar c }{m_p m_e }, \qquad \qquad \hbar = \frac{\Gamma m_p m_e }{ \alpha c }.&lt;/math&gt;

Bohr radius becomes equal
:&lt;math&gt;~ R_B = \frac{\Gamma m_p }{ \alpha^2 c^2 } = \frac{\Gamma m_p }{ V^2_B },&lt;/math&gt;

where &lt;math&gt;~ V_B = \alpha c &lt;/math&gt; is the orbital speed of the electron cloud at the first energy level.

Hence &lt;math&gt;~ V^2_B = \frac{\Gamma m_p }{ R_B }&lt;/math&gt;, and the kinetic energy of the electron, taking into account determination of strong gravitational constant, is equal to:
:&lt;math&gt;~ K = \frac{m_e V^2_B }{ 2 } = \frac{\Gamma m_p m_e }{ 2 R_B }=\frac { e^2}{8 \pi \varepsilon_0 R_B } = - \frac {W}{2} ,&lt;/math&gt;

where &lt;math&gt;~ W &lt;/math&gt;  is the potential energy of electron in the electric field of the nucleus of a hydrogen atom.

It turns out the virial theorem in the form &lt;math&gt;~ K = - \frac {W}{2} &lt;/math&gt;. The total electron energy is also found at the first energy level:

:&lt;math&gt;~ E = K+W = \frac {W}{2} = -K = -13.6 &lt;/math&gt; eV.

With the help of the constant &lt;math&gt; ~ \Gamma &lt;/math&gt; the [[w:Invariant mass#Rest energy | rest energy]] of proton in the form of a ball is equal to half of its potential energy of strong gravitational field in accordance with [[w:virial theorem |virial theorem]], &lt;ref&gt; [[User:Fedosin | Sergey Fedosin]], [https://www.lap-publishing.com/catalog/details//store/gb/book/978-3-659-57301-9/the-physical-theories-and-infinite-hierarchical-nesting-of-matter-volume-1 The physical theories and infinite hierarchical nesting of matter], Volume 1, LAP LAMBERT Academic Publishing, pages: 580, {{ISBN|978-3-659-57301-9}}. &lt;/ref&gt; if we assume that the binding energy &lt;math&gt; ~ E_b &lt;/math&gt; for the proton up to a sign is equal to the total energy of proton, and &lt;math&gt; ~ E_b &lt;/math&gt; becomes very close to relativistic energy in the form of rest energy:

: &lt;math&gt;~ m_p c^2 \approx E_b = -\frac {W_p}{2} = \frac{ k \Gamma m^2_p }{ 2R_p},&lt;/math&gt;

where &lt;math&gt; ~ R_p =8.73 \times 10^{-16} &lt;/math&gt; m  is the proton radius,
&lt;math&gt; ~ k=0.62 &lt;/math&gt; (in the hypothetical case of a uniform mass density of the proton there must be &lt;math&gt; ~ k = 0.6 &lt;/math&gt;). This implies that the mass of nucleons is determined by the energy of the strong gravitation according to the principle of [[w:mass–energy equivalence |mass–energy equivalence]].

If we assume that the magnetic moment of the proton is created by the maximum rotation of its positive charge distributed over the volume of the proton in the form of a ball, when the centripetal acceleration at the equator becomes equal to acceleration of strong gravitation, the formula for the magnetic moment is as follows:
: &lt;math&gt; ~ P_m = \delta e \sqrt {\Gamma m_p R_p}, &lt;/math&gt;

where &lt;math&gt; ~ P_m = 1.41 \times 10^{-26} &lt;/math&gt; J / T  is the magnetic moment of the proton,
&lt;math&gt; ~ \delta = 0.1875 &lt;/math&gt; (in the case of uniform density and charge should be &lt;math&gt; ~ \delta = 0.2 &lt;/math&gt;).

From the formulas for the energy and the magnetic moment the radius of the proton is determined in the self-consistent model. &lt;ref&gt; Fedosin S.G. [http://vixra.org/abs/1208.0006 The radius of the proton in the self-consistent model.] Hadronic Journal, Vol. 35, No. 4, pp. 349-363 (2012). &lt;/ref&gt;

The strong gravitational constant is also included in the formula describing the [[w:nuclear force |nuclear force]] through strong gravitation and [[Physics/Essays/Fedosin/Gravitational torsion field |gravitational torsion field]] of rotating particles. &lt;ref&gt;[http://sergf.ru/com.htm Comments to the book]: Fedosin S.G. Fizicheskie teorii i beskonechnaia vlozhennost’ materii. – Perm, 2009, 844 pages, Tabl. 21, Pic. 41, Ref. 289. {{ISBN|978-5-9901951-1-0}}. (in Russian). &lt;/ref&gt; A feature of the [[Physics/Essays/Fedosin/Gravitational induction |gravitational induction]] is that if two bodies rotate along one axis and come close by the force of gravitation, then these bodies will increase the angular velocity of its rotation. In this regard, it is assumed that the nucleons in atomic nuclei rotate at maximum speed. This may explain the equilibrium of the nucleons in atomic nuclei as a balance between the attractive force of strong gravitation and the strong force of the torsion field (of gravitomagnetic forces in [[Physics/Essays/Fedosin/Gravitoelectromagnetism|gravitoelectromagnetism]]). In particular, the [[Physics/Essays/Fedosin/Coupling constant |coupling constant]] is

:&lt;math&gt;\alpha_{pp}=  \frac{\beta \Gamma m^2_p }{\hbar c }=13.4 \beta &lt;/math&gt;,
where &lt;math&gt; ~ \beta &lt;/math&gt; is equal to 0.26 for the interaction of two nucleons, and tending to 1 for bodies with a lower mass density.

The constant &lt;math&gt;~\alpha_{pp}&lt;/math&gt; is close to [[w:coupling constant |coupling constant]] of [[Charges/Interactions/Strong|strong interaction]] of two nucleons in [[w:Standard Model |Standard Model]] 
:&lt;math&gt;\alpha_s=  \frac{ g^2_{N \pi}}{4\pi\hbar c } \approx 14.6&lt;/math&gt; , where &lt;math&gt;~g_{N \pi} &lt;/math&gt; is the constant of the pseudoscalar nucleon-pionic interaction.

[[Physics/Essays/Fedosin/Fine structure constant | Fine structure constant]] is coupling constant of electromagnetic interaction and may be written so:

:&lt;math &gt; ~ \alpha = \frac { \Gamma m_p m_e }{\hbar c }\approx \frac {1}{137.036}.&lt;/math&gt;

==Role of squared Avogadro number ==
Considering Avogadro number &lt;math&gt;N&lt;/math&gt; as a scaling factor, U. V. S. Seshavatharam and S. Lakshminarayana finally arrived at a value of &lt;ref&gt; U. V. S. Seshavatharam and S. Lakshminarayana. Unified model of universe and the atom. Book. {{ISBN|9783843393966}}, LAP LAMBERT Academic Publishing GmbH &amp; Co. KG, Germany, 2011 March 30.  &lt;/ref&gt; &lt;ref&gt; U. V. S. Seshavatharam and S. Lakshminarayana. Role of Avogadro number in grand unification. Hadronic Journal. Vol. 33, No 5, p. 513 (2010).  &lt;/ref&gt; &lt;ref&gt; U. V. S. Seshavatharam and S. Lakshminarayana. Atomic gravitational constant and the origin of elementary magnetic moments. To be published.  &lt;/ref&gt; 
&lt;math&gt; G_s = N^2G = 2.42 \times 10^{37} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.
It is noticed that in Hydrogen atom, ratio of total energy of electron and nuclear potential is equal to the electromagnetic and gravitational force ratio of electron where the operating gravitational constant is nothing but the atomic gravitational having a value N&lt;sup&gt;2&lt;/sup&gt;G. This is a direct confirmation of the existence of the atomic or nuclear gravitational constant in nuclear physics. Therefore, this subject can now be considered as part of the mainstream research in quantum gravity.

The central idea is: for mole number of particles, strength of gravity is &lt;math&gt;N.G&lt;/math&gt; and force required to bind &lt;math&gt;N&lt;/math&gt; particles is &lt;math&gt;\frac{c^4}{N.G}.&lt;/math&gt; Force required to bind one particle is &lt;math&gt;\frac{c^4}{N^2.G}.&lt;/math&gt; By considering this force magnitude as the characteristic weak force magnitude, it is observed that, &lt;math&gt;\ln \sqrt{\frac{e^2}{4 \pi \epsilon_0 G m_p^2}} \cong \sqrt{\frac{m_p}{m_e}-\ln\left(N^2\right)}&lt;/math&gt; where &lt;math&gt;m_p&lt;/math&gt; is the rest mass of proton and &lt;math&gt;m_e&lt;/math&gt; is the rest mass of electron. Obtained value of &lt;math&gt; G\cong \; 6.{\rm 6}66270{\rm 1}79\times {\rm 1}0^{-{\rm 1}1} {\rm \; m}^{{\rm 3}} {\rm Kg}^{{\rm -1}} {\rm sec}^{{\rm -2}.}&lt;/math&gt; Here the most important point to be emphasized is &lt;math&gt;\frac{c^4}{G}&lt;/math&gt; can be considered as the classical or upper limit of gravitational or electromagnetic force. It can be considered as the grand unified force. It is the origin of Planck scale and of the black hole astrophysics.

==Connection with usual gravitational constant==
With the help of [[Physics/Essays/Fedosin/Similarity of matter levels|similarity of matter levels]] and [[Physics/Essays/Fedosin/SPФ symmetry|SPФ symmetry]] in Theory of [[Physics/Essays/Fedosin/Infinite Hierarchical Nesting of Matter|Infinite Hierarchical Nesting of Matter]] the value of &lt;math&gt; ~ \Gamma &lt;/math&gt; can also be defined in terms of coefficients of similarity and the gravitational constant:

: &lt;math&gt;\Gamma = G \frac{ \Phi }{ P S^2},&lt;/math&gt;

where &lt;math&gt; ~ \Phi =1.62 \times 10^{57} &lt;/math&gt;, &lt;math&gt; ~ P= 1.4 \times 10^{19} &lt;/math&gt;, &lt;math&gt; ~ S= 0.23 &lt;/math&gt; are the coefficients of similarity in mass, size and speed, respectively, for the degenerate quantum objects at the atomic and stellar levels of matter.&lt;ref name=&quot;fed&quot;/&gt;  The powers of similarity coefficients in this equation correspond to dimension of gravitational constant according to [[w:Dimensional analysis |dimensional analysis]]. 

From the standpoint of Infinite Hierarchical Nesting of Matter and [[w:Le Sage's theory of gravitation |Le Sage's theory of gravitation]], the presence of two gravitational constants &lt;math&gt; ~ \Gamma &lt;/math&gt; and &lt;math&gt; ~ G &lt;/math&gt; reflects the difference between the properties of gravitons and properties of matter at different levels of matter. &lt;ref &gt; Fedosin S.G. [http://sergf.ru/mgen.htm  Model of Gravitational Interaction in the Concept of Gravitons]. Journal of Vectorial Relativity, Vol. 4, No. 1, pp. 1-24 (2009). &lt;/ref&gt; &lt;ref&gt; Fedosin S.G. [http://vixra.org/abs/1503.0127 The graviton field as the source of mass and gravitational force in the modernized Le Sage’s model.] Physical Science International Journal, ISSN: 2348-0130, Vol. 8, Issue 4, pp. 1-18 (2015). http://dx.doi.org/10.9734/PSIJ/2015/22197. &lt;/ref&gt;

In particular, for the strong gravitational constant and the ordinary gravitational constant it is possible to write similar relations, in which these constants are expressed in terms of the corresponding energy densities of gravitons’ fluxes in [[Physics/Essays/Fedosin/Electrogravitational vacuum |electrogravitational vacuum]] and the parameters of the densest object of the corresponding level of matter: &lt;ref name=&quot;cha&quot;&gt; Fedosin S.G. [https://www.ajol.info/index.php/jfas/article/view/168204  The charged component of the vacuum field as the source of electric force in the modernized Le Sage’s model.] Journal of Fundamental and Applied Sciences, Vol. 8, No. 3, pp. 971-1020 (2016). http://dx.doi.org/10.4314/jfas.v8i3.18, https://dx.doi.org/10.5281/zenodo.845357.&lt;/ref&gt;
:&lt;math&gt;~ \Gamma = \frac { \varepsilon_c \vartheta^2}{4 \pi M^2_n } , \qquad \qquad  G = \frac { \varepsilon_{cs} \vartheta^2_s}{4 \pi M^2_s } , &lt;/math&gt;

where &lt;math&gt;~ \varepsilon_c = 7.4 \cdot 10^{35}&lt;/math&gt; J/m³  is the energy density of the graviton fluxes for cubic distribution;
&lt;math&gt;~ \vartheta = 2.67 \cdot 10^{-30} &lt;/math&gt; m²  is the cross-section of interaction of the charged particles of the electrogravitational vacuum ([[Physics/Essays/Fedosin/Praon|praon]]s) with nucleons, which is very close in magnitude to the geometrical cross-section of the nucleon and is used to calculate the [[electric constant]]; &lt;math&gt;~ M_n &lt;/math&gt; is the mass of the nucleon;
&lt;math&gt; \varepsilon_{cs} = \varepsilon_c \frac {\Phi S^2}{ P^3} = 2.3 \cdot 10^{34}&lt;/math&gt; J/m³ is the energy density of the graviton fluxes at the stellar level for cubic distribution; &lt;math&gt;~ \vartheta_s = \vartheta P^2 = 5.2 \cdot 10^{8} &lt;/math&gt; m²  is the cross-section of interaction between the gravitons and a neutron star; &lt;math&gt;~ M_s = M_n \Phi = 2.7 \cdot 10^{30} &lt;/math&gt; kg is the mass of the neutron star.

At the matter level of [[Physics/Essays/Fedosin/Praon|praon]]s, its own strong gravitational constant &lt;math&gt;~G_{pr} &lt;/math&gt; must act. Considering that the coefficient of similarity in speed between the nucleon and praon levels of matter is &lt;math&gt;~S \approx 1 &lt;/math&gt;, we can write: 

: &lt;math&gt; G_{pr} = \Gamma \frac{ \Phi }{ P S^2} = \frac{ q^2_{pr}\beta}{4 \pi \varepsilon_{0} m^2_{pr} } =1.752 \cdot 10^{67}&lt;/math&gt; m&lt;sup&gt;3&lt;/sup&gt;•s&lt;sup&gt;–2&lt;/sup&gt;•kg&lt;sup&gt;–1&lt;/sup&gt;,

where &lt;math&gt;~ q_{pr} = 1.06 \cdot 10^{-57} &lt;/math&gt; C is the charge of the praon, &lt;math&gt;~ m_{pr} = 1 \cdot 10^{-84}&lt;/math&gt; kg is the mass of the praon, &lt;math&gt;~ \beta = \frac { m_p }{ m_e }= 1836.152&lt;/math&gt; is the proton to electron mass ratio.

==Connection with mass and unification of interaction==
The main object of unification is to understand the origin of elementary particles mass, (Dirac) magnetic moments and their forces. Right now and till today ‘string theory’ with 4 + 6 extra dimensions not in a position to explain the unification of gravitational and non-gravitational forces. More clearly speaking it is not in a position to bring down the planck scale to the nuclear size. Physicists say – if strength of strong interaction is unity, with reference to the strong interaction, strength of gravitation is 10&lt;sup&gt;−39&lt;/sup&gt;. The fundamental question to be answered is: is mass an inherent property of any elementary particle?

One can say: for any elementary particle mass is an induced property. This idea makes grand unification easy. Note that [[Theory of relativity/General relativity|general relativity]] does not throw any light on the ‘mass generation’ of charged particles. It only suggests that space-time is curved near the massive celestial objects. More over it couples the cosmic (dust) matter with geometry. But how matter is created? Why and how elementary particle possesses both charge and mass? Such types of questions are not discussed in the frame work of general relativity.

The first step in unification is to understand the origin of the [[w:Invariant mass |rest mass]] of a charged elementary particle. Second step is to understand the combined effects of its electromagnetic (or charged) and gravitational interactions. Third step is to understand its behavior with surroundings when it is created. Fourth step is to understand its behavior with cosmic space-time or other particles. Right from its birth to death, in all these steps the underlying fact is that whether it is a strongly interacting particle or weakly interacting particle, it is having some rest mass. To understand the first two steps somehow one can implement the gravitational constant in sub atomic physics.

To bring down the [[Physics/Essays/Fedosin/Planck mass|Planck mass]] scale to the observed elementary particles mass scale a large scale factor is required. Just like [[w:relative permeability |relative permeability]] and [[w:relative permittivity |relative permittivity]] by any suitable reason in atomic space if one is able to increase the value of classical gravitational constant, it helps in four ways. Observed elementary particles mass can be generated and grand unification can be achieved. Third important application is characteristic building block of the cosmological [[dark matter]] can be quantified in terms of fundamental physical constants. Fourth important application is – no [[w:extra dimensions |extra dimensions]] are required. Finally nuclear physics and quantum mechanics can be studied in the view of [[strong gravitation |strong nuclear gravity]] where nuclear charge and atomic gravitational constant play a crucial role in the nuclear space-time curvature, [[w:quantum chromodynamics |quantum chromodynamics]] and [[w:Color confinement |quark confinement]]. Not only that cosmology and particle physics can be studied in a unified way. In this connection it is suggested that square root of ratio of atomic gravitational constant and classical gravitational constant is equal to the Avogadro number. &lt;ref&gt; [http://www.journal-of-nuclear-physics.com/?p=316 AGNI – Avogadro's gravity for nuclear interactions.] Nuclear experiments blog: Journal of Nuclear Physics, Nov. 2010. &lt;/ref&gt; The Avogadro constant expresses the number of elementary entities per mole of substance and it has the value mol&lt;sup&gt;–1&lt;/sup&gt;. Avogadro's constant is a scaling factor between macroscopic and microscopic (atomic scale) observations of nature. It is an observed fact. The very unfortunate thing is that even though it is a large number it is neither implemented in cosmology nor implemented in grand unification. 

Modern physics is having hardly 100 years of ‘strong nuclear’ back ground. By Einstein’s time very little information was available on nuclear strong and weak forces. Avogadro hypothesis was proposed in 1811. Compared to modern nuclear physics,  Avogadro number is having 100 years of old history. Avogadro number may not be a fundamental physical constant but can be considered as a ‘scale factor’. But quantitatively it can be linked with the fundamental force ratios. Future thoughts and experiments may give some clue of it. Best present example is the ratio of planck mass and electron mass.  Considering this ratio automatically N&lt;sup&gt;2&lt;/sup&gt; comes into picture. It is noticed that in Hydrogen atom, ratio of total energy of electron and nuclear potential is equal to the electromagnetic and gravitational force ratio of electron where the operating gravitational constant is nothing but the atomic gravitational having a value N&lt;sup&gt;2&lt;/sup&gt;G. This is a direct confirmation of the existence of the atomoc or nuclear gravitational constant.  

Here the very important question to be answered is – which is more fundamental either &lt;math&gt;  G &lt;/math&gt; or &lt;math&gt; G_s &lt;/math&gt; ? It is proposed that both can be considered as the 'head' and 'tail' of matter coin. It can also be suggested that classical &lt;math&gt;  G &lt;/math&gt; is a consequence of the existence of atomic &lt;math&gt; G_s &lt;/math&gt;. It is known that there is a difference in between 'absolute findings' and 'absolute measurements'. Absolute findings can be understood where as 'absolute measurements' can not be made by nuclear experiments which are being conducted under the sky of universal gravity with unknown origin of elementary particles mass.

Till today there is no explanation for this fantastic and large difference between &lt;math&gt; G &lt;/math&gt; or &lt;math&gt; G_s &lt;/math&gt; or between gravitation and strong interaction, about 10&lt;sup&gt;−39&lt;/sup&gt;. It can be supposed that elementary particles construction is much more fundamental than the black hole's construction. If one wishes to unify electroweak, strong and gravitational interactions it is a must to implement the classical gravitational constant &lt;math&gt; G &lt;/math&gt; in the sub atomic physics. &lt;ref&gt; Seshavatharam, U. V. S.; Lakshminarayana, S. [http://adsabs.harvard.edu/abs/2010IJMPE..19..263S  Super symmetry in strong and weak interactions.] International journal of modern physics E, Issue 02, pp. 263-280, Feb.2010. &lt;/ref&gt; By any reason if one implements the Planck scale in elementary particle physics and nuclear physics automatically &lt;math&gt; G &lt;/math&gt; comes into subatomic physics. Then a large arbitrary number has to be considered as a proportionality constant. After that its physical significance has to be analyzed. Alternatively its equivalent 'strong atomic gravitational constant' can also be assumed. Some attempts have been done in physics history.

Whether it may be real or an equivalent if it is existing as a 'single constant' its physical significance can be understood. Nuclear size can be fitted with 'nuclear Schwarzschild radius'. Nucleus can be considered as 'strong nuclear black hole'. This idea requires a basic nuclear fermion! Nuclear binding energy constants can be generated directly. Proton-neutron stability can be studied. Origin of strong [[w:coupling constant |coupling constant]] and [[w:Fermi's interaction |Fermi's weak coupling constant]] can be understood. Charged lepton masses can be fitted. Such applications can be considered favorable for the proposed assumptions and further analysis can be carried out positively for understanding and developing this proposed 'Avogadro's strong nuclear gravity'.

Unification means: finding the similarities, finding the limiting physical constants, finding the key numbers, coupling the key physical constants, coupling the key physical concepts, coupling the key physical properties, minimizing the number of dimensions, minimizing the number of inputs and implementing the key physical constant or key number in different branches of physics. This is a very lengthy process. In all these cases observations, interpretations, experiments and imagination play a key role. The main difficulty is with interpretations and observations. As the interpretation changes physical concept changes, physical equation changes and finally the destiny changes. 

Note that human beings are part of this universal gravity. There are some natural restrictions to experiments. Seeing a black hole is highly speculative. But indirectly its significances can be well understood. In the similar way in nuclear and particle physics: any experimental setup which is being run under the influence of the proposed strong nuclear gravity, without knowing the probing particle’s massive origin, without knowing the massive origin of the nucleus: based on ‘grand unified scheme’ one may not be able to unearth the absolute findings. Note that observer, experimental setup and the probing particle all are under the same influence of universal gravity. When searching for an experimental proof in grand/final unification scheme or dark matter projects this fact may be considered positively for further analysis. 

To conclude it can be suggested that – existence of strong gravitational constant as Atomic gravitational constant is true and its consequences can be understood easily and can be implemented easily in grand unification program and dark matter projects.

== Notes ==
&lt;references/&gt;

== See also ==
* [[Strong gravitation]]
* [[Coupling constant]]
* [[Gravitational constant]]
* [[w:Gravitational coupling constant |Gravitational coupling constant]]
* [[Fine structure constant]]
* [[w:Dimensionless physical constant |Dimensionless physical constant]]
* [[Physics/Essays/Fedosin/Infinite Hierarchical Nesting of Matter|Infinite Hierarchical Nesting of Matter]]
* [[Physics/Essays/Fedosin/Similarity of matter levels|Similarity of matter levels]]
* [[Physics/Essays/Fedosin/SPФ symmetry|SPФ symmetry]]
* [[Quantization of parameters of cosmic systems]]
* [[Hydrogen system]]
* [[Stellar constants]]
* [[Gravitational model of strong interaction]]
* [[Model of quark quasiparticles]]
* [[Physics/Essays/Fedosin/Substantial electron model|Substantial electron model]]
* [[Physics/Essays/Fedosin/Substantial neutron model|Substantial neutron model]]
* [[Physics/Essays/Fedosin/Substantial proton model|Substantial proton model]]
* [[Physics/Essays/Fedosin/Substantial photon model|Substantial photon model]]
* [[Electrogravitational vacuum]]

==External links==
* [http://www.wikiznanie.ru/ru-wz/index.php/%D0%9F%D0%BE%D1%81%D1%82%D0%BE%D1%8F%D0%BD%D0%BD%D0%B0%D1%8F_%D1%81%D0%B8%D0%BB%D1%8C%D0%BD%D0%BE%D0%B9_%D0%B3%D1%80%D0%B0%D0%B2%D0%B8%D1%82%D0%B0%D1%86%D0%B8%D0%B8 Strong gravitational constant (in Russian)]

[[Category:Gravitation]]
[[Category:Fundamental constants]]</text>
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      <text bytes="35096" sha1="rvp2jsn9i1badgaizu3949nywy62g6v" xml:space="preserve">==Strong (nuclear) gravitation ==
In Astronomy the only one available characteristic empirical physical constant is the gravitational constant. Without completing the charge-mass unification or final unification: one cannot say, whether it is an ‘input to the unification’ or ‘output of unification’. The same idea can be applied to the atomic physical constants also. Sitting in a grand unified roof one cannot make an  ‘absolute measurement’ but can make an ‘absolute finding’. Up till now, no atomic model has implemented the gravitational constant in the atomic or nuclear physics. Then, whatever may be its magnitude, measuring its value from existing atomic principles is impossible. Its value has been measured in the lab only within a range of 1 cm to a few metres, whereas the observed nuclear size is 1.2 fermi. Until one measures the value of the gravitational constant in microscopic physics, the debate of strong (nuclear) gravitation can be considered positively. The idea of strong gravitation originally referred specifically to mathematical approach of Abdus Salam of unification of gravitation and quantum chromodynamics, but is now often used for any particle level gravitation approach. Now many persons are working on this subject. A main advantage of this subject is: it couples black hole physics and particle physics.

==Strong gravitational constant==
The '''strong gravitational constant''', denoted &lt;math&gt;~\;\; \Gamma &lt;/math&gt; or &lt;math&gt;~G_s &lt;/math&gt;, is a grand unified physical constant of strong gravitation, involved in calculation of gravitational attraction at the level of elementary particles and atoms.

According to [[w:Isaac Newton| Newton]]'s law of universal gravitation, the force of gravitational attraction between two massive points with masses &lt;math&gt; ~ m_1 &lt;/math&gt; and &lt;math&gt; ~ m_2 &lt;/math&gt;, located at a distance &lt;math&gt; ~ R &lt;/math&gt; between them, is:

: &lt;math&gt;F=G \frac{m_1 m_2}{R^2}.&lt;/math&gt;

The coefficient of proportionality &lt;math&gt; ~ G &lt;/math&gt; in this expression is called [[w:gravitational constant |gravitational constant]]. It is assumed, that in contrast to the usual force of gravity, at the level of elementary particles acts [[Physics/Essays/Fedosin/Strong gravitation |strong gravitation]]. In order to describe it &lt;math&gt; ~ G &lt;/math&gt; in the formula for gravitational force must be replaced on &lt;math&gt; ~ \Gamma &lt;/math&gt;:

: &lt;math&gt;F_{sg}=\Gamma \frac{m_1 m_2}{R^2}.&lt;/math&gt;

==Dimensions and magnitude==
The dimensions assigned to the strong gravitational constant may be found from the equation above — length cubed, divided by [[w:mass |mass]] and by time squared (in SI units, metres cubed per kilogram per second squared).

There are several ways to assess the value of &lt;math&gt; ~ \Gamma &lt;/math&gt;. J. Dufour, under the assumption that the strong gravitational constant depends on the type of objects, from the interaction of two deuterium nuclei determined, &lt;ref&gt; J. Dufour. [http://www.iscmns.org/CMNS/CMNS.htm &quot;Very sizeable increase of gravity at pico-meter distance: a novel working hypothesis to explain anomalous heat effects and apparent transmutations in certain metal hydrogen systems&quot;]. J. of condensed matter nuclear science, Vol. 1, pp. 47-61 (2007). &lt;/ref&gt; that  &lt;math&gt; G' = 2.06 \times 10^{25} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

Based on the analogy between hadrons and Kerr-Newman [[black hole]]s &lt;ref&gt;Strong Interactions, Gravitation and Cosmology. Abdus Salam Publ. in: NATO Advanced Study Institute, Erice, June16-July 6, 1972 ; in: High Energy Astrophysics and its Relation to Elementary Particle Physics, 441-452 MIT Press, Cambridge (1974). &lt;/ref&gt; Sivaram, C. and Sinha, K.P, &lt;ref&gt; Sivaram, C. and Sinha, K.P. Strong gravity, black holes, and hadrons. Physical Review D, Vol. 16, Issue 6, pp. 1975-1978 (1977). &lt;/ref&gt;  &lt;ref&gt;Salam A. and Sivaram C. Strong Gravity Approach to QCD and Confinement. Mod. Phys. Lett., v. A8(4), pp. 321-326 (1993). &lt;/ref&gt; and Raut, Usha and Shina, KP &lt;ref&gt;Raut, Usha and Shina, KP (1983) [http://eprints.iisc.ernet.in/13571/  Strong gravity and the fine structure constant.] In: Proceedings of the Indian Academy of Sciences Part A: Physical Sciences, 49 (2). pp. 352-358. &lt;/ref&gt; accepted the value &lt;math&gt; \Gamma = 6.7 \times 10^{27} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

This value of the strong gravitational constant allowed estimating the strong spin-torsion interaction between spinning protons. &lt;ref&gt;V. de Sabbata, C. Sivaram. [http://prints.iiap.res.in/bitstream/2248/4394/3/Strong%20spin-torsion%20 Strong Spin-Torsion Interaction between Spinning Protons.] Il Nuovo Cimento, Vol. 101A, N. 2, pp. 273-283 (1989). &lt;/ref&gt;

In paper of Mongan &lt;ref&gt;T. R.  Mongan. [http://th1.ihep.su/archive/gr-qc/070622.html Cold dark matter from &quot;strong gravity&quot;.] General Relativity &amp; Quantum Cosmology, 20 Jun 2007; [http://ru.arxiv.org/abs/0706.3050v2 arXiv:0706.3050v2.] &lt;/ref&gt; strong gravitational constant is &lt;math&gt; G_s = 1.1 \times 10^{28} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

According to [https://en.everybodywiki.com/Robert_Oldershaw Robert L. Oldershaw] &lt;ref&gt; Oldershaw R.L. [http://arxiv.org/abs/physics/0701132v3 Discrete Scale Relativity.] Astrophysics and Space Science, Vol. 311, N. 4, pp. 431-433 (2007). DOI: 10.107/s10509-007-9557-x. &lt;/ref&gt; value of the strong gravitational constant is &lt;math&gt; G_s = 2.18 \times 10^{28} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

As in Oldershaw’ paper, strong gravitational constant could be related &lt;ref&gt;Stone R.A. Quark Confinement and Force Unification. Progress in Physics, April 2010, Vol. 2, P. 19–20. &lt;/ref&gt; with the proton radius &lt;math&gt; ~ R_p &lt;/math&gt;, the proton mass &lt;math&gt; ~ m_p &lt;/math&gt; and the speed of light &lt;math&gt;~c &lt;/math&gt;: 
: &lt;math&gt;sG_p= \frac{R_p c^2}{2 m_p }= 2.4 \times 10^{28} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

According to Tennakone who identified the electron and the proton as black holes in the strong gravitational field, strong gravitational constant is: &lt;ref&gt;K. Tennakone. [http://prd.aps.org/abstract/PRD/v10/i6/p1722_1 Electron, muon, proton, and strong gravity.]  Phys. Rev. D, Volume 10, Issue 6, pp.1722-1725 (1974). &lt;/ref&gt;
: &lt;math&gt;\Gamma =  3.9 \times 10^{28} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

Zane Andrea Quintili finds a strong gravitational constant based on the similarity between the Planck mass and radius, and accordingly the mass and radius of the proton: &lt;ref&gt; Zane Andrea Quintili. [http://vixra.org/abs/1904.0540 Gravitational Field and Proton Radius]. vixra.org. (2019). &lt;/ref&gt;
: &lt;math&gt; G_q = \frac {8\hbar c }{m^2_p} =9.04 \times 10^{28} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

Recami et al &lt;ref&gt;Recami, E.; Ammiraju, P.; Hernandez, H.E.; Kretly, L.C.; Rodrigues, W.A., Jr. Elementary particles as micro-universes: a geometric approach to &quot;strong gravity&quot;. Apeiron, January 01, 1997. &lt;/ref&gt; &lt;ref&gt;Recami E. and Tonin-Zanchin V. The strong coupling constant: its theoretical derivation from a geometric approach to hadron structure. Found. Phys. Lett., v, 7(1), pp. 85-92 (1994). &lt;/ref&gt; define strong gravitational constant through the mass of the pion &lt;math&gt; ~ m_{\pi} &lt;/math&gt; as follows:

: &lt;math&gt;N\approx \frac{h c}{ m^2_{\pi} }= 3.2 \times 10^{30} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;,

where &lt;math&gt; ~ h &lt;/math&gt; – [[w:Planck constant |Planck constant]].

From this they derive [[w:coupling constant | constant of strong interaction]] of two nucleons in the following form: &lt;ref&gt;Erasmo Recami, Tonin-Zanchin, Antonino Del Popolo, Mario Gambera. [http://arxiv.org/abs/physics/0105080 The strong coupling constant,] Heavy Ion Physics, Vol. 10, pp. 345-349 (1999). &lt;/ref&gt;

:&lt;math&gt; \frac{ N g^2}{\hbar c } \approx 14&lt;/math&gt; , where &lt;math&gt;~g &lt;/math&gt; indicates a strong charge, &lt;math&gt; ~ \hbar &lt;/math&gt; is reduced Planck constant.

Stanislav Fisenko et all found &lt;ref&gt;Stanislav Fisenko &amp; Igor Fisenko. [http://www.ccsenet.org/journal/index.php/apr/article/download/8060/6060 The Conception of Thermonuclear Reactor on the Principle of Gravitational Confinement of Dense High-temperature Plasma.] Applied Physics Research, Vol. 2, No. 2, pp. 71-79 (2010). &lt;/ref&gt; &lt;ref&gt;S. I. Fisenko, M. M. Beilinson and B. G. Umanov. Some notes on the concept of “strong” gravitation and possibilities of its experimental investigation. Physics Letters A, Volume 148, Issues 8-9, pp. 405-407 (1990).&lt;/ref&gt; a spectrum of steady states of the electron in proper gravitational field (0.511 MeV …0.681 MeV) on the base of strong coupling constant
: &lt;math&gt;N= 5.1 \times 10^{31} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

U. V. S. Seshavatharam and S. Lakshminarayana &lt;ref&gt; U. V. S. Seshavatharam and S. Lakshminarayana. Strong nuclear gravitational constant and the origin of nuclear planck scale. Progress in Physics, vol. 3, pp. 31-38 (2010). [http://fs.gallup.unm.edu/PP-03-2010.pdf] &lt;/ref&gt; in determining &lt;math&gt; ~ G_s &lt;/math&gt; repelled from the Fermi constant, which led them to the value &lt;math&gt; G_s = 6.94 \times 10^{31} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

In the paper &lt;ref&gt;Perng J. J. [https://springerlink3.metapress.com/content/1503066144130716/resource-secured/?target=fulltext.pdf&amp;sid=qwdy0a45odlh3a55v4k45u45&amp;sh=www.springerlink.com Strong gravitation and elementary particles.] Nuovo Cimento, Lettere, Serie 2, vol. 23, N. 15, pp. 552-554 (1978). &lt;/ref&gt; strong gravitational constant equal to &lt;math&gt;\Gamma =2.77 \times 10^{32} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.

[[User:Fedosin | Sergey Fedosin]] entered the strong gravitational constant in 1999 on the basis of equality between the Coulomb electric force and gravitational force in the hydrogen atom on the [[w:Bohr radius |Bohr radius]]. This leads to the following expression for the value of the strong gravitational constant: &lt;ref name=&quot;fed&quot;&gt; Fedosin S.G. [http://lccn.loc.gov/2009457349 Fizika i filosofiia podobiia: ot preonov do metagalaktik,] Perm, (1999-06-09) 544 pp. {{ISBN|5-8131-0012-1}}. &lt;/ref&gt;

: &lt;math&gt;\Gamma= \frac{e^2}{4 \pi \varepsilon_{0} m_p m_e }=1.514 \times 10^{29} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;,

where &lt;math&gt; ~ e &lt;/math&gt; – [[w:elementary charge |elementary charge]], &lt;math&gt; ~ \pi &lt;/math&gt; – [[w:pi (number) | pi]], &lt;math&gt; ~ \varepsilon_{0} &lt;/math&gt; – [[electric constant]], &lt;math&gt; ~ m_p &lt;/math&gt; – the mass of [[w:proton |proton]], &lt;math&gt; ~ m_e &lt;/math&gt; – the mass of [[w:electron| electron]]. 

It is assumed that strong gravitation, as a universal force, acts on the matter of nucleons, hadrons, electrons and elementary particles, regardless of the type of these particles. In contrast, the standard approach considers that strong interaction does not affect electrons and other leptons.

The small mass and large charge of matter do not allow the electron to be entirely in some small volume near the nucleus, and it gets disklike axisymmetric shape, which is limited by size of atom. In the hydrogen atom electrical forces between the nucleus and matter of the electron are attractive, but they are compensated by the repulsion of the intrinsic charge of the electron. There are the centripetal force of rotation of the electron around the nucleus, and the gravitational attraction between massive nucleus and matter of the electron. All these forces are equal in magnitude. From here follows that the action of strong gravitation between the masses of nucleus and electron on the one hand, and the electric force between charges of the nucleus and the electron, on the other hand, allows to estimate the value of &lt;math&gt; ~ \Gamma &lt;/math&gt;.
If &lt;math&gt;~ R_B = \frac {\hbar }{ m_e \alpha c } &lt;/math&gt; is the Bohr radius, then the equality of forces gives:

: &lt;math&gt; \frac {\Gamma m_p m_e }{R^2_B} = \frac{e^2}{4 \pi \varepsilon_{0} R^2_B } .&lt;/math&gt;

[[Physics/Essays/Fedosin/Fine structure constant | Fine structure constant]] is
:&lt;math&gt; \alpha = \frac{e^2}{4 \pi \varepsilon_0 \hbar c}, &lt;/math&gt;

So that
: &lt;math&gt; \Gamma= \frac{\alpha \hbar c }{m_p m_e }, \qquad \qquad \hbar = \frac{\Gamma m_p m_e }{ \alpha c }.&lt;/math&gt;

Bohr radius becomes equal
:&lt;math&gt;~ R_B = \frac{\Gamma m_p }{ \alpha^2 c^2 } = \frac{\Gamma m_p }{ V^2_B },&lt;/math&gt;

where &lt;math&gt;~ V_B = \alpha c &lt;/math&gt; is the orbital speed of the electron cloud at the first energy level.

Hence &lt;math&gt;~ V^2_B = \frac{\Gamma m_p }{ R_B }&lt;/math&gt;, and the kinetic energy of the electron, taking into account determination of strong gravitational constant, is equal to:
:&lt;math&gt;~ K = \frac{m_e V^2_B }{ 2 } = \frac{\Gamma m_p m_e }{ 2 R_B }=\frac { e^2}{8 \pi \varepsilon_0 R_B } = - \frac {W}{2} ,&lt;/math&gt;

where &lt;math&gt;~ W &lt;/math&gt;  is the potential energy of electron in the electric field of the nucleus of a hydrogen atom.

It turns out the virial theorem in the form &lt;math&gt;~ K = - \frac {W}{2} &lt;/math&gt;. The total electron energy is also found at the first energy level:

:&lt;math&gt;~ E = K+W = \frac {W}{2} = -K = -13.6 &lt;/math&gt; eV.

With the help of the constant &lt;math&gt; ~ \Gamma &lt;/math&gt; the [[w:Invariant mass#Rest energy | rest energy]] of proton in the form of a ball is equal to half of its potential energy of strong gravitational field in accordance with [[w:virial theorem |virial theorem]], &lt;ref&gt; [[User:Fedosin | Sergey Fedosin]], [https://www.lap-publishing.com/catalog/details//store/gb/book/978-3-659-57301-9/the-physical-theories-and-infinite-hierarchical-nesting-of-matter-volume-1 The physical theories and infinite hierarchical nesting of matter], Volume 1, LAP LAMBERT Academic Publishing, pages: 580, {{ISBN|978-3-659-57301-9}}. &lt;/ref&gt; if we assume that the binding energy &lt;math&gt; ~ E_b &lt;/math&gt; for the proton up to a sign is equal to the total energy of proton, and &lt;math&gt; ~ E_b &lt;/math&gt; becomes very close to relativistic energy in the form of rest energy:

: &lt;math&gt;~ m_p c^2 \approx E_b = -\frac {W_p}{2} = \frac{ k \Gamma m^2_p }{ 2R_p},&lt;/math&gt;

where &lt;math&gt; ~ R_p =8.73 \times 10^{-16} &lt;/math&gt; m  is the proton radius,
&lt;math&gt; ~ k=0.62 &lt;/math&gt; (in the hypothetical case of a uniform mass density of the proton there must be &lt;math&gt; ~ k = 0.6 &lt;/math&gt;). This implies that the mass of nucleons is determined by the energy of the strong gravitation according to the principle of [[w:mass–energy equivalence |mass–energy equivalence]].

If we assume that the magnetic moment of the proton is created by the maximum rotation of its positive charge distributed over the volume of the proton in the form of a ball, when the centripetal acceleration at the equator becomes equal to acceleration of strong gravitation, the formula for the magnetic moment is as follows:
: &lt;math&gt; ~ P_m = \delta e \sqrt {\Gamma m_p R_p}, &lt;/math&gt;

where &lt;math&gt; ~ P_m = 1.41 \times 10^{-26} &lt;/math&gt; J / T  is the magnetic moment of the proton,
&lt;math&gt; ~ \delta = 0.1875 &lt;/math&gt; (in the case of uniform density and charge should be &lt;math&gt; ~ \delta = 0.2 &lt;/math&gt;).

From the formulas for the energy and the magnetic moment the radius of the proton is determined in the self-consistent model. &lt;ref&gt; Fedosin S.G. [http://vixra.org/abs/1208.0006 The radius of the proton in the self-consistent model.] Hadronic Journal, Vol. 35, No. 4, pp. 349-363 (2012). &lt;/ref&gt;

The strong gravitational constant is also included in the formula describing the [[w:nuclear force |nuclear force]] through strong gravitation and [[Physics/Essays/Fedosin/Gravitational torsion field |gravitational torsion field]] of rotating particles. &lt;ref&gt;[http://sergf.ru/com.htm Comments to the book]: Fedosin S.G. Fizicheskie teorii i beskonechnaia vlozhennost’ materii. – Perm, 2009, 844 pages, Tabl. 21, Pic. 41, Ref. 289. {{ISBN|978-5-9901951-1-0}}. (in Russian). &lt;/ref&gt; A feature of the [[Physics/Essays/Fedosin/Gravitational induction |gravitational induction]] is that if two bodies rotate along one axis and come close by the force of gravitation, then these bodies will increase the angular velocity of its rotation. In this regard, it is assumed that the nucleons in atomic nuclei rotate at maximum speed. This may explain the equilibrium of the nucleons in atomic nuclei as a balance between the attractive force of strong gravitation and the strong force of the torsion field (of gravitomagnetic forces in [[Physics/Essays/Fedosin/Gravitoelectromagnetism|gravitoelectromagnetism]]). In particular, the [[Physics/Essays/Fedosin/Coupling constant |coupling constant]] is

:&lt;math&gt;\alpha_{pp}=  \frac{\beta \Gamma m^2_p }{\hbar c }=13.4 \beta &lt;/math&gt;,
where &lt;math&gt; ~ \beta &lt;/math&gt; is equal to 0.26 for the interaction of two nucleons, and tending to 1 for bodies with a lower mass density.

The constant &lt;math&gt;~\alpha_{pp}&lt;/math&gt; is close to [[w:coupling constant |coupling constant]] of [[Charges/Interactions/Strong|strong interaction]] of two nucleons in [[w:Standard Model |Standard Model]] 
:&lt;math&gt;\alpha_s=  \frac{ g^2_{N \pi}}{4\pi\hbar c } \approx 14.6&lt;/math&gt; , where &lt;math&gt;~g_{N \pi} &lt;/math&gt; is the constant of the pseudoscalar nucleon-pionic interaction.

[[Physics/Essays/Fedosin/Fine structure constant | Fine structure constant]] is coupling constant of electromagnetic interaction and may be written so:

:&lt;math &gt; ~ \alpha = \frac { \Gamma m_p m_e }{\hbar c }\approx \frac {1}{137.036}.&lt;/math&gt;

==Role of squared Avogadro number ==
Considering Avogadro number &lt;math&gt;N&lt;/math&gt; as a scaling factor, U. V. S. Seshavatharam and S. Lakshminarayana finally arrived at a value of &lt;ref&gt; U. V. S. Seshavatharam and S. Lakshminarayana. Unified model of universe and the atom. Book. {{ISBN|9783843393966}}, LAP LAMBERT Academic Publishing GmbH &amp; Co. KG, Germany, 2011 March 30.  &lt;/ref&gt; &lt;ref&gt; U. V. S. Seshavatharam and S. Lakshminarayana. Role of Avogadro number in grand unification. Hadronic Journal. Vol. 33, No 5, p. 513 (2010).  &lt;/ref&gt; &lt;ref&gt; U. V. S. Seshavatharam and S. Lakshminarayana. Atomic gravitational constant and the origin of elementary magnetic moments. To be published.  &lt;/ref&gt; 
&lt;math&gt; G_s = N^2G = 2.42 \times 10^{37} \ \mbox{m}^3 \ \mbox{kg}^{-1} \ \mbox{s}^{-2}&lt;/math&gt;.
It is noticed that in Hydrogen atom, ratio of total energy of electron and nuclear potential is equal to the electromagnetic and gravitational force ratio of electron where the operating gravitational constant is nothing but the atomic gravitational having a value N&lt;sup&gt;2&lt;/sup&gt;G. This is a direct confirmation of the existence of the atomic or nuclear gravitational constant in nuclear physics. Therefore, this subject can now be considered as part of the mainstream research in quantum gravity.

The central idea is: for mole number of particles, strength of gravity is &lt;math&gt;N.G&lt;/math&gt; and force required to bind &lt;math&gt;N&lt;/math&gt; particles is &lt;math&gt;\frac{c^4}{N.G}.&lt;/math&gt; Force required to bind one particle is &lt;math&gt;\frac{c^4}{N^2.G}.&lt;/math&gt; By considering this force magnitude as the characteristic weak force magnitude, it is observed that, &lt;math&gt;\ln \sqrt{\frac{e^2}{4 \pi \epsilon_0 G m_p^2}} \cong \sqrt{\frac{m_p}{m_e}-\ln\left(N^2\right)}&lt;/math&gt; where &lt;math&gt;m_p&lt;/math&gt; is the rest mass of proton and &lt;math&gt;m_e&lt;/math&gt; is the rest mass of electron. Obtained value of &lt;math&gt; G\cong \; 6.{\rm 6}66270{\rm 1}79\times {\rm 1}0^{-{\rm 1}1} {\rm \; m}^{{\rm 3}} {\rm Kg}^{{\rm -1}} {\rm sec}^{{\rm -2}.}&lt;/math&gt; Here the most important point to be emphasized is &lt;math&gt;\frac{c^4}{G}&lt;/math&gt; can be considered as the classical or upper limit of gravitational or electromagnetic force. It can be considered as the grand unified force. It is the origin of Planck scale and of the black hole astrophysics.

==Connection with usual gravitational constant==
With the help of [[Physics/Essays/Fedosin/Similarity of matter levels|similarity of matter levels]] and [[Physics/Essays/Fedosin/SPФ symmetry|SPФ symmetry]] in Theory of [[Physics/Essays/Fedosin/Infinite Hierarchical Nesting of Matter|Infinite Hierarchical Nesting of Matter]] the value of &lt;math&gt; ~ \Gamma &lt;/math&gt; can also be defined in terms of coefficients of similarity and the gravitational constant:

: &lt;math&gt;\Gamma = G \frac{ \Phi }{ P S^2},&lt;/math&gt;

where &lt;math&gt; ~ \Phi =1.62 \times 10^{57} &lt;/math&gt;, &lt;math&gt; ~ P= 1.4 \times 10^{19} &lt;/math&gt;, &lt;math&gt; ~ S= 0.23 &lt;/math&gt; are the coefficients of similarity in mass, size and speed, respectively, for the degenerate quantum objects at the atomic and stellar levels of matter.&lt;ref name=&quot;fed&quot;/&gt;  The powers of similarity coefficients in this equation correspond to dimension of gravitational constant according to [[w:Dimensional analysis |dimensional analysis]]. 

From the standpoint of Infinite Hierarchical Nesting of Matter and [[w:Le Sage's theory of gravitation |Le Sage's theory of gravitation]], the presence of two gravitational constants &lt;math&gt; ~ \Gamma &lt;/math&gt; and &lt;math&gt; ~ G &lt;/math&gt; reflects the difference between the properties of gravitons and properties of matter at different levels of matter. &lt;ref &gt; Fedosin S.G. [http://sergf.ru/mgen.htm  Model of Gravitational Interaction in the Concept of Gravitons]. Journal of Vectorial Relativity, Vol. 4, No. 1, pp. 1-24 (2009). &lt;/ref&gt; &lt;ref&gt; Fedosin S.G. [http://vixra.org/abs/1503.0127 The graviton field as the source of mass and gravitational force in the modernized Le Sage’s model.] Physical Science International Journal, ISSN: 2348-0130, Vol. 8, Issue 4, pp. 1-18 (2015). http://dx.doi.org/10.9734/PSIJ/2015/22197. &lt;/ref&gt;

In particular, for the strong gravitational constant and the ordinary gravitational constant it is possible to write similar relations, in which these constants are expressed in terms of the corresponding energy densities of gravitons’ fluxes in [[Physics/Essays/Fedosin/Electrogravitational vacuum |electrogravitational vacuum]] and the parameters of the densest object of the corresponding level of matter: &lt;ref name=&quot;cha&quot;&gt; Fedosin S.G. [https://www.ajol.info/index.php/jfas/article/view/168204  The charged component of the vacuum field as the source of electric force in the modernized Le Sage’s model.] Journal of Fundamental and Applied Sciences, Vol. 8, No. 3, pp. 971-1020 (2016). http://dx.doi.org/10.4314/jfas.v8i3.18, https://dx.doi.org/10.5281/zenodo.845357.&lt;/ref&gt;
:&lt;math&gt;~ \Gamma = \frac { \varepsilon_c \vartheta^2}{4 \pi M^2_n } , \qquad \qquad  G = \frac { \varepsilon_{cs} \vartheta^2_s}{4 \pi M^2_s } , &lt;/math&gt;

where &lt;math&gt;~ \varepsilon_c = 7.4 \cdot 10^{35}&lt;/math&gt; J/m³  is the energy density of the graviton fluxes for cubic distribution;
&lt;math&gt;~ \vartheta = 2.67 \cdot 10^{-30} &lt;/math&gt; m²  is the cross-section of interaction of the charged particles of the electrogravitational vacuum ([[Physics/Essays/Fedosin/Praon|praon]]s) with nucleons, which is very close in magnitude to the geometrical cross-section of the nucleon and is used to calculate the [[electric constant]]; &lt;math&gt;~ M_n &lt;/math&gt; is the mass of the nucleon;
&lt;math&gt; \varepsilon_{cs} = \varepsilon_c \frac {\Phi S^2}{ P^3} = 2.3 \cdot 10^{34}&lt;/math&gt; J/m³ is the energy density of the graviton fluxes at the stellar level for cubic distribution; &lt;math&gt;~ \vartheta_s = \vartheta P^2 = 5.2 \cdot 10^{8} &lt;/math&gt; m²  is the cross-section of interaction between the gravitons and a neutron star; &lt;math&gt;~ M_s = M_n \Phi = 2.7 \cdot 10^{30} &lt;/math&gt; kg is the mass of the neutron star.

At the matter level of [[Physics/Essays/Fedosin/Praon|praon]]s, its own strong gravitational constant &lt;math&gt;~G_{pr} &lt;/math&gt; must act. Considering that the coefficient of similarity in speed between the nucleon and praon levels of matter is &lt;math&gt;~S \approx 1 &lt;/math&gt;, we can write: 

: &lt;math&gt; G_{pr} = \Gamma \frac{ \Phi }{ P S^2} = \frac{ q^2_{pr}\beta}{4 \pi \varepsilon_{0} m^2_{pr} } =1.752 \cdot 10^{67}&lt;/math&gt; m&lt;sup&gt;3&lt;/sup&gt;•s&lt;sup&gt;–2&lt;/sup&gt;•kg&lt;sup&gt;–1&lt;/sup&gt;,

where &lt;math&gt;~ q_{pr} = 1.06 \cdot 10^{-57} &lt;/math&gt; C is the charge of the praon, &lt;math&gt;~ m_{pr} = 1 \cdot 10^{-84}&lt;/math&gt; kg is the mass of the praon, &lt;math&gt;~ \beta = \frac { m_p }{ m_e }= 1836.152&lt;/math&gt; is the proton to electron mass ratio.

==Connection with mass and unification of interaction==
The main object of unification is to understand the origin of elementary particles mass, (Dirac) magnetic moments and their forces. Right now and till today ‘string theory’ with 4 + 6 extra dimensions not in a position to explain the unification of gravitational and non-gravitational forces. More clearly speaking it is not in a position to bring down the planck scale to the nuclear size. Physicists say – if strength of strong interaction is unity, with reference to the strong interaction, strength of gravitation is 10&lt;sup&gt;−39&lt;/sup&gt;. The fundamental question to be answered is: is mass an inherent property of any elementary particle?

One can say: for any elementary particle mass is an induced property. This idea makes grand unification easy. Note that [[Theory of relativity/General relativity|general relativity]] does not throw any light on the ‘mass generation’ of charged particles. It only suggests that space-time is curved near the massive celestial objects. More over it couples the cosmic (dust) matter with geometry. But how matter is created? Why and how elementary particle possesses both charge and mass? Such types of questions are not discussed in the frame work of general relativity.

The first step in unification is to understand the origin of the [[w:Invariant mass |rest mass]] of a charged elementary particle. Second step is to understand the combined effects of its electromagnetic (or charged) and gravitational interactions. Third step is to understand its behavior with surroundings when it is created. Fourth step is to understand its behavior with cosmic space-time or other particles. Right from its birth to death, in all these steps the underlying fact is that whether it is a strongly interacting particle or weakly interacting particle, it is having some rest mass. To understand the first two steps somehow one can implement the gravitational constant in sub atomic physics.

To bring down the [[Physics/Essays/Fedosin/Planck mass|Planck mass]] scale to the observed elementary particles mass scale a large scale factor is required. Just like [[w:relative permeability |relative permeability]] and [[w:relative permittivity |relative permittivity]] by any suitable reason in atomic space if one is able to increase the value of classical gravitational constant, it helps in four ways. Observed elementary particles mass can be generated and grand unification can be achieved. Third important application is characteristic building block of the cosmological [[dark matter]] can be quantified in terms of fundamental physical constants. Fourth important application is – no [[w:extra dimensions |extra dimensions]] are required. Finally nuclear physics and quantum mechanics can be studied in the view of [[strong gravitation |strong nuclear gravity]] where nuclear charge and atomic gravitational constant play a crucial role in the nuclear space-time curvature, [[w:quantum chromodynamics |quantum chromodynamics]] and [[w:Color confinement |quark confinement]]. Not only that cosmology and particle physics can be studied in a unified way. In this connection it is suggested that square root of ratio of atomic gravitational constant and classical gravitational constant is equal to the Avogadro number. &lt;ref&gt; [http://www.journal-of-nuclear-physics.com/?p=316 AGNI – Avogadro's gravity for nuclear interactions.] Nuclear experiments blog: Journal of Nuclear Physics, Nov. 2010. &lt;/ref&gt; The Avogadro constant expresses the number of elementary entities per mole of substance and it has the value mol&lt;sup&gt;–1&lt;/sup&gt;. Avogadro's constant is a scaling factor between macroscopic and microscopic (atomic scale) observations of nature. It is an observed fact. The very unfortunate thing is that even though it is a large number it is neither implemented in cosmology nor implemented in grand unification. 

Modern physics is having hardly 100 years of ‘strong nuclear’ back ground. By Einstein’s time very little information was available on nuclear strong and weak forces. Avogadro hypothesis was proposed in 1811. Compared to modern nuclear physics,  Avogadro number is having 100 years of old history. Avogadro number may not be a fundamental physical constant but can be considered as a ‘scale factor’. But quantitatively it can be linked with the fundamental force ratios. Future thoughts and experiments may give some clue of it. Best present example is the ratio of planck mass and electron mass.  Considering this ratio automatically N&lt;sup&gt;2&lt;/sup&gt; comes into picture. It is noticed that in Hydrogen atom, ratio of total energy of electron and nuclear potential is equal to the electromagnetic and gravitational force ratio of electron where the operating gravitational constant is nothing but the atomic gravitational having a value N&lt;sup&gt;2&lt;/sup&gt;G. This is a direct confirmation of the existence of the atomoc or nuclear gravitational constant.  

Here the very important question to be answered is – which is more fundamental either &lt;math&gt;  G &lt;/math&gt; or &lt;math&gt; G_s &lt;/math&gt; ? It is proposed that both can be considered as the 'head' and 'tail' of matter coin. It can also be suggested that classical &lt;math&gt;  G &lt;/math&gt; is a consequence of the existence of atomic &lt;math&gt; G_s &lt;/math&gt;. It is known that there is a difference in between 'absolute findings' and 'absolute measurements'. Absolute findings can be understood where as 'absolute measurements' can not be made by nuclear experiments which are being conducted under the sky of universal gravity with unknown origin of elementary particles mass.

Till today there is no explanation for this fantastic and large difference between &lt;math&gt; G &lt;/math&gt; or &lt;math&gt; G_s &lt;/math&gt; or between gravitation and strong interaction, about 10&lt;sup&gt;−39&lt;/sup&gt;. It can be supposed that elementary particles construction is much more fundamental than the black hole's construction. If one wishes to unify electroweak, strong and gravitational interactions it is a must to implement the classical gravitational constant &lt;math&gt; G &lt;/math&gt; in the sub atomic physics. &lt;ref&gt; Seshavatharam, U. V. S.; Lakshminarayana, S. [http://adsabs.harvard.edu/abs/2010IJMPE..19..263S  Super symmetry in strong and weak interactions.] International journal of modern physics E, Issue 02, pp. 263-280, Feb.2010. &lt;/ref&gt; By any reason if one implements the Planck scale in elementary particle physics and nuclear physics automatically &lt;math&gt; G &lt;/math&gt; comes into subatomic physics. Then a large arbitrary number has to be considered as a proportionality constant. After that its physical significance has to be analyzed. Alternatively its equivalent 'strong atomic gravitational constant' can also be assumed. Some attempts have been done in physics history.

Whether it may be real or an equivalent if it is existing as a 'single constant' its physical significance can be understood. Nuclear size can be fitted with 'nuclear Schwarzschild radius'. Nucleus can be considered as 'strong nuclear black hole'. This idea requires a basic nuclear fermion! Nuclear binding energy constants can be generated directly. Proton-neutron stability can be studied. Origin of strong [[w:coupling constant |coupling constant]] and [[w:Fermi's interaction |Fermi's weak coupling constant]] can be understood. Charged lepton masses can be fitted. Such applications can be considered favorable for the proposed assumptions and further analysis can be carried out positively for understanding and developing this proposed 'Avogadro's strong nuclear gravity'.

Unification means: finding the similarities, finding the limiting physical constants, finding the key numbers, coupling the key physical constants, coupling the key physical concepts, coupling the key physical properties, minimizing the number of dimensions, minimizing the number of inputs and implementing the key physical constant or key number in different branches of physics. This is a very lengthy process. In all these cases observations, interpretations, experiments and imagination play a key role. The main difficulty is with interpretations and observations. As the interpretation changes physical concept changes, physical equation changes and finally the destiny changes. 

Note that human beings are part of this universal gravity. There are some natural restrictions to experiments. Seeing a black hole is highly speculative. But indirectly its significances can be well understood. In the similar way in nuclear and particle physics: any experimental setup which is being run under the influence of the proposed strong nuclear gravity, without knowing the probing particle’s massive origin, without knowing the massive origin of the nucleus: based on ‘grand unified scheme’ one may not be able to unearth the absolute findings. Note that observer, experimental setup and the probing particle all are under the same influence of universal gravity. When searching for an experimental proof in grand/final unification scheme or dark matter projects this fact may be considered positively for further analysis. 

To conclude it can be suggested that – existence of strong gravitational constant as Atomic gravitational constant is true and its consequences can be understood easily and can be implemented easily in grand unification program and dark matter projects.

== Notes ==
&lt;references/&gt;

== See also ==
* [[Physics/Essays/Fedosin/Strong gravitation |Strong gravitation]]
* [[Physics/Essays/Fedosin/Coupling constant |Coupling constant]]
* [[w:Gravitational coupling constant |Gravitational coupling constant]]
* [[Physics/Essays/Fedosin/Fine structure constant|Fine structure constant]]
* [[w:Dimensionless physical constant |Dimensionless physical constant]]
* [[Physics/Essays/Fedosin/Infinite Hierarchical Nesting of Matter|Infinite Hierarchical Nesting of Matter]]
* [[Physics/Essays/Fedosin/Similarity of matter levels|Similarity of matter levels]]
* [[Physics/Essays/Fedosin/SPФ symmetry|SPФ symmetry]]
* [[Physics/Essays/Fedosin/Quantization of parameters of cosmic systems |Quantization of parameters of cosmic systems]]
* [[Physics/Essays/Fedosin/Hydrogen system |Hydrogen system]]
* [[Physics/Essays/Fedosin/Stellar constants |Stellar constants]]
* [[Physics/Essays/Fedosin/Gravitational model of strong interaction |Gravitational model of strong interaction]]
* [[Physics/Essays/Fedosin/Model of quark quasiparticles |Model of quark quasiparticles]]
* [[Physics/Essays/Fedosin/Substantial electron model|Substantial electron model]]
* [[Physics/Essays/Fedosin/Substantial neutron model|Substantial neutron model]]
* [[Physics/Essays/Fedosin/Substantial proton model|Substantial proton model]]
* [[Physics/Essays/Fedosin/Substantial photon model|Substantial photon model]]
* [[Physics/Essays/Fedosin/Electrogravitational vacuum |Electrogravitational vacuum]]

==External links==
* [http://www.wikiznanie.ru/ru-wz/index.php/%D0%9F%D0%BE%D1%81%D1%82%D0%BE%D1%8F%D0%BD%D0%BD%D0%B0%D1%8F_%D1%81%D0%B8%D0%BB%D1%8C%D0%BD%D0%BE%D0%B9_%D0%B3%D1%80%D0%B0%D0%B2%D0%B8%D1%82%D0%B0%D1%86%D0%B8%D0%B8 Strong gravitational constant (in Russian)]

[[Category:Gravitation]]
[[Category:Fundamental constants]]</text>
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  <page>
    <title>Complex Analysis</title>
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      <text bytes="12093" sha1="gly08dcq2712g7alxcjp12pcwopny79" xml:space="preserve">[[File:Wiki2Reveal Logo.png|146px|thumb|Course contains [[v:en:Wiki2Reveal|Wiki2Reveal]] Slides]]
[[File:Mapping f z equal 1 over z.gif|thumb|Moving the argument of function &lt;math&gt;f&lt;/math&gt; in the complex number plane. The point &lt;math&gt;z&lt;/math&gt; has a blue color and &lt;math&gt;f(z)= \frac{1}{z}&lt;/math&gt;  is marked in red color. &lt;math&gt;z&lt;/math&gt; is moved on a curve with  &lt;math&gt;\gamma(t)=t\cdot e^{it}&lt;/math&gt;.]]
[[File:Image of path 1 over z.webm|thumb|Image of path in the complex numbers for the function &lt;math&gt;f(z)=\frac{1}{z}&lt;/math&gt;]]
'''Complex analysis''' is a study of functions of a complex variable. This is a one quarter course in complex analysis at the undergraduate level.

==Articles==

* [[Algebra II]]
* [[Dummy variable]]
* [[Materials Science and Engineering/Equations/Quantum Mechanics]]

== Slides for Lectures ==

=== Chapter 1 - Intoduction ===
* '''[[Complex Numbers/From real to complex numbers|Complex Numbers]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Numbers/From%20real%20to%20complex%20numbers&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Complex%20Numbers&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
** [[w:en:Heine–Borel_theorem|Heine-Borel Theorem]]
* '''[[Riemann sphere|Riemann sphere]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Riemann%20sphere&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Riemann%20sphere&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex_Analysis/Exponentiation_and_square_root|Exponentiation and roots]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex_Analysis/Exponentiation_and_square_root&amp;author=Complex_Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Exponentiation_and_square_root&amp;coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]

=== Chapter 2 - Topological Foundations ===
* '''[[Complex Analysis/Sequences and series|Sequences and series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Sequences%20and%20series&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Sequences%20and%20series&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* [[/Power series/]]
* '''[[Inverse-producing extensions of Topological Algebras/topological algebra|Topological algebra]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Inverse-producing%20extensions%20of%20Topological%20Algebras/topological%20algebra&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=topological%20algebra&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* [[w:en:Topological space|Topological space]] - Definition: [[Norms, metrics, topology#Definition:_topology|Topology]] 
* '''[[Norms, metrics, topology|Norms, metrics, topology]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Norms,%20metrics,%20topology&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Norms,%20metrics,%20topology&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]

=== Chapter 3 - Complex Derivative ===
* '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Holomorphic%20function&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Holomorphic%20function&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Partial derivative|Partial Derivative]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Partial%20derivative&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Partial%20Derivative&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Cauchy-Riemann Equations|Cauchy-Riemann Equations (CRE)]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Cauchy-Riemann%20Equations&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Cauchy-Riemann%20Equations&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Application of Cauchy-Riemann Equations|Application of Cauchy-Riemann Equations]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Application%20of%20Cauchy-Riemann%20Equations&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Application%20of%20Cauchy-Riemann%20Equations&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]

=== Chapter 4 - Curves and Line Integrals ===
* '''[[Line integral|Line integral in &lt;math&gt;\mathbb{R}^n&lt;/math&gt;]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Line%20integral&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Line%20integral&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[/Curves/|Curves]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Curves&amp;author=Complex_Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Curves&amp;coursetitle=Complex%20Analysis  Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
** [[w:en:Holomorphic function|Wikipedia: holomorphic function]] 
** [[w:en:Integral|Wikipedia:Integral ]]
* '''[[Complex_Analysis/Paths|Paths]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Paths&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Paths&amp;coursetitle=Complex%20Analysis  Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Path_Integral|Path integral]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Path%20Integral&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Path_Integral&amp;coursetitle=Complex%20Analysis  Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
** [[w:en:Curve integral |Wikipedia: Curve integral]]
** [[w:en:Continuity|Continuity]] and [[w:en:Limit of a sequence|Limit of a sequence]]
** [[/Inequalities/]]
** [[Complex Analysis/rectifiable curve|rectifiable curve]] - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/rectifiable%20curve&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=rectifiable%20curve&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]

* '''[[Complex Analysis/Trace|Trace of Curve]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Trace&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Trace&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]

* '''[[Complex Analysis/Curve Integral|Curve Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Curve%20Integral&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Curve%20Integral&amp;coursetitle=Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]]

=== Chapter 5 - Holomorphic Functions ===
* '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Holomorphic%20function&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Holomorphic%20function&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
** [[Holomorphism/Criteria|Criteria]] - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Holomorphism/Criteria&amp;author=Course:Complex_Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Criteria&amp;coursetitle=Complex_Analysis slideset]) [[File:Wiki2Reveal Logo.png|35px]]
** [[w:en:Holomorphic_function#.C3.84quivalent_properties_of_holomorphic_functions_of_one_variable|Wikipedia: Holomorphic function criteria]]
** [[/Differences from real differentiability/]]
** [[w:Conformal_mapping|conformal mappings]]&lt;math&gt;(\ast)&lt;/math&gt;, 

* '''[[Complex Analysis/Path of Integration|Path of Integration]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Path%20of%20Integration&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Path%20of%20Integration&amp;coursetitle=Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]]

* '''[[Complex Analysis/Goursat's Lemma (Details)|Goursat's Lemma (Details)]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Goursat%27s%20Lemma%20(Details)&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Goursat%27s%20Lemma%20(Details)&amp;coursetitle=Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]]

* '''[[/Cauchy's Integral Theorem for Disks/]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex+Analysis/Cauchy%27s+Integral+Theorem_for_Disks&amp;author=Complex+Analysis&amp;audioslide=yes&amp;language=en Slide Set]) [[File:Wiki2Reveal Logo.png|35px]],
* '''[[Complex Analysis/Identity Theorem|Identity Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Identity%20Theorem&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes Slide Set]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Liouville's Theorem|Liouville's Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Liouville%27s%20Theorem&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Liouville%27s%20Theorem&amp;coursetitle=Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]]
=== Complex Analysis Part 2 ===

*'''[[/Chain/|Chain of Paths]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=ComplexAnalysis/Chain&amp;author=ComplexAnalysis&amp;language=de&amp;audioslide=yes&amp;shorttitle=Chain&amp;coursetitle=ComplexAnalysis Slides]) [[File:Wiki2Reveal Logo.png|35px]]

*'''[[/cycle/]]''' ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=cycle_and_Chain&amp;author=ComplexAnalysis&amp;audioslide=yes&amp;language=de Slides]) [[File:Wiki2Reveal Logo.png|35px]]

*[[Laurent Series|From Taylor Series to Laurent Series]], ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Laurent-Series&amp;author=ComplexAnalysis&amp;audioslide=yes&amp;language=de Slides]) [[File:Wiki2Reveal Logo.png|35px]]

*[[Cauchy Integral Theorem|Cauchy's Integral Theorem]] for cycles ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Integral_of_Cauchy&amp;author=ComplexAnalysis&amp;audioslide=yes&amp;language=de Slides]) [[File:Wiki2Reveal Logo.png|35px]]

*[[Cauchy Integral Formula|Cauchy Integral Formula]] for cycles ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Integralformula_of_Cauchy&amp;author=ComplexAnalysis&amp;audioslide=yes&amp;language=de Slides]) [[File:Wiki2Reveal Logo.png|35px]]


==Lectures==
* [[/Cauchy-Riemann equations/]]
* [[Cauchy Theorem for a triangle]]
* [[Complex analytic function]]
* [[Complex Numbers]]
* [[Divergent series]]
* [[Estimation lemma]]
* [[Fourier series]]
* [[Fourier transform]]
* [[Fourier transforms]]
* [[Laplace transform]]
* [[Riemann hypothesis]]
* [[The Real and Complex Number System]]
* [[Warping functions]]

==Sample exams==

[[/Sample Midterm Exam 1/]]
[[/Sample Midterm Exam 2/]]

==See also==
* [[Boundary Value Problems]]
* [[Introduction to Elasticity]]
* [[The Prime Sequence Problem]]
* [[Wikipedia: Complex analysis]]
*[[Complex number]]
[[Category:Complex analysis| ]]
[[Category:Mathematics courses]]
[[Category:Mathematics]]

&lt;noinclude&gt;
[[de:Kurs:Funktionentheorie]]
&lt;/noinclude&gt;</text>
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[[File:Mapping f z equal 1 over z.gif|thumb|Moving the argument of function &lt;math&gt;f&lt;/math&gt; in the complex number plane. The point &lt;math&gt;z&lt;/math&gt; has a blue color and &lt;math&gt;f(z)= \frac{1}{z}&lt;/math&gt;  is marked in red color. &lt;math&gt;z&lt;/math&gt; is moved on a curve with  &lt;math&gt;\gamma(t)=t\cdot e^{it}&lt;/math&gt;.]]
[[File:Image of path 1 over z.webm|thumb|Image of path in the complex numbers for the function &lt;math&gt;f(z)=\frac{1}{z}&lt;/math&gt;]]
'''Complex analysis''' is a study of functions of a complex variable. This is a one quarter course in complex analysis at the undergraduate level.

==Articles==

* [[Algebra II]]
* [[Dummy variable]]
* [[Materials Science and Engineering/Equations/Quantum Mechanics]]

== Slides for Lectures ==

=== Chapter 1 - Intoduction ===
* '''[[Complex Numbers/From real to complex numbers|Complex Numbers]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Numbers/From%20real%20to%20complex%20numbers&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Complex%20Numbers&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
** [[w:en:Heine–Borel_theorem|Heine-Borel Theorem]]
* '''[[Riemann sphere|Riemann sphere]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Riemann%20sphere&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Riemann%20sphere&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex_Analysis/Exponentiation_and_square_root|Exponentiation and roots]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex_Analysis/Exponentiation_and_square_root&amp;author=Complex_Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Exponentiation_and_square_root&amp;coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]

=== Chapter 2 - Topological Foundations ===
* '''[[Complex Analysis/Sequences and series|Sequences and series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Sequences%20and%20series&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Sequences%20and%20series&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* [[/Power series/]]
* '''[[Inverse-producing extensions of Topological Algebras/topological algebra|Topological algebra]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Inverse-producing%20extensions%20of%20Topological%20Algebras/topological%20algebra&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=topological%20algebra&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* [[w:en:Topological space|Topological space]] - Definition: [[Norms, metrics, topology#Definition:_topology|Topology]] 
* '''[[Norms, metrics, topology|Norms, metrics, topology]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Norms,%20metrics,%20topology&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Norms,%20metrics,%20topology&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]

=== Chapter 3 - Complex Derivative ===
* '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Holomorphic%20function&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Holomorphic%20function&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Partial derivative|Partial Derivative]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Partial%20derivative&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Partial%20Derivative&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Cauchy-Riemann Equations|Cauchy-Riemann Equations (CRE)]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Cauchy-Riemann%20Equations&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Cauchy-Riemann%20Equations&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Application of Cauchy-Riemann Equations|Application of Cauchy-Riemann Equations]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Application%20of%20Cauchy-Riemann%20Equations&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Application%20of%20Cauchy-Riemann%20Equations&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]

=== Chapter 4 - Curves and Line Integrals ===
* '''[[Line integral|Line integral in &lt;math&gt;\mathbb{R}^n&lt;/math&gt;]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Line%20integral&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Line%20integral&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[/Curves/|Curves]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Curves&amp;author=Complex_Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Curves&amp;coursetitle=Complex%20Analysis  Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
** [[w:en:Holomorphic function|Wikipedia: holomorphic function]] 
** [[w:en:Integral|Wikipedia:Integral ]]
* '''[[Complex_Analysis/Paths|Paths]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Paths&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Paths&amp;coursetitle=Complex%20Analysis  Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Path_Integral|Path integral]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Path%20Integral&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Path_Integral&amp;coursetitle=Complex%20Analysis  Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
** [[w:en:Curve integral |Wikipedia: Curve integral]]
** [[w:en:Continuity|Continuity]] and [[w:en:Limit of a sequence|Limit of a sequence]]
** [[/Inequalities/]]
** [[Complex Analysis/rectifiable curve|rectifiable curve]] - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/rectifiable%20curve&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=rectifiable%20curve&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]

* '''[[Complex Analysis/Trace|Trace of Curve]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Trace&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Trace&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]

* '''[[Complex Analysis/Curve Integral|Curve Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Curve%20Integral&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Curve%20Integral&amp;coursetitle=Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]]

=== Chapter 5 - Holomorphic Functions ===
* '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Holomorphic%20function&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Holomorphic%20function&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
** [[Holomorphism/Criteria|Criteria]] - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Holomorphism/Criteria&amp;author=Course:Complex_Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Criteria&amp;coursetitle=Complex_Analysis slideset]) [[File:Wiki2Reveal Logo.png|35px]]
** [[w:en:Holomorphic_function#.C3.84quivalent_properties_of_holomorphic_functions_of_one_variable|Wikipedia: Holomorphic function criteria]]
** [[/Differences from real differentiability/]]
** [[w:Conformal_mapping|conformal mappings]]&lt;math&gt;(\ast)&lt;/math&gt;, 

* '''[[Complex Analysis/Path of Integration|Path of Integration]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Path%20of%20Integration&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Path%20of%20Integration&amp;coursetitle=Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]]

* '''[[Complex Analysis/Goursat's Lemma (Details)|Goursat's Lemma (Details)]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Goursat%27s%20Lemma%20(Details)&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Goursat%27s%20Lemma%20(Details)&amp;coursetitle=Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]]

* '''[[/Cauchy's Integral Theorem for Disks/]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex+Analysis/Cauchy%27s+Integral+Theorem_for_Disks&amp;author=Complex+Analysis&amp;audioslide=yes&amp;language=en Slide Set]) [[File:Wiki2Reveal Logo.png|35px]],
* '''[[Complex Analysis/Identity Theorem|Identity Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Identity%20Theorem&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes Slide Set]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Liouville's Theorem|Liouville's Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Liouville%27s%20Theorem&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Liouville%27s%20Theorem&amp;coursetitle=Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]]
=== Complex Analysis Part 2 ===

*'''[[/Chain/|Chain of Paths]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=ComplexAnalysis/Chain&amp;author=ComplexAnalysis&amp;language=de&amp;audioslide=yes&amp;shorttitle=Chain&amp;coursetitle=ComplexAnalysis Slides]) [[File:Wiki2Reveal Logo.png|35px]]

*'''[[/cycle/]]''' ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=cycle_and_Chain&amp;author=ComplexAnalysis&amp;audioslide=yes&amp;language=de Slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Laurent Series|Laurent Series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Laurent%20Series&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Laurent%20Series&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
  

*[[Cauchy Integral Theorem|Cauchy's Integral Theorem]] for cycles ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Integral_of_Cauchy&amp;author=ComplexAnalysis&amp;audioslide=yes&amp;language=de Slides]) [[File:Wiki2Reveal Logo.png|35px]]

*[[Cauchy Integral Formula|Cauchy Integral Formula]] for cycles ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Integralformula_of_Cauchy&amp;author=ComplexAnalysis&amp;audioslide=yes&amp;language=de Slides]) [[File:Wiki2Reveal Logo.png|35px]]

==Lectures==
* [[/Cauchy-Riemann equations/]]
* [[Cauchy Theorem for a triangle]]
* [[Complex analytic function]]
* [[Complex Numbers]]
* [[Divergent series]]
* [[Estimation lemma]]
* [[Fourier series]]
* [[Fourier transform]]
* [[Fourier transforms]]
* [[Laplace transform]]
* [[Riemann hypothesis]]
* [[The Real and Complex Number System]]
* [[Warping functions]]

==Sample exams==

[[/Sample Midterm Exam 1/]]
[[/Sample Midterm Exam 2/]]

==See also==
* [[Boundary Value Problems]]
* [[Introduction to Elasticity]]
* [[The Prime Sequence Problem]]
* [[Wikipedia: Complex analysis]]
*[[Complex number]]
[[Category:Complex analysis| ]]
[[Category:Mathematics courses]]
[[Category:Mathematics]]

&lt;noinclude&gt;
[[de:Kurs:Funktionentheorie]]
&lt;/noinclude&gt;</text>
      <sha1>otctaxbo17p1a8v0no7wv4x84o0k6a2</sha1>
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    <revision>
      <id>2692257</id>
      <parentid>2692249</parentid>
      <timestamp>2024-12-17T08:55:08Z</timestamp>
      <contributor>
        <username>Bert Niehaus</username>
        <id>2387134</id>
      </contributor>
      <comment>/* Chapter 4 - Curves and Line Integrals */</comment>
      <origin>2692257</origin>
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      <text bytes="12149" sha1="meerp02zfct11x4325o1tbzd7ewcvk5" xml:space="preserve">[[File:Wiki2Reveal Logo.png|146px|thumb|Course contains [[v:en:Wiki2Reveal|Wiki2Reveal]] Slides]]
[[File:Mapping f z equal 1 over z.gif|thumb|Moving the argument of function &lt;math&gt;f&lt;/math&gt; in the complex number plane. The point &lt;math&gt;z&lt;/math&gt; has a blue color and &lt;math&gt;f(z)= \frac{1}{z}&lt;/math&gt;  is marked in red color. &lt;math&gt;z&lt;/math&gt; is moved on a curve with  &lt;math&gt;\gamma(t)=t\cdot e^{it}&lt;/math&gt;.]]
[[File:Image of path 1 over z.webm|thumb|Image of path in the complex numbers for the function &lt;math&gt;f(z)=\frac{1}{z}&lt;/math&gt;]]
'''Complex analysis''' is a study of functions of a complex variable. This is a one quarter course in complex analysis at the undergraduate level.

==Articles==

* [[Algebra II]]
* [[Dummy variable]]
* [[Materials Science and Engineering/Equations/Quantum Mechanics]]

== Slides for Lectures ==

=== Chapter 1 - Intoduction ===
* '''[[Complex Numbers/From real to complex numbers|Complex Numbers]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Numbers/From%20real%20to%20complex%20numbers&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Complex%20Numbers&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
** [[w:en:Heine–Borel_theorem|Heine-Borel Theorem]]
* '''[[Riemann sphere|Riemann sphere]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Riemann%20sphere&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Riemann%20sphere&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex_Analysis/Exponentiation_and_square_root|Exponentiation and roots]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex_Analysis/Exponentiation_and_square_root&amp;author=Complex_Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Exponentiation_and_square_root&amp;coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]

=== Chapter 2 - Topological Foundations ===
* '''[[Complex Analysis/Sequences and series|Sequences and series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Sequences%20and%20series&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Sequences%20and%20series&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* [[/Power series/]]
* '''[[Inverse-producing extensions of Topological Algebras/topological algebra|Topological algebra]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Inverse-producing%20extensions%20of%20Topological%20Algebras/topological%20algebra&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=topological%20algebra&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* [[w:en:Topological space|Topological space]] - Definition: [[Norms, metrics, topology#Definition:_topology|Topology]] 
* '''[[Norms, metrics, topology|Norms, metrics, topology]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Norms,%20metrics,%20topology&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Norms,%20metrics,%20topology&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]

=== Chapter 3 - Complex Derivative ===
* '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Holomorphic%20function&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Holomorphic%20function&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Partial derivative|Partial Derivative]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Partial%20derivative&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Partial%20Derivative&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Cauchy-Riemann Equations|Cauchy-Riemann Equations (CRE)]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Cauchy-Riemann%20Equations&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Cauchy-Riemann%20Equations&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Application of Cauchy-Riemann Equations|Application of Cauchy-Riemann Equations]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Application%20of%20Cauchy-Riemann%20Equations&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Application%20of%20Cauchy-Riemann%20Equations&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]

=== Chapter 4 - Curves and Line Integrals ===
* '''[[Line integral|Line integral in &lt;math&gt;\mathbb{R}^n&lt;/math&gt;]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Line%20integral&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Line%20integral&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[/Curves/|Curves]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Curves&amp;author=Complex_Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Curves&amp;coursetitle=Complex%20Analysis  Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
** [[w:en:Holomorphic function|Wikipedia: holomorphic function]] 
** [[w:en:Integral|Wikipedia:Integral ]]
* '''[[Complex_Analysis/Paths|Paths]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Paths&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Paths&amp;coursetitle=Complex%20Analysis  Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]

* '''[[Path Integral|Path Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Path%20Integral&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Path%20Integral&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
  
** [[w:en:Curve integral |Wikipedia: Curve integral]]
** [[w:en:Continuity|Continuity]] and [[w:en:Limit of a sequence|Limit of a sequence]]
** [[/Inequalities/]]
** [[Complex Analysis/rectifiable curve|rectifiable curve]] - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/rectifiable%20curve&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=rectifiable%20curve&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]

* '''[[Complex Analysis/Trace|Trace of Curve]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Trace&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Trace&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]

* '''[[Complex Analysis/Curve Integral|Curve Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Curve%20Integral&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Curve%20Integral&amp;coursetitle=Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]]

=== Chapter 5 - Holomorphic Functions ===
* '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Holomorphic%20function&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Holomorphic%20function&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
** [[Holomorphism/Criteria|Criteria]] - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Holomorphism/Criteria&amp;author=Course:Complex_Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Criteria&amp;coursetitle=Complex_Analysis slideset]) [[File:Wiki2Reveal Logo.png|35px]]
** [[w:en:Holomorphic_function#.C3.84quivalent_properties_of_holomorphic_functions_of_one_variable|Wikipedia: Holomorphic function criteria]]
** [[/Differences from real differentiability/]]
** [[w:Conformal_mapping|conformal mappings]]&lt;math&gt;(\ast)&lt;/math&gt;, 

* '''[[Complex Analysis/Path of Integration|Path of Integration]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Path%20of%20Integration&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Path%20of%20Integration&amp;coursetitle=Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]]

* '''[[Complex Analysis/Goursat's Lemma (Details)|Goursat's Lemma (Details)]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Goursat%27s%20Lemma%20(Details)&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Goursat%27s%20Lemma%20(Details)&amp;coursetitle=Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]]

* '''[[/Cauchy's Integral Theorem for Disks/]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex+Analysis/Cauchy%27s+Integral+Theorem_for_Disks&amp;author=Complex+Analysis&amp;audioslide=yes&amp;language=en Slide Set]) [[File:Wiki2Reveal Logo.png|35px]],
* '''[[Complex Analysis/Identity Theorem|Identity Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Identity%20Theorem&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes Slide Set]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Liouville's Theorem|Liouville's Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Liouville%27s%20Theorem&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Liouville%27s%20Theorem&amp;coursetitle=Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]]
=== Complex Analysis Part 2 ===

*'''[[/Chain/|Chain of Paths]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=ComplexAnalysis/Chain&amp;author=ComplexAnalysis&amp;language=de&amp;audioslide=yes&amp;shorttitle=Chain&amp;coursetitle=ComplexAnalysis Slides]) [[File:Wiki2Reveal Logo.png|35px]]

*'''[[/cycle/]]''' ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=cycle_and_Chain&amp;author=ComplexAnalysis&amp;audioslide=yes&amp;language=de Slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Laurent Series|Laurent Series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Laurent%20Series&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Laurent%20Series&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
  

*[[Cauchy Integral Theorem|Cauchy's Integral Theorem]] for cycles ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Integral_of_Cauchy&amp;author=ComplexAnalysis&amp;audioslide=yes&amp;language=de Slides]) [[File:Wiki2Reveal Logo.png|35px]]

*[[Cauchy Integral Formula|Cauchy Integral Formula]] for cycles ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Integralformula_of_Cauchy&amp;author=ComplexAnalysis&amp;audioslide=yes&amp;language=de Slides]) [[File:Wiki2Reveal Logo.png|35px]]

==Lectures==
* [[/Cauchy-Riemann equations/]]
* [[Cauchy Theorem for a triangle]]
* [[Complex analytic function]]
* [[Complex Numbers]]
* [[Divergent series]]
* [[Estimation lemma]]
* [[Fourier series]]
* [[Fourier transform]]
* [[Fourier transforms]]
* [[Laplace transform]]
* [[Riemann hypothesis]]
* [[The Real and Complex Number System]]
* [[Warping functions]]

==Sample exams==

[[/Sample Midterm Exam 1/]]
[[/Sample Midterm Exam 2/]]

==See also==
* [[Boundary Value Problems]]
* [[Introduction to Elasticity]]
* [[The Prime Sequence Problem]]
* [[Wikipedia: Complex analysis]]
*[[Complex number]]
[[Category:Complex analysis| ]]
[[Category:Mathematics courses]]
[[Category:Mathematics]]

&lt;noinclude&gt;
[[de:Kurs:Funktionentheorie]]
&lt;/noinclude&gt;</text>
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    <revision>
      <id>2692260</id>
      <parentid>2692257</parentid>
      <timestamp>2024-12-17T09:17:27Z</timestamp>
      <contributor>
        <username>Eshaa2024</username>
        <id>2993595</id>
      </contributor>
      <comment>/* Chapter 5 - Holomorphic Functions */</comment>
      <origin>2692260</origin>
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[[File:Mapping f z equal 1 over z.gif|thumb|Moving the argument of function &lt;math&gt;f&lt;/math&gt; in the complex number plane. The point &lt;math&gt;z&lt;/math&gt; has a blue color and &lt;math&gt;f(z)= \frac{1}{z}&lt;/math&gt;  is marked in red color. &lt;math&gt;z&lt;/math&gt; is moved on a curve with  &lt;math&gt;\gamma(t)=t\cdot e^{it}&lt;/math&gt;.]]
[[File:Image of path 1 over z.webm|thumb|Image of path in the complex numbers for the function &lt;math&gt;f(z)=\frac{1}{z}&lt;/math&gt;]]
'''Complex analysis''' is a study of functions of a complex variable. This is a one quarter course in complex analysis at the undergraduate level.

==Articles==

* [[Algebra II]]
* [[Dummy variable]]
* [[Materials Science and Engineering/Equations/Quantum Mechanics]]

== Slides for Lectures ==

=== Chapter 1 - Intoduction ===
* '''[[Complex Numbers/From real to complex numbers|Complex Numbers]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Numbers/From%20real%20to%20complex%20numbers&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Complex%20Numbers&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
** [[w:en:Heine–Borel_theorem|Heine-Borel Theorem]]
* '''[[Riemann sphere|Riemann sphere]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Riemann%20sphere&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Riemann%20sphere&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex_Analysis/Exponentiation_and_square_root|Exponentiation and roots]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex_Analysis/Exponentiation_and_square_root&amp;author=Complex_Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Exponentiation_and_square_root&amp;coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]

=== Chapter 2 - Topological Foundations ===
* '''[[Complex Analysis/Sequences and series|Sequences and series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Sequences%20and%20series&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Sequences%20and%20series&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* [[/Power series/]]
* '''[[Inverse-producing extensions of Topological Algebras/topological algebra|Topological algebra]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Inverse-producing%20extensions%20of%20Topological%20Algebras/topological%20algebra&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=topological%20algebra&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* [[w:en:Topological space|Topological space]] - Definition: [[Norms, metrics, topology#Definition:_topology|Topology]] 
* '''[[Norms, metrics, topology|Norms, metrics, topology]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Norms,%20metrics,%20topology&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Norms,%20metrics,%20topology&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]

=== Chapter 3 - Complex Derivative ===
* '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Holomorphic%20function&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Holomorphic%20function&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Partial derivative|Partial Derivative]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Partial%20derivative&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Partial%20Derivative&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Cauchy-Riemann Equations|Cauchy-Riemann Equations (CRE)]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Cauchy-Riemann%20Equations&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Cauchy-Riemann%20Equations&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Application of Cauchy-Riemann Equations|Application of Cauchy-Riemann Equations]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Application%20of%20Cauchy-Riemann%20Equations&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Application%20of%20Cauchy-Riemann%20Equations&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]

=== Chapter 4 - Curves and Line Integrals ===
* '''[[Line integral|Line integral in &lt;math&gt;\mathbb{R}^n&lt;/math&gt;]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Line%20integral&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Line%20integral&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[/Curves/|Curves]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Curves&amp;author=Complex_Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Curves&amp;coursetitle=Complex%20Analysis  Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
** [[w:en:Holomorphic function|Wikipedia: holomorphic function]] 
** [[w:en:Integral|Wikipedia:Integral ]]
* '''[[Complex_Analysis/Paths|Paths]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Paths&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Paths&amp;coursetitle=Complex%20Analysis  Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]

* '''[[Path Integral|Path Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Path%20Integral&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Path%20Integral&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
  
** [[w:en:Curve integral |Wikipedia: Curve integral]]
** [[w:en:Continuity|Continuity]] and [[w:en:Limit of a sequence|Limit of a sequence]]
** [[/Inequalities/]]
** [[Complex Analysis/rectifiable curve|rectifiable curve]] - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/rectifiable%20curve&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=rectifiable%20curve&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]

* '''[[Complex Analysis/Trace|Trace of Curve]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Trace&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Trace&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]

* '''[[Complex Analysis/Curve Integral|Curve Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Curve%20Integral&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Curve%20Integral&amp;coursetitle=Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]]

=== Chapter 5 - Holomorphic Functions ===
* '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Holomorphic%20function&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Holomorphic%20function&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
** [[Holomorphism/Criteria|Criteria]] - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Holomorphism/Criteria&amp;author=Course:Complex_Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Criteria&amp;coursetitle=Complex_Analysis slideset]) [[File:Wiki2Reveal Logo.png|35px]]
** [[w:en:Holomorphic_function#.C3.84quivalent_properties_of_holomorphic_functions_of_one_variable|Wikipedia: Holomorphic function criteria]]
** [[/Differences from real differentiability/]]
** [[w:Conformal_mapping|conformal mappings]]&lt;math&gt;(\ast)&lt;/math&gt;, 

* '''[[Complex Analysis/Path of Integration|Path of Integration]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Path%20of%20Integration&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Path%20of%20Integration&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]

* '''[[Complex Analysis/Goursat's Lemma (Details)|Goursat's Lemma (Details)]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Goursat%27s%20Lemma%20(Details)&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Goursat%27s%20Lemma%20(Details)&amp;coursetitle=Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]]

* '''[[/Cauchy's Integral Theorem for Disks/]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex+Analysis/Cauchy%27s+Integral+Theorem_for_Disks&amp;author=Complex+Analysis&amp;audioslide=yes&amp;language=en Slide Set]) [[File:Wiki2Reveal Logo.png|35px]],
* '''[[Complex Analysis/Identity Theorem|Identity Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Identity%20Theorem&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes Slide Set]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Liouville's Theorem|Liouville's Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Liouville%27s%20Theorem&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Liouville%27s%20Theorem&amp;coursetitle=Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]]


=== Complex Analysis Part 2 ===

*'''[[/Chain/|Chain of Paths]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=ComplexAnalysis/Chain&amp;author=ComplexAnalysis&amp;language=de&amp;audioslide=yes&amp;shorttitle=Chain&amp;coursetitle=ComplexAnalysis Slides]) [[File:Wiki2Reveal Logo.png|35px]]

*'''[[/cycle/]]''' ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=cycle_and_Chain&amp;author=ComplexAnalysis&amp;audioslide=yes&amp;language=de Slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Laurent Series|Laurent Series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Laurent%20Series&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Laurent%20Series&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
  

*[[Cauchy Integral Theorem|Cauchy's Integral Theorem]] for cycles ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Integral_of_Cauchy&amp;author=ComplexAnalysis&amp;audioslide=yes&amp;language=de Slides]) [[File:Wiki2Reveal Logo.png|35px]]

*[[Cauchy Integral Formula|Cauchy Integral Formula]] for cycles ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Integralformula_of_Cauchy&amp;author=ComplexAnalysis&amp;audioslide=yes&amp;language=de Slides]) [[File:Wiki2Reveal Logo.png|35px]]

==Lectures==
* [[/Cauchy-Riemann equations/]]
* [[Cauchy Theorem for a triangle]]
* [[Complex analytic function]]
* [[Complex Numbers]]
* [[Divergent series]]
* [[Estimation lemma]]
* [[Fourier series]]
* [[Fourier transform]]
* [[Fourier transforms]]
* [[Laplace transform]]
* [[Riemann hypothesis]]
* [[The Real and Complex Number System]]
* [[Warping functions]]

==Sample exams==

[[/Sample Midterm Exam 1/]]
[[/Sample Midterm Exam 2/]]

==See also==
* [[Boundary Value Problems]]
* [[Introduction to Elasticity]]
* [[The Prime Sequence Problem]]
* [[Wikipedia: Complex analysis]]
*[[Complex number]]
[[Category:Complex analysis| ]]
[[Category:Mathematics courses]]
[[Category:Mathematics]]

&lt;noinclude&gt;
[[de:Kurs:Funktionentheorie]]
&lt;/noinclude&gt;</text>
      <sha1>o3dlxw43nhqrvxmlf9ur1f0j66p43c7</sha1>
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    <revision>
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        <id>2993595</id>
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      <comment>/* Chapter 5 - Holomorphic Functions */</comment>
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      <text bytes="12200" sha1="2g5hz0apof29kh0qndq5gthlhvktxve" xml:space="preserve">[[File:Wiki2Reveal Logo.png|146px|thumb|Course contains [[v:en:Wiki2Reveal|Wiki2Reveal]] Slides]]
[[File:Mapping f z equal 1 over z.gif|thumb|Moving the argument of function &lt;math&gt;f&lt;/math&gt; in the complex number plane. The point &lt;math&gt;z&lt;/math&gt; has a blue color and &lt;math&gt;f(z)= \frac{1}{z}&lt;/math&gt;  is marked in red color. &lt;math&gt;z&lt;/math&gt; is moved on a curve with  &lt;math&gt;\gamma(t)=t\cdot e^{it}&lt;/math&gt;.]]
[[File:Image of path 1 over z.webm|thumb|Image of path in the complex numbers for the function &lt;math&gt;f(z)=\frac{1}{z}&lt;/math&gt;]]
'''Complex analysis''' is a study of functions of a complex variable. This is a one quarter course in complex analysis at the undergraduate level.

==Articles==

* [[Algebra II]]
* [[Dummy variable]]
* [[Materials Science and Engineering/Equations/Quantum Mechanics]]

== Slides for Lectures ==

=== Chapter 1 - Intoduction ===
* '''[[Complex Numbers/From real to complex numbers|Complex Numbers]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Numbers/From%20real%20to%20complex%20numbers&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Complex%20Numbers&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
** [[w:en:Heine–Borel_theorem|Heine-Borel Theorem]]
* '''[[Riemann sphere|Riemann sphere]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Riemann%20sphere&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Riemann%20sphere&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex_Analysis/Exponentiation_and_square_root|Exponentiation and roots]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex_Analysis/Exponentiation_and_square_root&amp;author=Complex_Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Exponentiation_and_square_root&amp;coursetitle=Complex_Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]

=== Chapter 2 - Topological Foundations ===
* '''[[Complex Analysis/Sequences and series|Sequences and series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Sequences%20and%20series&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Sequences%20and%20series&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* [[/Power series/]]
* '''[[Inverse-producing extensions of Topological Algebras/topological algebra|Topological algebra]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Inverse-producing%20extensions%20of%20Topological%20Algebras/topological%20algebra&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=topological%20algebra&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* [[w:en:Topological space|Topological space]] - Definition: [[Norms, metrics, topology#Definition:_topology|Topology]] 
* '''[[Norms, metrics, topology|Norms, metrics, topology]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Norms,%20metrics,%20topology&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Norms,%20metrics,%20topology&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]

=== Chapter 3 - Complex Derivative ===
* '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Holomorphic%20function&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Holomorphic%20function&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Partial derivative|Partial Derivative]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Partial%20derivative&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Partial%20Derivative&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Cauchy-Riemann Equations|Cauchy-Riemann Equations (CRE)]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Cauchy-Riemann%20Equations&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Cauchy-Riemann%20Equations&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Application of Cauchy-Riemann Equations|Application of Cauchy-Riemann Equations]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Application%20of%20Cauchy-Riemann%20Equations&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Application%20of%20Cauchy-Riemann%20Equations&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]

=== Chapter 4 - Curves and Line Integrals ===
* '''[[Line integral|Line integral in &lt;math&gt;\mathbb{R}^n&lt;/math&gt;]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Line%20integral&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Line%20integral&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[/Curves/|Curves]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Curves&amp;author=Complex_Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Curves&amp;coursetitle=Complex%20Analysis  Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
** [[w:en:Holomorphic function|Wikipedia: holomorphic function]] 
** [[w:en:Integral|Wikipedia:Integral ]]
* '''[[Complex_Analysis/Paths|Paths]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Paths&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Paths&amp;coursetitle=Complex%20Analysis  Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]

* '''[[Path Integral|Path Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Path%20Integral&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Path%20Integral&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
  
** [[w:en:Curve integral |Wikipedia: Curve integral]]
** [[w:en:Continuity|Continuity]] and [[w:en:Limit of a sequence|Limit of a sequence]]
** [[/Inequalities/]]
** [[Complex Analysis/rectifiable curve|rectifiable curve]] - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/rectifiable%20curve&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=rectifiable%20curve&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]

* '''[[Complex Analysis/Trace|Trace of Curve]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Trace&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Trace&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]

* '''[[Complex Analysis/Curve Integral|Curve Integral]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Curve%20Integral&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Curve%20Integral&amp;coursetitle=Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]]

=== Chapter 5 - Holomorphic Functions ===
* '''[[Holomorphic function|Holomorphic function]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Holomorphic%20function&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Holomorphic%20function&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
** [[Holomorphism/Criteria|Criteria]] - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Holomorphism/Criteria&amp;author=Course:Complex_Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Criteria&amp;coursetitle=Complex_Analysis slideset]) [[File:Wiki2Reveal Logo.png|35px]]
** [[w:en:Holomorphic_function#.C3.84quivalent_properties_of_holomorphic_functions_of_one_variable|Wikipedia: Holomorphic function criteria]]
** [[/Differences from real differentiability/]]
** [[w:Conformal_mapping|conformal mappings]]&lt;math&gt;(\ast)&lt;/math&gt;, 

* '''[[Complex Analysis/Path of Integration|Path of Integration]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Path%20of%20Integration&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Path%20of%20Integration&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]

* '''[[Complex Analysis/Goursat's Lemma (Details)|Goursat's Lemma (Details)]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Goursat%27s%20Lemma%20(Details)&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Goursat%27s%20Lemma%20(Details)&amp;coursetitle=Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]]

* '''[[/Cauchy's Integral Theorem for Disks/]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex+Analysis/Cauchy%27s+Integral+Theorem_for_Disks&amp;author=Complex+Analysis&amp;audioslide=yes&amp;language=en Slide Set]) [[File:Wiki2Reveal Logo.png|35px]],
* '''[[Complex Analysis/Identity Theorem|Identity Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Identity%20Theorem&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Identity%20Theorem&amp;coursetitle= Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Complex Analysis/Liouville's Theorem|Liouville's Theorem]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Complex%20Analysis/Liouville%27s%20Theorem&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Liouville%27s%20Theorem&amp;coursetitle=Complex%20Analysis Slide Set]) [[File:Wiki2Reveal Logo.png|35px]]


=== Complex Analysis Part 2 ===

*'''[[/Chain/|Chain of Paths]]''' - ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=ComplexAnalysis/Chain&amp;author=ComplexAnalysis&amp;language=de&amp;audioslide=yes&amp;shorttitle=Chain&amp;coursetitle=ComplexAnalysis Slides]) [[File:Wiki2Reveal Logo.png|35px]]

*'''[[/cycle/]]''' ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=cycle_and_Chain&amp;author=ComplexAnalysis&amp;audioslide=yes&amp;language=de Slides]) [[File:Wiki2Reveal Logo.png|35px]]
* '''[[Laurent Series|Laurent Series]]''' - ([https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Laurent%20Series&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Laurent%20Series&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]) [[File:Wiki2Reveal Logo.png|35px]]
  

*[[Cauchy Integral Theorem|Cauchy's Integral Theorem]] for cycles ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Integral_of_Cauchy&amp;author=ComplexAnalysis&amp;audioslide=yes&amp;language=de Slides]) [[File:Wiki2Reveal Logo.png|35px]]

*[[Cauchy Integral Formula|Cauchy Integral Formula]] for cycles ([https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Integralformula_of_Cauchy&amp;author=ComplexAnalysis&amp;audioslide=yes&amp;language=de Slides]) [[File:Wiki2Reveal Logo.png|35px]]

==Lectures==
* [[/Cauchy-Riemann equations/]]
* [[Cauchy Theorem for a triangle]]
* [[Complex analytic function]]
* [[Complex Numbers]]
* [[Divergent series]]
* [[Estimation lemma]]
* [[Fourier series]]
* [[Fourier transform]]
* [[Fourier transforms]]
* [[Laplace transform]]
* [[Riemann hypothesis]]
* [[The Real and Complex Number System]]
* [[Warping functions]]

==Sample exams==

[[/Sample Midterm Exam 1/]]
[[/Sample Midterm Exam 2/]]

==See also==
* [[Boundary Value Problems]]
* [[Introduction to Elasticity]]
* [[The Prime Sequence Problem]]
* [[Wikipedia: Complex analysis]]
*[[Complex number]]
[[Category:Complex analysis| ]]
[[Category:Mathematics courses]]
[[Category:Mathematics]]

&lt;noinclude&gt;
[[de:Kurs:Funktionentheorie]]
&lt;/noinclude&gt;</text>
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    <title>Physics/Essays/Fedosin/Strong gravitation</title>
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      <text bytes="18077" sha1="12e4mkckbwgf04g8en74sx43kk5gq63" xml:space="preserve">'''Strong gravitation''' is [[w:fundamental interaction| fundamental gravitational interaction]] at the level of [[w:elementary particle| elementary particles]], one of the components of the [[Charges/Interactions/Strong|strong interaction]] in physics according to the [[Physics/Essays/Fedosin/Gravitational model of strong interaction |gravitational model of strong interaction]]. It is assumed that strong gravitation and electromagnetic forces are responsible for the formation and integrity of the matter of elementary particles and [[w:atomic nucleus| atomic nuclei]], and also participates in the interactions between electrons and nuclei in atoms and molecules. For describing of strong gravitation equations of [[Physics/Essays/Fedosin/Lorentz-invariant theory of gravitation |Lorentz-invariant theory of gravitation]] are used.

==History==
After the discovery of the [[electron]] in 1897, of the [[w:proton| proton]] in 1919, and of the [[w:neutron |neutron]] in 1932, and of their compositions in the form of atomic nuclei, atoms and molecules, it became necessary to describe the forces acting between the particles and binding their matter. In most cases, the behavior of the electron and proton, placed in the external electromagnetic field, is satisfactorily described by electromagnetic forces. This led to the standard electromagnetic model of the atom. As for the interaction of [[w:nucleon| nucleon]]s in atomic nuclei, the hypothesis of the Japanese physicist H. Yukawa was initially accepted about the binding between the particles by means of [[w:meson| meson]]s, mostly of [[w:pion| pion]]s. Then, in the framework of the quark theory all hadrons began to be considered to be composed of [[w:quark| quark]]s.

However the idea, that the fundamental interaction between a set of elementary particles must occur due to the action of another set of elementary particles, belongs to the atomistic theory, but it contradicts the Theory of [[Physics/Essays/Fedosin/Infinite Hierarchical Nesting of Matter|Infinite Hierarchical Nesting of Matter]]. Indeed, the reactions between elementary particles follow the laws of conservation of energy, momentum and electric charge; the matter, energy-momentum and the charge of one type of particles transforms into the corresponding quantities of other particles, but this does not mean that the carrier and the cause of the interactions are the elementary particles themselves. The interaction of nucleons with each other by means of pions hardly agrees with quarks and [[w:gluon |gluon]]s, which are used to describe the integrity of hadrons, due to the problem of non-observability of quarks in the free state and the uncertainty of transformation of the forces between the quarks inside each of the nucleons into the [[Charges/Interactions/Strong|strong interaction]] between different nucleons in the atomic nucleus. The introduction of [[virtual particle]]s with their exotic properties (short lifetime, the simultaneous generation of particles and antiparticles, etc.) does not save the situation. Thus, the abstract explanation of the [[Charges/Interactions/Electromagnetics|electromagnetic interaction]] of two charges with the help of virtual [[w:photon |photon]]s as the field quanta still remains the statement which is not supported by the concrete model of the interaction process.

Among the attempts to explain the strong interaction in connection with gravitation there is a hypothesis that in the model of hadronic bags the hadrons are de Sitter microuniverses, in which quarks are enclosed. The radius of hadrons corresponding to the radii of these microuniverses, is associated with the strong gravitational constant and the corresponding [[w:cosmological constant| cosmological constant]].&lt;ref&gt;Salam, A., and Strathdee, J. Confinement Through Tensor Gauge Fields. Physical Review D, 1978, Vol.18, Issue 12, P. 4596-4609. &lt;/ref&gt; To explain the properties of hadrons in the assumption of strong gravitational interaction the analogies between hadrons and Kerr — Newman [[black hole]]s are described. &lt;ref&gt; Sivaram, C. and Sinha, K.P. Strong gravity, black holes, and hadrons. Physical Review D, 1977, Vol. 16, Issue 6, P. 1975-1978. &lt;/ref&gt; &lt;ref&gt;Recami, E. and Castorina, P. On Quark Confinement: Hadrons as «Strong Black- Holes». Letters Nuovo Cimento, 1976, Vol. 15, No 10, P. 347-350. &lt;/ref&gt; &lt;ref&gt; Pavsic, M. (1978). Unified Theory Of Strong And Gravitational Interactions. Nuovo Cimento B, Vol. 48, P. 205-253. &lt;/ref&gt; &lt;ref&gt; Oldershaw R. L. [http://arxiv.org/abs/astro-ph/0701006v4 Hadrons as Kerr-Newman Black Holes.]  arXiv:astro-ph/0701006v4, 30 Dec 2006. &lt;/ref&gt;

In 1999 [[User:Fedosin | Sergey Fedosin]], based on the [[Physics/Essays/Fedosin/Similarity of matter levels|similarity of matter levels]], [[Physics/Essays/Fedosin/SPФ symmetry|SPФ symmetry]] and [[w:Le Sage's theory of gravitation| Le Sage's theory of gravitation]], according to which black holes are not admitted, postulated the existence of strong gravitation as the fundamental force at the atomic level and found the value of [[strong gravitational constant]] &lt;math&gt;~\Gamma= 1{.}514 \cdot 10^{29}&lt;/math&gt; m&lt;sup&gt;3&lt;/sup&gt;•s&lt;sup&gt;–2&lt;/sup&gt;•kg&lt;sup&gt;–1&lt;/sup&gt;. &lt;ref name=&quot;fed&quot;&gt;Fedosin S.G. [http://lccn.loc.gov/2009457349 Fizika i filosofiia podobiia ot preonov do metagalaktik], Perm, pages 544, 1999. {{ISBN|5-8131-0012-1}}. &lt;/ref&gt;

== Applications ==
=== Hadrons ===
The equality between the rest energy of the proton and its total energy, which due to the [[w:virial theorem| virial theorem]] is approximately equal to the half of the [[w:potential energy| potential energy]] of the strong gravitational field, allows us to estimate the radius of the proton &lt;math&gt;~R_p &lt;/math&gt;:
: &lt;math&gt; m_p c^2 = \frac{ k \Gamma {m_p}^2}{ 2 R_p },&lt;/math&gt;
: &lt;math&gt; R_p = \frac{ k \Gamma m_p}{2 c^2 }=0.87 \cdot 10^{-15}&lt;/math&gt; m,

here &lt;math&gt;~ m_p &lt;/math&gt; is the proton mass, &lt;math&gt;~ c &lt;/math&gt; is the [[w:speed of light |speed of light]], &lt;math&gt;~ k &lt;/math&gt; is the coefficient depending on the distribution of matter, in the case of the uniform mass density of the proton &lt;math&gt;~ k=0.6 &lt;/math&gt;. According to the self-consistent model &lt;ref name=com&gt; [http://sergf.ru/com.htm Comments to the book:] Fedosin S.G. Fizicheskie teorii i beskonechnaia vlozhennost’ materii. – Perm, 2009, 844 pages, {{ISBN|978-5-9901951-1-0}}. (in Russian).&lt;/ref&gt; &lt;ref&gt; Fedosin S.G. [http://vixra.org/abs/1208.0006 The radius of the proton in the self-consistent model.] Hadronic Journal, Vol. 35, No. 4, pp. 349-363 (2012). http://dx.doi.org/10.5281/zenodo.889451. &lt;/ref&gt; for the proton &lt;math&gt;~ k=0.62 &lt;/math&gt;.

The obtained value &lt;math&gt;~R_p &lt;/math&gt; coincides with the experimentally obtained sizes of the proton and the neutron, &lt;ref&gt;Hofstadter, Robert, [http://nobelprize.org/nobel_prizes/physics/laureates/1961/hofstadter-lecture.pdf The electron-scattering method and its application to the structure of nuclei and nucleons,] Nobel Lecture (December 11, 1961).&lt;/ref&gt; confirming the validity of the idea of strong gravitation. At the same time the given equality implies the explanation of the essence of the rest energy of bodies as the energy associated with the strong gravitation of the nucleons of the bodies’ matter. According to the [[w:mass–energy equivalence |mass–energy equivalence]], the rest energy of the nucleon is proportional to its mass. On the other hand, the total energy of the nucleon includes the energy of the strong gravitational field which is proportional to the squared mass, and the internal energy of the nucleon matter which is proportional to the matter mass in the expression for the [[w:kinetic energy |kinetic energy]]. As a result, the total energy is proportional only to the mass just as well as the rest energy.

On the comparison of the maximum angular momentum of the strong gravitational field and the angular momentum of the proton with the uniform matter distribution another estimate of the proton radius is based: &lt;ref&gt;[[User:Fedosin |Fedosin S.G.]] [http://lccn.loc.gov/2009457352 Sovremennye problemy fiziki: v poiskakh novykh printsipov,] Editorial URSS, Moskva (2002). &lt;/ref&gt; 

: &lt;math&gt; R_p = \frac{ 5 \Gamma m_p}{21 c^2 }=0.67 \cdot 10^{-15}&lt;/math&gt; m.

As the model of emerging of strong gravitation the modernized [[w:Le Sage's theory of gravitation |Le Sage's theory of gravitation]] is used, which becomes universal taking into account the Theory of [[Physics/Essays/Fedosin/Infinite Hierarchical Nesting of Matter|Infinite Hierarchical Nesting of Matter]]. &lt;ref &gt; Fedosin S.G. [http://sergf.ru/mgen.htm  Model of Gravitational Interaction in the Concept of Gravitons].  Journal of Vectorial Relativity, Vol. 4, No. 1, pp.1-24 (2009). http://dx.doi.org/10.5281/zenodo.890886.&lt;/ref&gt; &lt;ref&gt; Fedosin S.G. [http://vixra.org/abs/1503.0127 The graviton field as the source of mass and gravitational force in the modernized Le Sage’s model.] Physical Science International Journal, ISSN: 2348-0130, Vol. 8, Issue 4, pp. 1-18 (2015). http://dx.doi.org/10.9734/PSIJ/2015/22197. &lt;/ref&gt;

At the stellar [[Physics/Essays/Fedosin/Similarity of matter levels|scale level of matter]] the analogues of nucleons are [[Physics/Essays/Fedosin/Neutron star|neutron star]]s, the integrity of which is maintained by the ordinary force of gravitation and the pressure force in the matter arising from the repulsion of the nucleons from each other. Similarly, in the matter of nucleons the compensation of the strong gravitation and the internal pressure force takes place (see the [[Physics/Essays/Fedosin/Substantial neutron model|substantial neutron model]] and the [[Physics/Essays/Fedosin/Substantial proton model|substantial proton model]]). In this picture, for the stability of nucleons and describing their properties [[w:quark |quark]]s are not required, in contrast to the standard [[w:quantum chromodynamics| quantum chromodynamics]]. At the same time in the [[model of quark quasiparticles]] the quarks are seen not as real particles inside hadrons but as quasiparticles, the constituent elements of the hadrons’ matter which carry the mass, charge and magnetic moment. This ensures the observed symmetry of hadron properties. In turn, the quarks themselves can be reduced to the combinations of two hadronic phases of the matter. &lt;ref name=&quot;fs1&quot;&gt; Sergey Fedosin, [https://www.lap-publishing.com/catalog/details//store/gb/book/978-3-659-57301-9/the-physical-theories-and-infinite-hierarchical-nesting-of-matter-volume-1 The physical theories and infinite hierarchical nesting of matter], Volume 1, LAP LAMBERT Academic Publishing, pages: 580, {{ISBN|978-3-659-57301-9}}.&lt;/ref&gt; The analysis of the Regge hadron families also shows that they can be explained by taking into account the quantization of the [[w:spin| spin]] and the matter state of the particles, retained by the strong gravitational field.

=== Electron ===
Strong gravitation significantly affects the construction of the model of the electron, leading to the [[Physics/Essays/Fedosin/Substantial electron model|substantial model]] of this particle. In particular, the electron charge is so large, that strong gravitation is not able to keep the electron matter from the Coulomb electric force of repulsion of the charges. Therefore, the stability of the electron in the atom is possible only in the form of a scattered electron cloud (disc) and due to the forces of attraction to the nucleus from strong gravitation and the nuclear charge. Another fact, the quantization of energy levels and of the orbital angular momentum of the electron in the atom, is explained based on the condition that the flux of kinetic energy of the motion of electron matter around the nucleus is equal to the sum of the energy fluxes from the strong gravitation and the electromagnetic field. &lt;ref name=&quot;fs1&quot;/&gt; This leads to the stationary states of the electron in the atom, in which it does not produce emission. For the hydrogen atom it is also found that the magnetic energy of the nucleus in the magnetic field of the electron equals the energy of the nucleus’ spin in the [[gravitational torsion field|torsion field]] of strong gravitation of the electron in case of limiting rotation of the nucleus. &lt;ref name=com/&gt;

=== The interaction of nucleons in the atomic nucleus===
The experiments with the scattering of nucleons on each other allow us to estimate the effective potential of [[Charges/Interactions/Strong|strong interaction]] acting between these particles. &lt;ref&gt; Ishii N., Aoki S., Hatsuda T. [http://arxiv.org/abs/nucl-th/0611096v1 The Nuclear Force from Lattice QCD]. – arXiv: nucl-th / 0611096 v1, 28 Nov 2006. &lt;/ref&gt; As the distance decreases the interaction force increases rapidly. To describe this force the [[gravitational model of strong interaction]] is used, in which the nuclear forces are the sum of the attraction from the strong gravitation, the repulsion of the nucleon spins due to the torsion field of strong gravitation, as well as from the action of electromagnetic forces. At short distances, the repulsive force of the spins dominates, which is inversely proportional to the fourth and then the fifth degree of the distance. At large distances, there is attraction of the nucleons, mainly from the strong gravitation. At distances close to the radius of the nucleon, the neutron and the proton are in the equilibrium state, which gives the [[w:deuteron |deuteron]] as the simplest atomic nucleus with two nucleons. &lt;ref name=&quot;fs1&quot;/&gt; Taking into account the strong gravitation allows us to construct the model of simplest nuclei and their geometric configuration, as well as to explain the dependence of the specific binding energy of atomic nuclei on their atomic number due to the saturation effect of the strong gravitational energy and the increase of the electrical repulsive energy of protons.

=== Strange particles===
In quantum chromodynamics, it is assumed that the long lifetime, inherent in some hadrons, is due to the presence of strange quarks in them. However, the models of strange particles can be constructed similarly to the models of atomic nuclei, by connecting nucleons and mesons under the influence of strong gravitation. &lt;ref name=com/&gt; The composition of some strange hadrons is described in the [[model of quark quasiparticles]].

=== Interatomic interaction===
The interaction of atoms leads to the formation of molecules, as well as simple and molecular substances. In contrast to the nucleons in atomic nuclei, in the interaction of atoms the strong gravitation acts between the nuclei of all atoms as well as between the electrons, complementing the electromagnetic forces. In this case the electron discs, surrounding the atomic nuclei, due to the rapid rotation in them of the matter, which is charged and oriented by the magnetic field, have the possibility to shield the gravitational forces between the nuclei, reducing them to the level of electrical forces. The equilibrium of atoms in molecules and in substances is achieved in case of the balance of gravitational and electromagnetic forces. With increasing of the distance between the atoms, the so-called [[w:Van der Waals force |Van der Waals force]] occurs between them in the form of the attraction rapidly decreasing with the distance. The estimate with the help of the Le Sage's theory of gravitation gives the radius of action of strong gravitation in the matter with the density of the order of Earth's density, about 0.7 m. &lt;ref name=&quot;fs1&quot;/&gt;

=== Photon ===
The [[Physics/Essays/Fedosin/Substantial photon model|substantial photon model]] assumes that the photon consists of [[Physics/Essays/Fedosin/Praon|praon]]s bonded to each other by means of strong gravitation.&lt;ref name=cc&gt;Fedosin S.G. [https://www.ajol.info/index.php/jfas/article/view/168204  The charged component of the vacuum field as the source of electric force in the modernized Le Sage’s model.] Journal of Fundamental and Applied Sciences, Vol. 8, No. 3, pp. 971-1020 (2016). http://dx.doi.org/10.4314/jfas.v8i3.18, https://dx.doi.org/10.5281/zenodo.845357. &lt;/ref&gt; &lt;ref name=sub&gt;Fedosin S.G. [https://www.ajol.info/index.php/jfas/article/view/151053 The substantial model of the photon]. Journal of Fundamental and Applied Sciences, Vol. 9, No. 1, pp. 411-467 (2017). http://dx.doi.org/10.4314/jfas.v9i1.25. &lt;/ref&gt; This leads to the fact that the photon has a rest mass, as well as a magnetic moment.

==References==
{{reflist}}

== See also ==
* [[Physics/Essays/Fedosin/Infinite Hierarchical Nesting of Matter|Infinite Hierarchical Nesting of Matter]]
* [[Physics/Essays/Fedosin/Similarity of matter levels|Similarity of matter levels]]
* [[Physics/Essays/Fedosin/SPФ symmetry|SPФ symmetry]]
* [[Physics/Essays/Fedosin/Quantization of parameters of cosmic systems |Quantization of parameters of cosmic systems]]
* [[Physics/Essays/Fedosin/Hydrogen system |Hydrogen system]]
* [[Physics/Essays/Fedosin/Strong gravitational constant|Strong gravitational constant]]
* [[Charges/Interactions/Strong|Strong interaction]]
* [[Gravitational torsion field]]
* [[Physics/Essays/Fedosin/Gravitational model of strong interaction |gravitational model of strong interaction]]
* [[Physics/Essays/Fedosin/Model of quark quasiparticles |Model of quark quasiparticles]]
* [[Physics/Essays/Fedosin/Substantial electron model|Substantial electron model]]
* [[Physics/Essays/Fedosin/Substantial neutron model|Substantial neutron model]]
* [[Physics/Essays/Fedosin/Substantial proton model|Substantial proton model]]
* [[Physics/Essays/Fedosin/Electrogravitational vacuum |Electrogravitational vacuum]]

== External links ==
* [http://www.wikiznanie.ru/ru-wz/index.php/%D0%A1%D0%B8%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D0%B3%D1%80%D0%B0%D0%B2%D0%B8%D1%82%D0%B0%D1%86%D0%B8%D1%8F Strong gravitation in Russian]


{{Fundamental Forces}}
{{Theories of gravitation}}

[[Category:Concepts in physics]] [[Category:Particle physics]] [[Category:Theory of infinite nesting of matter]] [[Category:Covariant theory of gravitation]] [[Category: Metric theory of relativity]]</text>
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      <text bytes="18162" sha1="f8errue0q5a736v9p0urbbzya07ww1s" xml:space="preserve">'''Strong gravitation''' is [[w:fundamental interaction| fundamental gravitational interaction]] at the level of [[w:elementary particle| elementary particles]], one of the components of the [[Charges/Interactions/Strong|strong interaction]] in physics according to the [[Physics/Essays/Fedosin/Gravitational model of strong interaction |gravitational model of strong interaction]]. It is assumed that strong gravitation and electromagnetic forces are responsible for the formation and integrity of the matter of elementary particles and [[w:atomic nucleus| atomic nuclei]], and also participates in the interactions between electrons and nuclei in atoms and molecules. For describing of strong gravitation equations of [[Physics/Essays/Fedosin/Lorentz-invariant theory of gravitation |Lorentz-invariant theory of gravitation]] are used.

==History==
After the discovery of the [[w:electron |electron]] in 1897, of the [[w:proton| proton]] in 1919, and of the [[w:neutron |neutron]] in 1932, and of their compositions in the form of atomic nuclei, atoms and molecules, it became necessary to describe the forces acting between the particles and binding their matter. In most cases, the behavior of the electron and proton, placed in the external electromagnetic field, is satisfactorily described by electromagnetic forces. This led to the standard electromagnetic model of the atom. As for the interaction of [[w:nucleon| nucleon]]s in atomic nuclei, the hypothesis of the Japanese physicist H. Yukawa was initially accepted about the binding between the particles by means of [[w:meson| meson]]s, mostly of [[w:pion| pion]]s. Then, in the framework of the quark theory all hadrons began to be considered to be composed of [[w:quark| quark]]s.

However the idea, that the fundamental interaction between a set of elementary particles must occur due to the action of another set of elementary particles, belongs to the atomistic theory, but it contradicts the Theory of [[Physics/Essays/Fedosin/Infinite Hierarchical Nesting of Matter|Infinite Hierarchical Nesting of Matter]]. Indeed, the reactions between elementary particles follow the laws of conservation of energy, momentum and electric charge; the matter, energy-momentum and the charge of one type of particles transforms into the corresponding quantities of other particles, but this does not mean that the carrier and the cause of the interactions are the elementary particles themselves. The interaction of nucleons with each other by means of pions hardly agrees with quarks and [[w:gluon |gluon]]s, which are used to describe the integrity of hadrons, due to the problem of non-observability of quarks in the free state and the uncertainty of transformation of the forces between the quarks inside each of the nucleons into the [[Charges/Interactions/Strong|strong interaction]] between different nucleons in the atomic nucleus. The introduction of [[w:virtual particle |virtual particle]]s with their exotic properties (short lifetime, the simultaneous generation of particles and antiparticles, etc.) does not save the situation. Thus, the abstract explanation of the [[Charges/Interactions/Electromagnetics|electromagnetic interaction]] of two charges with the help of virtual [[w:photon |photon]]s as the field quanta still remains the statement which is not supported by the concrete model of the interaction process.

Among the attempts to explain the strong interaction in connection with gravitation there is a hypothesis that in the model of hadronic bags the hadrons are de Sitter microuniverses, in which quarks are enclosed. The radius of hadrons corresponding to the radii of these microuniverses, is associated with the strong gravitational constant and the corresponding [[w:cosmological constant| cosmological constant]].&lt;ref&gt;Salam, A., and Strathdee, J. Confinement Through Tensor Gauge Fields. Physical Review D, 1978, Vol.18, Issue 12, P. 4596-4609. &lt;/ref&gt; To explain the properties of hadrons in the assumption of strong gravitational interaction the analogies between hadrons and Kerr — Newman [[black hole]]s are described. &lt;ref&gt; Sivaram, C. and Sinha, K.P. Strong gravity, black holes, and hadrons. Physical Review D, 1977, Vol. 16, Issue 6, P. 1975-1978. &lt;/ref&gt; &lt;ref&gt;Recami, E. and Castorina, P. On Quark Confinement: Hadrons as «Strong Black- Holes». Letters Nuovo Cimento, 1976, Vol. 15, No 10, P. 347-350. &lt;/ref&gt; &lt;ref&gt; Pavsic, M. (1978). Unified Theory Of Strong And Gravitational Interactions. Nuovo Cimento B, Vol. 48, P. 205-253. &lt;/ref&gt; &lt;ref&gt; Oldershaw R. L. [http://arxiv.org/abs/astro-ph/0701006v4 Hadrons as Kerr-Newman Black Holes.]  arXiv:astro-ph/0701006v4, 30 Dec 2006. &lt;/ref&gt;

In 1999 [[User:Fedosin | Sergey Fedosin]], based on the [[Physics/Essays/Fedosin/Similarity of matter levels|similarity of matter levels]], [[Physics/Essays/Fedosin/SPФ symmetry|SPФ symmetry]] and [[w:Le Sage's theory of gravitation| Le Sage's theory of gravitation]], according to which black holes are not admitted, postulated the existence of strong gravitation as the fundamental force at the atomic level and found the value of [[Physics/Essays/Fedosin/Strong gravitational constant|strong gravitational constant]] &lt;math&gt;~\Gamma= 1{.}514 \cdot 10^{29}&lt;/math&gt; m&lt;sup&gt;3&lt;/sup&gt;•s&lt;sup&gt;–2&lt;/sup&gt;•kg&lt;sup&gt;–1&lt;/sup&gt;. &lt;ref name=&quot;fed&quot;&gt;Fedosin S.G. [http://lccn.loc.gov/2009457349 Fizika i filosofiia podobiia ot preonov do metagalaktik], Perm, pages 544, 1999. {{ISBN|5-8131-0012-1}}. &lt;/ref&gt;

== Applications ==
=== Hadrons ===
The equality between the rest energy of the proton and its total energy, which due to the [[w:virial theorem| virial theorem]] is approximately equal to the half of the [[w:potential energy| potential energy]] of the strong gravitational field, allows us to estimate the radius of the proton &lt;math&gt;~R_p &lt;/math&gt;:
: &lt;math&gt; m_p c^2 = \frac{ k \Gamma {m_p}^2}{ 2 R_p },&lt;/math&gt;
: &lt;math&gt; R_p = \frac{ k \Gamma m_p}{2 c^2 }=0.87 \cdot 10^{-15}&lt;/math&gt; m,

here &lt;math&gt;~ m_p &lt;/math&gt; is the proton mass, &lt;math&gt;~ c &lt;/math&gt; is the [[w:speed of light |speed of light]], &lt;math&gt;~ k &lt;/math&gt; is the coefficient depending on the distribution of matter, in the case of the uniform mass density of the proton &lt;math&gt;~ k=0.6 &lt;/math&gt;. According to the self-consistent model &lt;ref name=com&gt; [http://sergf.ru/com.htm Comments to the book:] Fedosin S.G. Fizicheskie teorii i beskonechnaia vlozhennost’ materii. – Perm, 2009, 844 pages, {{ISBN|978-5-9901951-1-0}}. (in Russian).&lt;/ref&gt; &lt;ref&gt; Fedosin S.G. [http://vixra.org/abs/1208.0006 The radius of the proton in the self-consistent model.] Hadronic Journal, Vol. 35, No. 4, pp. 349-363 (2012). http://dx.doi.org/10.5281/zenodo.889451. &lt;/ref&gt; for the proton &lt;math&gt;~ k=0.62 &lt;/math&gt;.

The obtained value &lt;math&gt;~R_p &lt;/math&gt; coincides with the experimentally obtained sizes of the proton and the neutron, &lt;ref&gt;Hofstadter, Robert, [http://nobelprize.org/nobel_prizes/physics/laureates/1961/hofstadter-lecture.pdf The electron-scattering method and its application to the structure of nuclei and nucleons,] Nobel Lecture (December 11, 1961).&lt;/ref&gt; confirming the validity of the idea of strong gravitation. At the same time the given equality implies the explanation of the essence of the rest energy of bodies as the energy associated with the strong gravitation of the nucleons of the bodies’ matter. According to the [[w:mass–energy equivalence |mass–energy equivalence]], the rest energy of the nucleon is proportional to its mass. On the other hand, the total energy of the nucleon includes the energy of the strong gravitational field which is proportional to the squared mass, and the internal energy of the nucleon matter which is proportional to the matter mass in the expression for the [[w:kinetic energy |kinetic energy]]. As a result, the total energy is proportional only to the mass just as well as the rest energy.

On the comparison of the maximum angular momentum of the strong gravitational field and the angular momentum of the proton with the uniform matter distribution another estimate of the proton radius is based: &lt;ref&gt;[[User:Fedosin |Fedosin S.G.]] [http://lccn.loc.gov/2009457352 Sovremennye problemy fiziki: v poiskakh novykh printsipov,] Editorial URSS, Moskva (2002). &lt;/ref&gt; 

: &lt;math&gt; R_p = \frac{ 5 \Gamma m_p}{21 c^2 }=0.67 \cdot 10^{-15}&lt;/math&gt; m.

As the model of emerging of strong gravitation the modernized [[w:Le Sage's theory of gravitation |Le Sage's theory of gravitation]] is used, which becomes universal taking into account the Theory of [[Physics/Essays/Fedosin/Infinite Hierarchical Nesting of Matter|Infinite Hierarchical Nesting of Matter]]. &lt;ref &gt; Fedosin S.G. [http://sergf.ru/mgen.htm  Model of Gravitational Interaction in the Concept of Gravitons].  Journal of Vectorial Relativity, Vol. 4, No. 1, pp.1-24 (2009). http://dx.doi.org/10.5281/zenodo.890886.&lt;/ref&gt; &lt;ref&gt; Fedosin S.G. [http://vixra.org/abs/1503.0127 The graviton field as the source of mass and gravitational force in the modernized Le Sage’s model.] Physical Science International Journal, ISSN: 2348-0130, Vol. 8, Issue 4, pp. 1-18 (2015). http://dx.doi.org/10.9734/PSIJ/2015/22197. &lt;/ref&gt;

At the stellar [[Physics/Essays/Fedosin/Similarity of matter levels|scale level of matter]] the analogues of nucleons are [[Physics/Essays/Fedosin/Neutron star|neutron star]]s, the integrity of which is maintained by the ordinary force of gravitation and the pressure force in the matter arising from the repulsion of the nucleons from each other. Similarly, in the matter of nucleons the compensation of the strong gravitation and the internal pressure force takes place (see the [[Physics/Essays/Fedosin/Substantial neutron model|substantial neutron model]] and the [[Physics/Essays/Fedosin/Substantial proton model|substantial proton model]]). In this picture, for the stability of nucleons and describing their properties [[w:quark |quark]]s are not required, in contrast to the standard [[w:quantum chromodynamics| quantum chromodynamics]]. At the same time in the [[model of quark quasiparticles]] the quarks are seen not as real particles inside hadrons but as quasiparticles, the constituent elements of the hadrons’ matter which carry the mass, charge and magnetic moment. This ensures the observed symmetry of hadron properties. In turn, the quarks themselves can be reduced to the combinations of two hadronic phases of the matter. &lt;ref name=&quot;fs1&quot;&gt; Sergey Fedosin, [https://www.lap-publishing.com/catalog/details//store/gb/book/978-3-659-57301-9/the-physical-theories-and-infinite-hierarchical-nesting-of-matter-volume-1 The physical theories and infinite hierarchical nesting of matter], Volume 1, LAP LAMBERT Academic Publishing, pages: 580, {{ISBN|978-3-659-57301-9}}.&lt;/ref&gt; The analysis of the Regge hadron families also shows that they can be explained by taking into account the quantization of the [[w:spin| spin]] and the matter state of the particles, retained by the strong gravitational field.

=== Electron ===
Strong gravitation significantly affects the construction of the model of the electron, leading to the [[Physics/Essays/Fedosin/Substantial electron model|substantial model]] of this particle. In particular, the electron charge is so large, that strong gravitation is not able to keep the electron matter from the Coulomb electric force of repulsion of the charges. Therefore, the stability of the electron in the atom is possible only in the form of a scattered electron cloud (disc) and due to the forces of attraction to the nucleus from strong gravitation and the nuclear charge. Another fact, the quantization of energy levels and of the orbital angular momentum of the electron in the atom, is explained based on the condition that the flux of kinetic energy of the motion of electron matter around the nucleus is equal to the sum of the energy fluxes from the strong gravitation and the electromagnetic field. &lt;ref name=&quot;fs1&quot;/&gt; This leads to the stationary states of the electron in the atom, in which it does not produce emission. For the hydrogen atom it is also found that the magnetic energy of the nucleus in the magnetic field of the electron equals the energy of the nucleus’ spin in the [[gravitational torsion field|torsion field]] of strong gravitation of the electron in case of limiting rotation of the nucleus. &lt;ref name=com/&gt;

=== The interaction of nucleons in the atomic nucleus===
The experiments with the scattering of nucleons on each other allow us to estimate the effective potential of [[Charges/Interactions/Strong|strong interaction]] acting between these particles. &lt;ref&gt; Ishii N., Aoki S., Hatsuda T. [http://arxiv.org/abs/nucl-th/0611096v1 The Nuclear Force from Lattice QCD]. – arXiv: nucl-th / 0611096 v1, 28 Nov 2006. &lt;/ref&gt; As the distance decreases the interaction force increases rapidly. To describe this force the [[gravitational model of strong interaction]] is used, in which the nuclear forces are the sum of the attraction from the strong gravitation, the repulsion of the nucleon spins due to the torsion field of strong gravitation, as well as from the action of electromagnetic forces. At short distances, the repulsive force of the spins dominates, which is inversely proportional to the fourth and then the fifth degree of the distance. At large distances, there is attraction of the nucleons, mainly from the strong gravitation. At distances close to the radius of the nucleon, the neutron and the proton are in the equilibrium state, which gives the [[w:deuteron |deuteron]] as the simplest atomic nucleus with two nucleons. &lt;ref name=&quot;fs1&quot;/&gt; Taking into account the strong gravitation allows us to construct the model of simplest nuclei and their geometric configuration, as well as to explain the dependence of the specific binding energy of atomic nuclei on their atomic number due to the saturation effect of the strong gravitational energy and the increase of the electrical repulsive energy of protons.

=== Strange particles===
In quantum chromodynamics, it is assumed that the long lifetime, inherent in some hadrons, is due to the presence of strange quarks in them. However, the models of strange particles can be constructed similarly to the models of atomic nuclei, by connecting nucleons and mesons under the influence of strong gravitation. &lt;ref name=com/&gt; The composition of some strange hadrons is described in the [[model of quark quasiparticles]].

=== Interatomic interaction===
The interaction of atoms leads to the formation of molecules, as well as simple and molecular substances. In contrast to the nucleons in atomic nuclei, in the interaction of atoms the strong gravitation acts between the nuclei of all atoms as well as between the electrons, complementing the electromagnetic forces. In this case the electron discs, surrounding the atomic nuclei, due to the rapid rotation in them of the matter, which is charged and oriented by the magnetic field, have the possibility to shield the gravitational forces between the nuclei, reducing them to the level of electrical forces. The equilibrium of atoms in molecules and in substances is achieved in case of the balance of gravitational and electromagnetic forces. With increasing of the distance between the atoms, the so-called [[w:Van der Waals force |Van der Waals force]] occurs between them in the form of the attraction rapidly decreasing with the distance. The estimate with the help of the Le Sage's theory of gravitation gives the radius of action of strong gravitation in the matter with the density of the order of Earth's density, about 0.7 m. &lt;ref name=&quot;fs1&quot;/&gt;

=== Photon ===
The [[Physics/Essays/Fedosin/Substantial photon model|substantial photon model]] assumes that the photon consists of [[Physics/Essays/Fedosin/Praon|praon]]s bonded to each other by means of strong gravitation.&lt;ref name=cc&gt;Fedosin S.G. [https://www.ajol.info/index.php/jfas/article/view/168204  The charged component of the vacuum field as the source of electric force in the modernized Le Sage’s model.] Journal of Fundamental and Applied Sciences, Vol. 8, No. 3, pp. 971-1020 (2016). http://dx.doi.org/10.4314/jfas.v8i3.18, https://dx.doi.org/10.5281/zenodo.845357. &lt;/ref&gt; &lt;ref name=sub&gt;Fedosin S.G. [https://www.ajol.info/index.php/jfas/article/view/151053 The substantial model of the photon]. Journal of Fundamental and Applied Sciences, Vol. 9, No. 1, pp. 411-467 (2017). http://dx.doi.org/10.4314/jfas.v9i1.25. &lt;/ref&gt; This leads to the fact that the photon has a rest mass, as well as a magnetic moment.

==References==
{{reflist}}

== See also ==
* [[Physics/Essays/Fedosin/Infinite Hierarchical Nesting of Matter|Infinite Hierarchical Nesting of Matter]]
* [[Physics/Essays/Fedosin/Similarity of matter levels|Similarity of matter levels]]
* [[Physics/Essays/Fedosin/SPФ symmetry|SPФ symmetry]]
* [[Physics/Essays/Fedosin/Quantization of parameters of cosmic systems |Quantization of parameters of cosmic systems]]
* [[Physics/Essays/Fedosin/Hydrogen system |Hydrogen system]]
* [[Physics/Essays/Fedosin/Strong gravitational constant|Strong gravitational constant]]
* [[Charges/Interactions/Strong|Strong interaction]]
* [[Gravitational torsion field]]
* [[Physics/Essays/Fedosin/Gravitational model of strong interaction |gravitational model of strong interaction]]
* [[Physics/Essays/Fedosin/Model of quark quasiparticles |Model of quark quasiparticles]]
* [[Physics/Essays/Fedosin/Substantial electron model|Substantial electron model]]
* [[Physics/Essays/Fedosin/Substantial neutron model|Substantial neutron model]]
* [[Physics/Essays/Fedosin/Substantial proton model|Substantial proton model]]
* [[Physics/Essays/Fedosin/Electrogravitational vacuum |Electrogravitational vacuum]]

== External links ==
* [http://www.wikiznanie.ru/ru-wz/index.php/%D0%A1%D0%B8%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D0%B3%D1%80%D0%B0%D0%B2%D0%B8%D1%82%D0%B0%D1%86%D0%B8%D1%8F Strong gravitation in Russian]


{{Fundamental Forces}}
{{Theories of gravitation}}

[[Category:Concepts in physics]] [[Category:Particle physics]] [[Category:Theory of infinite nesting of matter]] [[Category:Covariant theory of gravitation]] [[Category: Metric theory of relativity]]</text>
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      <text bytes="18336" sha1="t7rrrini5fz80dnsz1ltd0x7bj9ucwp" xml:space="preserve">'''Strong gravitation''' is [[w:fundamental interaction| fundamental gravitational interaction]] at the level of [[w:elementary particle| elementary particles]], one of the components of the [[Charges/Interactions/Strong|strong interaction]] in physics according to the [[Physics/Essays/Fedosin/Gravitational model of strong interaction |gravitational model of strong interaction]]. It is assumed that strong gravitation and electromagnetic forces are responsible for the formation and integrity of the matter of elementary particles and [[w:atomic nucleus| atomic nuclei]], and also participates in the interactions between electrons and nuclei in atoms and molecules. For describing of strong gravitation equations of [[Physics/Essays/Fedosin/Lorentz-invariant theory of gravitation |Lorentz-invariant theory of gravitation]] are used.

==History==
After the discovery of the [[w:electron |electron]] in 1897, of the [[w:proton| proton]] in 1919, and of the [[w:neutron |neutron]] in 1932, and of their compositions in the form of atomic nuclei, atoms and molecules, it became necessary to describe the forces acting between the particles and binding their matter. In most cases, the behavior of the electron and proton, placed in the external electromagnetic field, is satisfactorily described by electromagnetic forces. This led to the standard electromagnetic model of the atom. As for the interaction of [[w:nucleon| nucleon]]s in atomic nuclei, the hypothesis of the Japanese physicist H. Yukawa was initially accepted about the binding between the particles by means of [[w:meson| meson]]s, mostly of [[w:pion| pion]]s. Then, in the framework of the quark theory all hadrons began to be considered to be composed of [[w:quark| quark]]s.

However the idea, that the fundamental interaction between a set of elementary particles must occur due to the action of another set of elementary particles, belongs to the atomistic theory, but it contradicts the Theory of [[Physics/Essays/Fedosin/Infinite Hierarchical Nesting of Matter|Infinite Hierarchical Nesting of Matter]]. Indeed, the reactions between elementary particles follow the laws of conservation of energy, momentum and electric charge; the matter, energy-momentum and the charge of one type of particles transforms into the corresponding quantities of other particles, but this does not mean that the carrier and the cause of the interactions are the elementary particles themselves. The interaction of nucleons with each other by means of pions hardly agrees with quarks and [[w:gluon |gluon]]s, which are used to describe the integrity of hadrons, due to the problem of non-observability of quarks in the free state and the uncertainty of transformation of the forces between the quarks inside each of the nucleons into the [[Charges/Interactions/Strong|strong interaction]] between different nucleons in the atomic nucleus. The introduction of [[w:virtual particle |virtual particle]]s with their exotic properties (short lifetime, the simultaneous generation of particles and antiparticles, etc.) does not save the situation. Thus, the abstract explanation of the [[Charges/Interactions/Electromagnetics|electromagnetic interaction]] of two charges with the help of virtual [[w:photon |photon]]s as the field quanta still remains the statement which is not supported by the concrete model of the interaction process.

Among the attempts to explain the strong interaction in connection with gravitation there is a hypothesis that in the model of hadronic bags the hadrons are de Sitter microuniverses, in which quarks are enclosed. The radius of hadrons corresponding to the radii of these microuniverses, is associated with the strong gravitational constant and the corresponding [[w:cosmological constant| cosmological constant]].&lt;ref&gt;Salam, A., and Strathdee, J. Confinement Through Tensor Gauge Fields. Physical Review D, 1978, Vol.18, Issue 12, P. 4596-4609. &lt;/ref&gt; To explain the properties of hadrons in the assumption of strong gravitational interaction the analogies between hadrons and Kerr — Newman [[black hole]]s are described. &lt;ref&gt; Sivaram, C. and Sinha, K.P. Strong gravity, black holes, and hadrons. Physical Review D, 1977, Vol. 16, Issue 6, P. 1975-1978. &lt;/ref&gt; &lt;ref&gt;Recami, E. and Castorina, P. On Quark Confinement: Hadrons as «Strong Black- Holes». Letters Nuovo Cimento, 1976, Vol. 15, No 10, P. 347-350. &lt;/ref&gt; &lt;ref&gt; Pavsic, M. (1978). Unified Theory Of Strong And Gravitational Interactions. Nuovo Cimento B, Vol. 48, P. 205-253. &lt;/ref&gt; &lt;ref&gt; Oldershaw R. L. [http://arxiv.org/abs/astro-ph/0701006v4 Hadrons as Kerr-Newman Black Holes.]  arXiv:astro-ph/0701006v4, 30 Dec 2006. &lt;/ref&gt;

In 1999 [[User:Fedosin | Sergey Fedosin]], based on the [[Physics/Essays/Fedosin/Similarity of matter levels|similarity of matter levels]], [[Physics/Essays/Fedosin/SPФ symmetry|SPФ symmetry]] and [[w:Le Sage's theory of gravitation| Le Sage's theory of gravitation]], according to which black holes are not admitted, postulated the existence of strong gravitation as the fundamental force at the atomic level and found the value of [[Physics/Essays/Fedosin/Strong gravitational constant|strong gravitational constant]] &lt;math&gt;~\Gamma= 1{.}514 \cdot 10^{29}&lt;/math&gt; m&lt;sup&gt;3&lt;/sup&gt;•s&lt;sup&gt;–2&lt;/sup&gt;•kg&lt;sup&gt;–1&lt;/sup&gt;. &lt;ref name=&quot;fed&quot;&gt;Fedosin S.G. [http://lccn.loc.gov/2009457349 Fizika i filosofiia podobiia ot preonov do metagalaktik], Perm, pages 544, 1999. {{ISBN|5-8131-0012-1}}. &lt;/ref&gt;

== Applications ==
=== Hadrons ===
The equality between the rest energy of the proton and its total energy, which due to the [[w:virial theorem| virial theorem]] is approximately equal to the half of the [[w:potential energy| potential energy]] of the strong gravitational field, allows us to estimate the radius of the proton &lt;math&gt;~R_p &lt;/math&gt;:
: &lt;math&gt; m_p c^2 = \frac{ k \Gamma {m_p}^2}{ 2 R_p },&lt;/math&gt;
: &lt;math&gt; R_p = \frac{ k \Gamma m_p}{2 c^2 }=0.87 \cdot 10^{-15}&lt;/math&gt; m,

here &lt;math&gt;~ m_p &lt;/math&gt; is the proton mass, &lt;math&gt;~ c &lt;/math&gt; is the [[w:speed of light |speed of light]], &lt;math&gt;~ k &lt;/math&gt; is the coefficient depending on the distribution of matter, in the case of the uniform mass density of the proton &lt;math&gt;~ k=0.6 &lt;/math&gt;. According to the self-consistent model &lt;ref name=com&gt; [http://sergf.ru/com.htm Comments to the book:] Fedosin S.G. Fizicheskie teorii i beskonechnaia vlozhennost’ materii. – Perm, 2009, 844 pages, {{ISBN|978-5-9901951-1-0}}. (in Russian).&lt;/ref&gt; &lt;ref&gt; Fedosin S.G. [http://vixra.org/abs/1208.0006 The radius of the proton in the self-consistent model.] Hadronic Journal, Vol. 35, No. 4, pp. 349-363 (2012). http://dx.doi.org/10.5281/zenodo.889451. &lt;/ref&gt; for the proton &lt;math&gt;~ k=0.62 &lt;/math&gt;.

The obtained value &lt;math&gt;~R_p &lt;/math&gt; coincides with the experimentally obtained sizes of the proton and the neutron, &lt;ref&gt;Hofstadter, Robert, [http://nobelprize.org/nobel_prizes/physics/laureates/1961/hofstadter-lecture.pdf The electron-scattering method and its application to the structure of nuclei and nucleons,] Nobel Lecture (December 11, 1961).&lt;/ref&gt; confirming the validity of the idea of strong gravitation. At the same time the given equality implies the explanation of the essence of the rest energy of bodies as the energy associated with the strong gravitation of the nucleons of the bodies’ matter. According to the [[w:mass–energy equivalence |mass–energy equivalence]], the rest energy of the nucleon is proportional to its mass. On the other hand, the total energy of the nucleon includes the energy of the strong gravitational field which is proportional to the squared mass, and the internal energy of the nucleon matter which is proportional to the matter mass in the expression for the [[w:kinetic energy |kinetic energy]]. As a result, the total energy is proportional only to the mass just as well as the rest energy.

On the comparison of the maximum angular momentum of the strong gravitational field and the angular momentum of the proton with the uniform matter distribution another estimate of the proton radius is based: &lt;ref&gt;[[User:Fedosin |Fedosin S.G.]] [http://lccn.loc.gov/2009457352 Sovremennye problemy fiziki: v poiskakh novykh printsipov,] Editorial URSS, Moskva (2002). &lt;/ref&gt; 

: &lt;math&gt; R_p = \frac{ 5 \Gamma m_p}{21 c^2 }=0.67 \cdot 10^{-15}&lt;/math&gt; m.

As the model of emerging of strong gravitation the modernized [[w:Le Sage's theory of gravitation |Le Sage's theory of gravitation]] is used, which becomes universal taking into account the Theory of [[Physics/Essays/Fedosin/Infinite Hierarchical Nesting of Matter|Infinite Hierarchical Nesting of Matter]]. &lt;ref &gt; Fedosin S.G. [http://sergf.ru/mgen.htm  Model of Gravitational Interaction in the Concept of Gravitons].  Journal of Vectorial Relativity, Vol. 4, No. 1, pp.1-24 (2009). http://dx.doi.org/10.5281/zenodo.890886.&lt;/ref&gt; &lt;ref&gt; Fedosin S.G. [http://vixra.org/abs/1503.0127 The graviton field as the source of mass and gravitational force in the modernized Le Sage’s model.] Physical Science International Journal, ISSN: 2348-0130, Vol. 8, Issue 4, pp. 1-18 (2015). http://dx.doi.org/10.9734/PSIJ/2015/22197. &lt;/ref&gt;

At the stellar [[Physics/Essays/Fedosin/Similarity of matter levels|scale level of matter]] the analogues of nucleons are [[Physics/Essays/Fedosin/Neutron star|neutron star]]s, the integrity of which is maintained by the ordinary force of gravitation and the pressure force in the matter arising from the repulsion of the nucleons from each other. Similarly, in the matter of nucleons the compensation of the strong gravitation and the internal pressure force takes place (see the [[Physics/Essays/Fedosin/Substantial neutron model|substantial neutron model]] and the [[Physics/Essays/Fedosin/Substantial proton model|substantial proton model]]). In this picture, for the stability of nucleons and describing their properties [[w:quark |quark]]s are not required, in contrast to the standard [[w:quantum chromodynamics| quantum chromodynamics]]. At the same time in the [[Physics/Essays/Fedosin/Model of quark quasiparticles |model of quark quasiparticles]] the quarks are seen not as real particles inside hadrons but as quasiparticles, the constituent elements of the hadrons’ matter which carry the mass, charge and magnetic moment. This ensures the observed symmetry of hadron properties. In turn, the quarks themselves can be reduced to the combinations of two hadronic phases of the matter. &lt;ref name=&quot;fs1&quot;&gt; Sergey Fedosin, [https://www.lap-publishing.com/catalog/details//store/gb/book/978-3-659-57301-9/the-physical-theories-and-infinite-hierarchical-nesting-of-matter-volume-1 The physical theories and infinite hierarchical nesting of matter], Volume 1, LAP LAMBERT Academic Publishing, pages: 580, {{ISBN|978-3-659-57301-9}}.&lt;/ref&gt; The analysis of the Regge hadron families also shows that they can be explained by taking into account the quantization of the [[w:spin| spin]] and the matter state of the particles, retained by the strong gravitational field.

=== Electron ===
Strong gravitation significantly affects the construction of the model of the electron, leading to the [[Physics/Essays/Fedosin/Substantial electron model|substantial model]] of this particle. In particular, the electron charge is so large, that strong gravitation is not able to keep the electron matter from the Coulomb electric force of repulsion of the charges. Therefore, the stability of the electron in the atom is possible only in the form of a scattered electron cloud (disc) and due to the forces of attraction to the nucleus from strong gravitation and the nuclear charge. Another fact, the quantization of energy levels and of the orbital angular momentum of the electron in the atom, is explained based on the condition that the flux of kinetic energy of the motion of electron matter around the nucleus is equal to the sum of the energy fluxes from the strong gravitation and the electromagnetic field. &lt;ref name=&quot;fs1&quot;/&gt; This leads to the stationary states of the electron in the atom, in which it does not produce emission. For the hydrogen atom it is also found that the magnetic energy of the nucleus in the magnetic field of the electron equals the energy of the nucleus’ spin in the [[gravitational torsion field|torsion field]] of strong gravitation of the electron in case of limiting rotation of the nucleus. &lt;ref name=com/&gt;

=== The interaction of nucleons in the atomic nucleus===
The experiments with the scattering of nucleons on each other allow us to estimate the effective potential of [[Charges/Interactions/Strong|strong interaction]] acting between these particles. &lt;ref&gt; Ishii N., Aoki S., Hatsuda T. [http://arxiv.org/abs/nucl-th/0611096v1 The Nuclear Force from Lattice QCD]. – arXiv: nucl-th / 0611096 v1, 28 Nov 2006. &lt;/ref&gt; As the distance decreases the interaction force increases rapidly. To describe this force the [[Physics/Essays/Fedosin/Gravitational model of strong interaction |gravitational model of strong interaction]] is used, in which the nuclear forces are the sum of the attraction from the strong gravitation, the repulsion of the nucleon spins due to the torsion field of strong gravitation, as well as from the action of electromagnetic forces. At short distances, the repulsive force of the spins dominates, which is inversely proportional to the fourth and then the fifth degree of the distance. At large distances, there is attraction of the nucleons, mainly from the strong gravitation. At distances close to the radius of the nucleon, the neutron and the proton are in the equilibrium state, which gives the [[w:deuteron |deuteron]] as the simplest atomic nucleus with two nucleons. &lt;ref name=&quot;fs1&quot;/&gt; Taking into account the strong gravitation allows us to construct the model of simplest nuclei and their geometric configuration, as well as to explain the dependence of the specific binding energy of atomic nuclei on their atomic number due to the saturation effect of the strong gravitational energy and the increase of the electrical repulsive energy of protons.

=== Strange particles===
In quantum chromodynamics, it is assumed that the long lifetime, inherent in some hadrons, is due to the presence of strange quarks in them. However, the models of strange particles can be constructed similarly to the models of atomic nuclei, by connecting nucleons and mesons under the influence of strong gravitation. &lt;ref name=com/&gt; The composition of some strange hadrons is described in the [[Physics/Essays/Fedosin/Model of quark quasiparticles |model of quark quasiparticles]].

=== Interatomic interaction===
The interaction of atoms leads to the formation of molecules, as well as simple and molecular substances. In contrast to the nucleons in atomic nuclei, in the interaction of atoms the strong gravitation acts between the nuclei of all atoms as well as between the electrons, complementing the electromagnetic forces. In this case the electron discs, surrounding the atomic nuclei, due to the rapid rotation in them of the matter, which is charged and oriented by the magnetic field, have the possibility to shield the gravitational forces between the nuclei, reducing them to the level of electrical forces. The equilibrium of atoms in molecules and in substances is achieved in case of the balance of gravitational and electromagnetic forces. With increasing of the distance between the atoms, the so-called [[w:Van der Waals force |Van der Waals force]] occurs between them in the form of the attraction rapidly decreasing with the distance. The estimate with the help of the Le Sage's theory of gravitation gives the radius of action of strong gravitation in the matter with the density of the order of Earth's density, about 0.7 m. &lt;ref name=&quot;fs1&quot;/&gt;

=== Photon ===
The [[Physics/Essays/Fedosin/Substantial photon model|substantial photon model]] assumes that the photon consists of [[Physics/Essays/Fedosin/Praon|praon]]s bonded to each other by means of strong gravitation.&lt;ref name=cc&gt;Fedosin S.G. [https://www.ajol.info/index.php/jfas/article/view/168204  The charged component of the vacuum field as the source of electric force in the modernized Le Sage’s model.] Journal of Fundamental and Applied Sciences, Vol. 8, No. 3, pp. 971-1020 (2016). http://dx.doi.org/10.4314/jfas.v8i3.18, https://dx.doi.org/10.5281/zenodo.845357. &lt;/ref&gt; &lt;ref name=sub&gt;Fedosin S.G. [https://www.ajol.info/index.php/jfas/article/view/151053 The substantial model of the photon]. Journal of Fundamental and Applied Sciences, Vol. 9, No. 1, pp. 411-467 (2017). http://dx.doi.org/10.4314/jfas.v9i1.25. &lt;/ref&gt; This leads to the fact that the photon has a rest mass, as well as a magnetic moment.

==References==
{{reflist}}

== See also ==
* [[Physics/Essays/Fedosin/Infinite Hierarchical Nesting of Matter|Infinite Hierarchical Nesting of Matter]]
* [[Physics/Essays/Fedosin/Similarity of matter levels|Similarity of matter levels]]
* [[Physics/Essays/Fedosin/SPФ symmetry|SPФ symmetry]]
* [[Physics/Essays/Fedosin/Quantization of parameters of cosmic systems |Quantization of parameters of cosmic systems]]
* [[Physics/Essays/Fedosin/Hydrogen system |Hydrogen system]]
* [[Physics/Essays/Fedosin/Strong gravitational constant|Strong gravitational constant]]
* [[Charges/Interactions/Strong|Strong interaction]]
* [[Gravitational torsion field]]
* [[Physics/Essays/Fedosin/Gravitational model of strong interaction |Gravitational model of strong interaction]]
* [[Physics/Essays/Fedosin/Model of quark quasiparticles |Model of quark quasiparticles]]
* [[Physics/Essays/Fedosin/Substantial electron model|Substantial electron model]]
* [[Physics/Essays/Fedosin/Substantial neutron model|Substantial neutron model]]
* [[Physics/Essays/Fedosin/Substantial proton model|Substantial proton model]]
* [[Physics/Essays/Fedosin/Electrogravitational vacuum |Electrogravitational vacuum]]

== External links ==
* [http://www.wikiznanie.ru/ru-wz/index.php/%D0%A1%D0%B8%D0%BB%D1%8C%D0%BD%D0%B0%D1%8F_%D0%B3%D1%80%D0%B0%D0%B2%D0%B8%D1%82%D0%B0%D1%86%D0%B8%D1%8F Strong gravitation in Russian]


{{Fundamental Forces}}
{{Theories of gravitation}}

[[Category:Concepts in physics]] [[Category:Particle physics]] [[Category:Theory of infinite nesting of matter]] [[Category:Covariant theory of gravitation]] [[Category: Metric theory of relativity]]</text>
      <sha1>t7rrrini5fz80dnsz1ltd0x7bj9ucwp</sha1>
    </revision>
  </page>
  <page>
    <title>Wikiversity:Newsletters/Tech News</title>
    <ns>4</ns>
    <id>162205</id>
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      <id>2692218</id>
      <parentid>2691083</parentid>
      <timestamp>2024-12-16T22:24:50Z</timestamp>
      <contributor>
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        <id>983498</id>
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      <comment>/* Tech News: 2024-51 */ new section</comment>
      <origin>2692218</origin>
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      <format>text/x-wiki</format>
      <text bytes="389126" sha1="omfvcmbcwty47ui2wxc1v2scgjy5lm6" xml:space="preserve">{{Archive box|[[/2014/]] · [[/2015/]] · [[/2016/]] · [[/2017/]] · [[/2018/]] · [[/2019/]] · [[/2020/]] · [[/2021/]] · [[/2022/]]}}
__TOC__
{{Clear}}

== Tech News: 2023-02 ==

&lt;section begin=&quot;technews-2023-W02&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/02|Translations]] are available.

'''Recent changes'''
* You can use tags to filter edits in the recent changes feed or on your watchlist. You can now use tags to filter out edits you don't want to see. Previously you could only use tags to focus on the edits with those tags. [https://phabricator.wikimedia.org/T174349]
* [[Special:WhatLinksHere|Special:WhatLinksHere]] shows all pages that link to a specific page. There is now a [https://wlh.toolforge.org prototype] for how to sort those pages alphabetically. You can see the discussion in the [[phab:T4306|Phabricator ticket]].
* You can now use the [[mw:Special:MyLanguage/Extension:Thanks|thanks]] function on your watchlist and the user contribution page. [https://phabricator.wikimedia.org/T51541]
* A wiki page can be moved to give it a new name. You can now get a dropdown menu with common reasons when you move a page. This is so you don't have to write the explanation every time. [https://phabricator.wikimedia.org/T325257]
* [[m:Special:MyLanguage/Matrix.org|Matrix]] is a chat tool. You can now use &lt;code&gt;matrix:&lt;/code&gt; to create Matrix links on wiki pages. [https://phabricator.wikimedia.org/T326021]
* You can filter out translations when you look at the recent changes on multilingual wikis. This didn't hide translation pages. You can now also hide subpages which are translation pages. [https://phabricator.wikimedia.org/T233493]

'''Changes later this week'''
* [[m:Special:MyLanguage/Real Time Preview for Wikitext|Realtime preview for wikitext]] is a tool which lets editors preview the page when they edit wikitext. It will be enabled for all users of the 2010 wikitext editor. You will find it in the editor toolbar.
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2023-01-10|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s5.dblist targeted wikis]) and on {{#time:j xg|2023-01-12|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s6.dblist targeted wikis]).
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.18|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-01-10|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-01-11|en}}. It will be on all wikis from {{#time:j xg|2023-01-12|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/02|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W02&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:07, 10 January 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24342971 --&gt;

== Tech News: 2023-03 ==

&lt;section begin=&quot;technews-2023-W03&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/03|Translations]] are available.

'''Problems'''
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] The URLs in &quot;{{int:last}}&quot; links on page history now contain &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;diff=prev&amp;oldid=[revision ID]&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; in place of &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;diff=[revision ID]&amp;oldid=[revision ID]&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. This is to fix a problem with links pointing to incorrect diffs when history was filtered by a tag. Some user scripts may break as a result of this change. [https://phabricator.wikimedia.org/T243569]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.19|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-01-17|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-01-18|en}}. It will be on all wikis from {{#time:j xg|2023-01-19|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* Some [[mw:Special:MyLanguage/Talk pages project/Usability|changes to the appearance of talk pages]] have only been available on &lt;code&gt;{{ns:1}}:&lt;/code&gt; and &lt;code&gt;{{ns:3}}:&lt;/code&gt; namespaces.  These will be extended to other talk namespaces, such as &lt;code&gt;{{ns:5}}:&lt;/code&gt;.  They will continue to be unavailable in non-talk namespaces, including &lt;code&gt;{{ns:4}}:&lt;/code&gt; pages (e.g., at the Village Pump).  You can [[Special:Preferences#mw-prefsection-editing-discussion|change your preferences]] ([[Special:Preferences#mw-prefsection-betafeatures|beta feature]]). [https://phabricator.wikimedia.org/T325417]
*On Wikisources, when an image is zoomed or panned in the Page: namespace, the same zoom and pan settings will be remembered for all Page: namespace pages that are linked to a particular Index: namespace page. [https://gerrit.wikimedia.org/r/c/mediawiki/extensions/ProofreadPage/+/868841]
* The Vector 2022 skin will become the default for the English Wikipedia desktop users. The change will take place on January 18 at 15:00 UTC. [[:en:w:Wikipedia:Vector 2022|Learn more]].

'''Future changes'''
* The 2023 edition of the [[m:Special:MyLanguage/Community Wishlist Survey 2023|Community Wishlist Survey]], which invites contributors to make technical proposals and vote for tools and improvements, starts next week on 23 January 2023 at 18:00 UTC. You can start drafting your proposals in [[m:Community Wishlist Survey/Sandbox|the CWS sandbox]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/03|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W03&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:10, 17 January 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24381020 --&gt;

== Tech News: 2023-04 ==

&lt;section begin=&quot;technews-2023-W04&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/04|Translations]] are available.

'''Problems'''
* Last week, for ~15 minutes, all wikis were unreachable for logged-in users and non-cached pages. This was caused by a timing issue. [https://wikitech.wikimedia.org/wiki/Incidents/2023-01-17_MediaWiki]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.20|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-01-24|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-01-25|en}}. It will be on all wikis from {{#time:j xg|2023-01-26|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* If you have the Beta Feature for [[mw:Special:MyLanguage/Talk pages project|DiscussionTools]] enabled, the appearance of talk pages will add more information about discussion activity. [https://www.mediawiki.org/wiki/Special:MyLanguage/Talk_pages_project/Usability#Status][https://phabricator.wikimedia.org/T317907]
* The 2023 edition of the [[m:Special:MyLanguage/Community Wishlist Survey 2023|Community Wishlist Survey]] (CWS), which invites contributors to make technical proposals and vote for tools and improvements, starts on Monday 23 January 2023 at [https://zonestamp.toolforge.org/1674496814 18:00 UTC]. 

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/04|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W04&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:46, 23 January 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24418874 --&gt;

== Tech News: 2023-05 ==

&lt;section begin=&quot;technews-2023-W05&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/05|Translations]] are available.

'''Problems'''
* Last week, for ~15 minutes, some users were unable to log in or edit pages. This was caused by a problem with session storage. [https://wikitech.wikimedia.org/wiki/Incidents/2023-01-24_sessionstore_quorum_issues]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.21|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-01-31|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-02-01|en}}. It will be on all wikis from {{#time:j xg|2023-02-02|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).

'''Future changes'''
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Wikis that use localized numbering schemes for references need to add new CSS. This will help to show citation numbers the same way in all reading and editing modes. If your wiki would prefer to do it yourselves, please see the [[mw:Special:MyLanguage/Parsoid/Parser Unification/Cite CSS|details and example CSS to copy from]], and also add your wiki to the list. Otherwise, the developers will directly help out starting the week of February 5.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/05|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W05&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:05, 31 January 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24455949 --&gt;

== Tech News: 2023-06 ==

&lt;section begin=&quot;technews-2023-W06&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/06|Translations]] are available.

'''Recent changes'''
* In the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Vector 2022 skin]], logged-out users using the full-width toggle will be able to see the setting of their choice even after refreshing pages or opening new ones. This only applies to wikis where Vector 2022 is the default. [https://phabricator.wikimedia.org/T321498]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.22|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-02-07|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-02-08|en}}. It will be on all wikis from {{#time:j xg|2023-02-09|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* Previously, we announced when some wikis would be in read-only for a few minutes because of a switch of their main database. These switches will not be announced any more, as the read-only time has become non-significant. Switches will continue to happen at 7AM UTC on Tuesdays and Thursdays. [https://phabricator.wikimedia.org/T292543#8568433]
* Across all the wikis, in the Vector 2022 skin, logged-in users will see the page-related links such as &quot;What links here&quot; in a [[mw:Special:MyLanguage/Reading/Web/Desktop_Improvements/Features/Page_tools|new side menu]]. It will be displayed on the other side of the screen. This change had previously been made on Czech, English, and Vietnamese Wikipedias. [https://phabricator.wikimedia.org/T328692]
*[[m:Special:MyLanguage/Community Wishlist Survey 2023|Community Wishlist Survey 2023]] will stop receiving new proposals on [https://zonestamp.toolforge.org/1675706431 Monday, 6 February 2023, at 18:00 UTC]. Proposers should complete any edits by then, to give time for [[m:Special:MyLanguage/Community_Wishlist_Survey/Help_us|translations]] and review. Voting will begin on Friday, 10 February.

'''Future changes'''
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Gadgets and user scripts will be changing to load on desktop and mobile sites. Previously they would only load on the desktop site. It is recommended that wiki administrators audit the [[MediaWiki:Gadgets-definition|gadget definitions]] prior to this change, and add &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;skins=…&lt;/code&gt;&lt;/bdi&gt; for any gadgets which should not load on mobile. [https://phabricator.wikimedia.org/T328610 More details are available].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/06|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W06&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 10:21, 6 February 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24491749 --&gt;

== Tech News: 2023-07 ==

&lt;section begin=&quot;technews-2023-W07&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/07|Translations]] are available.

'''Problems'''
* On wikis where patrolled edits are enabled, changes made to the [[mw:Special:MyLanguage/Growth/Communities/How to configure the mentors' list|mentor list]] by autopatrolled mentors are not correctly marked as patrolled. It will be fixed later this week. [https://phabricator.wikimedia.org/T328444]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.23|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-02-14|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-02-15|en}}. It will be on all wikis from {{#time:j xg|2023-02-16|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* The Reply tool and other parts of [[mw:Special:MyLanguage/Help:DiscussionTools#Mobile|DiscussionTools]] will be deployed for all editors using the mobile site.  You can [[mw:Special:MyLanguage/Talk_pages_project/Mobile#Status_Updates|read more about this decision]]. [https://phabricator.wikimedia.org/T298060]

'''Future changes'''
* All wikis will be read-only for a few minutes on March 1. This is planned for [https://zonestamp.toolforge.org/1677679222 14:00 UTC]. More information will be published in Tech News and will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T328287][https://phabricator.wikimedia.org/T327920][https://wikitech.wikimedia.org/wiki/Deployments]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/07|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W07&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:48, 14 February 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24540832 --&gt;

== Tech News: 2023-08 ==

&lt;section begin=&quot;technews-2023-W08&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/08|Translations]] are available.

'''Problems'''
* Last week, during planned maintenance of Cloud Services, unforeseen complications forced the team to turn off all tools for 2–3 hours to prevent data corruption. Work is ongoing to prevent similar problems in the future. [https://phabricator.wikimedia.org/T329535] 

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.23|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-02-21|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-02-22|en}}. It will be on all wikis from {{#time:j xg|2023-02-23|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
*The voting phase for the [[m:Special:MyLanguage/Community Wishlist Survey 2023|Community Wishlist Survey 2023]] ends on [https://zonestamp.toolforge.org/1677261621 24 February at 18:00 UTC]. The results of the survey will be announced on 28 February.

'''Future changes'''
* All wikis will be read-only for a few minutes on March 1. This is planned for [https://zonestamp.toolforge.org/1677679222 14:00 UTC]. More information will be published in Tech News and will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T328287][https://phabricator.wikimedia.org/T327920][https://wikitech.wikimedia.org/wiki/Deployments]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/08|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W08&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:57, 21 February 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24570514 --&gt;

== Tech News: 2023-09 ==

&lt;section begin=&quot;technews-2023-W09&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/09|Translations]] are available.

'''Problems'''
* Last week, in some areas of the world, there were problems with loading pages for 20 minutes and saving edits for 55 minutes. These issues were caused by a problem with our caching servers due to unforseen events during a routine maintenance task. [https://wikitech.wikimedia.org/wiki/Incidents/2023-02-22_wiki_outage][https://wikitech.wikimedia.org/wiki/Incidents/2023-02-22_read_only] 

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.25|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-02-28|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-03-01|en}}. It will be on all wikis from {{#time:j xg|2023-03-02|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* All wikis will be read-only for a few minutes on March 1. This is planned for [https://zonestamp.toolforge.org/1677679222 14:00 UTC]. [https://meta.wikimedia.org/wiki/Special:MyLanguage/Tech/Server_switch]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/09|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W09&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:47, 27 February 2023 (UTC)
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== Tech News: 2023-10 ==

&lt;section begin=&quot;technews-2023-W10&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/10|Translations]] are available.

'''Recent changes'''
* The Community Wishlist Survey 2023 edition has been concluded. Community Tech has [[m:Special:MyLanguage/Community Wishlist Survey 2023/Results|published the results]] of the survey and will provide an update on what is next in April 2023.
* On wikis which use [[mw:Special:MyLanguage/Writing_systems|LanguageConverter]] to handle multiple writing systems, articles which used custom conversion rules in the wikitext (primarily on Chinese Wikipedia) would have these rules applied inconsistently in the table of contents, especially in the Vector 2022 skin. This has now been fixed. [https://phabricator.wikimedia.org/T306862]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.26|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-03-07|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-03-08|en}}. It will be on all wikis from {{#time:j xg|2023-03-09|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* A search system has been added to the [[Special:Preferences|Preferences screen]]. This will let you find different options more easily. Making it work on mobile devices will happen soon. [https://phabricator.wikimedia.org/T313804]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/10|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W10&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:49, 6 March 2023 (UTC)
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== Tech News: 2023-11 ==

&lt;section begin=&quot;technews-2023-W11&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/11|Translations]] are available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.27|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-03-14|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-03-15|en}}. It will be on all wikis from {{#time:j xg|2023-03-16|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* Starting on Wednesday, a new set of Wikipedias will get &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]&quot; ({{int:project-localized-name-cbk_zamwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-cdowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-cewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-cebwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-chwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-chrwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-chywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ckbwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-cowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-csbwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-cuwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-cvwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-cywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-itwiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T304542][https://phabricator.wikimedia.org/T304550] 

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/11|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W11&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:20, 13 March 2023 (UTC)
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== Tech News: 2023-12 ==

&lt;section begin=&quot;technews-2023-W12&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/12|Translations]] are available.

'''Problems'''
* Last week, some users experienced issues loading image thumbnails. This was due to incorrectly cached images. [https://phabricator.wikimedia.org/T331820]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.1|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-03-21|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-03-22|en}}. It will be on all wikis from {{#time:j xg|2023-03-23|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] A link to the user's [[{{#special:CentralAuth}}]] page will appear on [[{{#special:Contributions}}]] — some user scripts which previously added this link may cause conflicts. This feature request was [[:m:Community Wishlist Survey 2023/Admins and patrollers/Add link to CentralAuth on Special:Contributions|voted #17 in the 2023 Community Wishlist Survey]].
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] The [[{{#special:AbuseFilter}}]] edit window will be resizable and larger by default. This feature request was [[:m:Community Wishlist Survey 2023/Anti-harassment/Make the AbuseFilter edit window resizable and larger by default|voted #80 in the 2023 Community Wishlist Survey]].
* There will be a new option for Administrators when they are unblocking a user, to add the unblocked user’s user page to their watchlist. This will work both via [[{{#special:Unblock}}]] and via the API. [https://phabricator.wikimedia.org/T257662]

'''Meetings'''
* You can join the next meeting with the Wikipedia mobile apps teams. During the meeting, we will discuss the current features and future roadmap. The meeting will be on [https://zonestamp.toolforge.org/1679677204 24 March at 17:00 (UTC)]. See [[mw:Special:MyLanguage/Wikimedia Apps/Office Hours|details and how to join]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/12|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W12&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:25, 21 March 2023 (UTC)
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== Tech News: 2023-13 ==

&lt;section begin=&quot;technews-2023-W13&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/13|Translations]] are available.

'''Recent changes'''
* The [[:mw:Special:MyLanguage/Extension:AbuseFilter|AbuseFilter]] condition limit was increased from 1000 to 2000. [https://phabricator.wikimedia.org/T309609]
* [[:m:Special:MyLanguage/Global AbuseFilter#Locally disabled actions|Some Global AbuseFilter]] actions will no longer apply to local projects. [https://phabricator.wikimedia.org/T332521]
* Desktop users are now able to subscribe to talk pages by clicking on the {{int:discussiontools-newtopicssubscription-button-subscribe-label}} link in the {{int:toolbox}} menu. If you subscribe to a talk page, you receive [[mw:Special:MyLanguage/Notifications|notifications]] when new topics are started on that talk page. This is separate from putting the page on your watchlist or subscribing to a single discussion. [https://phabricator.wikimedia.org/T263821]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.2|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-03-28|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-03-29|en}}. It will be on all wikis from {{#time:j xg|2023-03-30|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).

'''Future changes'''
* You will be able to choose [[mw:Special:MyLanguage/VisualEditor/Diffs|visual diffs]] on all [[m:Special:MyLanguage/Help:Page history|history pages]] at the Wiktionaries and Wikipedias. [https://phabricator.wikimedia.org/T314588]
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] The legacy [[mw:Mobile Content Service|Mobile Content Service]] is going away in July 2023. Developers are encouraged to switch to Parsoid or another API before then to ensure service continuity. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/4MVQQTONJT7FJAXNVOFV3WWVVMCHRINE/]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/13|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W13&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:13, 28 March 2023 (UTC)
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== Tech News: 2023-14 ==

&lt;section begin=&quot;technews-2023-W14&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/14|Translations]] are available.

'''Recent changes'''
* The system for automatically creating categories for the [[mw:Special:MyLanguage/Extension:Babel|Babel]] extension has had several important changes and fixes. One of them allows you to insert templates for automatic category descriptions on creation, allowing you to categorize the new categories. [https://phabricator.wikimedia.org/T211665][https://phabricator.wikimedia.org/T64714][https://phabricator.wikimedia.org/T170654][https://phabricator.wikimedia.org/T184941][https://phabricator.wikimedia.org/T33074]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.3|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-04-04|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-04-05|en}}. It will be on all wikis from {{#time:j xg|2023-04-06|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* Some older [[w:en:Web browser|Web browsers]] will stop being able to use [[w:en:JavaScript|JavaScript]] on Wikimedia wikis from this week. This mainly affects users of Internet Explorer 11. If you have an old web browser on your computer you can try to upgrade to a newer version. [https://phabricator.wikimedia.org/T178356]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The deprecated &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;jquery.hoverIntent&lt;/code&gt;&lt;/bdi&gt; module has been removed. This module could be used by gadgets and user scripts, to create an artificial delay in how JavaScript responds to a hover event. Gadgets and user scripts should now use jQuery &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;hover()&lt;/code&gt;&lt;/bdi&gt; or &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;on()&lt;/code&gt;&lt;/bdi&gt; instead. Examples can be found in the [[mw:Special:MyLanguage/ResourceLoader/Migration_guide_(users)#jquery.hoverIntent|migration guide]]. [https://phabricator.wikimedia.org/T311194]
* Some of the links in [[{{#special:SpecialPages}}]] will be re-arranged. There will be a clearer separation between links that relate to all users, and links related to your own user account. [https://phabricator.wikimedia.org/T333242]
* You will be able to hide the [[mw:Special:MyLanguage/Talk pages project/Replying|Reply button]] in archived discussion pages with a new &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;__ARCHIVEDTALK__&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; magic word.  There will also be a new &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;.mw-archivedtalk&lt;/code&gt;&lt;/bdi&gt; CSS class for hiding the Reply button in individual sections on a page. [https://phabricator.wikimedia.org/T249293][https://phabricator.wikimedia.org/T295553][https://gerrit.wikimedia.org/r/c/mediawiki/extensions/DiscussionTools/+/738221]

'''Future changes'''
* The Vega software that creates data visualizations in pages, such as graphs, will be upgraded to the newest version in the future. Graphs that still use the very old version 1.5 syntax may stop working properly. Most existing uses have been found and updated, but you can help to check, and to update any local documentation. [[phab:T260542|Examples of how to find and fix these graphs are available]]. 

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/14|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W14&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:39, 3 April 2023 (UTC)
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== Tech News: 2023-15 ==

&lt;section begin=&quot;technews-2023-W15&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/15|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] In the visual editor, it is now possible to edit captions of images in galleries without opening the gallery dialog. This feature request was [[:m:Community Wishlist Survey 2023/Editing/Editable gallery captions in Visual Editor|voted #61 in the 2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T190224]
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] You can now receive notifications when another user edits your user page. See the &quot;{{int:Echo-category-title-edit-user-page}}&quot; option in [[Special:Preferences#mw-prefsection-echo|your Preferences]]. This feature request was [[:m:Community Wishlist Survey 2023/Anti-harassment/Notifications for user page edits|voted #3 in the 2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T3876]

'''Problems'''
* There was a problem with all types of CentralNotice banners still being shown to logged-in users even if they had [[Special:Preferences#mw-prefsection-centralnotice-banners|turned off]] specific banner types. This has now been fixed. [https://phabricator.wikimedia.org/T331671]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.4|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-04-11|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-04-12|en}}. It will be on all wikis from {{#time:j xg|2023-04-13|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* Starting on Wednesday, a new set of Wikipedias will get &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]&quot; ({{int:project-localized-name-arywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-dawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-dinwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-dsbwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-eewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-elwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-emlwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-eowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-etwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-euwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-extwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tumwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ffwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-fiwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-fiu_vrowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-fjwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-fowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-frpwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-frrwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-furwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-gawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-gcrwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-gdwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-glwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-glkwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-gnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-gomwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-gotwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-guwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-gvwiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T304551][https://phabricator.wikimedia.org/T308133]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/15|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W15&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:05, 10 April 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24851886 --&gt;

== Tech News: 2023-16 ==

&lt;section begin=&quot;technews-2023-W16&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/16|Translations]] are available.

'''Recent changes'''
* You can now see [[mw:Special:MyLanguage/Help:Extension:Kartographer#Show_nearby_articles|nearby articles on a Kartographer map]] with the button for the new feature &quot;{{int:Kartographer-sidebar-nearbybutton}}&quot;. Six wikis have been testing this feature since October. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Geoinformation/Nearby_articles#Implementation][https://phabricator.wikimedia.org/T334079]
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] The [[m:Special:GlobalWatchlist|Special:GlobalWatchlist]] page now has links for &quot;{{int:globalwatchlist-markpageseen}}&quot; for each entry. This feature request was [[m:Community Wishlist Survey 2023/Notifications, Watchlists and Talk Pages/Button to mark a single change as read in the global watch list|voted #161 in the 2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T334246]

'''Problems'''
* At Wikimedia Commons, some thumbnails have not been getting replaced correctly after a new version of the image is uploaded. This should be fixed later this week. [https://phabricator.wikimedia.org/T331138][https://phabricator.wikimedia.org/T333042]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] For the last few weeks, some external tools had inconsistent problems with logging-in with OAuth. This has now been fixed. [https://phabricator.wikimedia.org/T332650]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.5|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-04-18|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-04-19|en}}. It will be on all wikis from {{#time:j xg|2023-04-20|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/16|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W16&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:54, 18 April 2023 (UTC)
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== Tech News: 2023-17 ==

&lt;section begin=&quot;technews-2023-W17&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/17|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] The date-selection menu on pages such as [[{{#special:Contributions}}]] will now show year-ranges that are in the current and past decade, instead of the current and future decade. This feature request was [[m:Community Wishlist Survey 2023/Miscellaneous/Change year range shown in date selection popup|voted #145 in the 2023 Community Wishlist Survey]].  [https://phabricator.wikimedia.org/T334316]

'''Problems'''
* Due to security issues with the [[mw:Special:MyLanguage/Extension:Graph|Graph extension]], graphs have been disabled in all Wikimedia projects. Wikimedia Foundation teams are working to respond to these vulnerabilities. [https://phabricator.wikimedia.org/T334940]
* For a few days, it was not possible to save some kinds of edits on the mobile version of a wiki. This has been fixed. [https://phabricator.wikimedia.org/T334797][https://phabricator.wikimedia.org/T334799][https://phabricator.wikimedia.org/T334794]

'''Changes later this week'''
* All wikis will be read-only for a few minutes on April 26. This is planned for [https://zonestamp.toolforge.org/1682517653 14:00 UTC]. [https://meta.wikimedia.org/wiki/Special:MyLanguage/Tech/Server_switch]
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.6|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-04-25|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-04-26|en}}. It will be on all wikis from {{#time:j xg|2023-04-27|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).

'''Future changes'''
* The Editing team plans an A/B test for [[mw:Special:MyLanguage/Talk pages project/Usability|a usability analysis of the Talk page project]]. The [[mw:Special:MyLanguage/Talk pages project/Usability/Analysis|planned measurements are available]]. Your wiki [[phab:T332946|may be invited to participate]]. Please suggest improvements to the measurement plan at [[mw:Talk:Talk pages project/Usability|the discussion page]].
* [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2023-2024|The Wikimedia Foundation annual plan 2023-2024 draft is open for comment and input]] until May 19. The final plan will be published in July 2023 on Meta-wiki.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/17|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W17&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:03, 24 April 2023 (UTC)
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== Tech News: 2023-18 ==

&lt;section begin=&quot;technews-2023-W18&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/18|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] The content attribution tools [[mw:Special:MyLanguage/Who Wrote That?|Who Wrote That?]], [[xtools:authorship|XTools Authorship]], and [[xtools:blame|XTools Blame]] now support the French and Italian Wikipedias. More languages will be added in the near future. This is part of the [[m:Community Wishlist Survey 2023/Reading/Extend &quot;Who Wrote That?&quot; tool to more wikis|#7 wish in the 2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T243711][https://phabricator.wikimedia.org/T270490][https://phabricator.wikimedia.org/T334891]
* The [[:commons:Special:MyLanguage/Commons:Video2commons|Video2commons]] tool has been updated. This fixed several bugs related to YouTube uploads. [https://github.com/toolforge/video2commons/pull/162/commits]
* The [[{{#special:Preferences}}]] page has been redesigned on mobile web. The new design makes it easier to browse the different categories and settings at low screen widths. You can also now access the page via a link in the Settings menu in the mobile web sidebar. [https://www.mediawiki.org/wiki/Moderator_Tools/Content_moderation_on_mobile_web/Preferences]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.7|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-05-02|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-05-03|en}}. It will be on all wikis from {{#time:j xg|2023-05-04|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/18|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W18&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:45, 2 May 2023 (UTC)
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== Tech News: 2023-19 ==

&lt;section begin=&quot;technews-2023-W19&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/19|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] When you close an image that is displayed via MediaViewer, it will now return to the wiki page instead of going back in your browser history. This feature request was [[m:Community Wishlist Survey 2023/Reading/Return to the article when closing the MediaViewer|voted #65 in the 2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T236591]
* The [[mw:Special:MyLanguage/Extension:SyntaxHighlight|SyntaxHighlight]] extension now supports &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;wikitext&lt;/code&gt;&lt;/bdi&gt; as a selected language. Old alternatives that were used to highlight wikitext, such as &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;html5&lt;/code&gt;&lt;/bdi&gt;, &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;moin&lt;/code&gt;&lt;/bdi&gt;, and &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;html+handlebars&lt;/code&gt;&lt;/bdi&gt;, can now be replaced. [https://phabricator.wikimedia.org/T29828]
* [[mw:Special:MyLanguage/Manual:Creating pages with preloaded text|Preloading text to new pages/sections]] now supports preloading from localized MediaWiki interface messages. [https://cs.wikipedia.org/wiki/User_talk:Martin_Urbanec_(WMF)?action=edit&amp;section=new&amp;preload=MediaWiki:July Here is an example] at the {{int:project-localized-name-cswiki/en}} that uses &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;preload=MediaWiki:July&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. [https://phabricator.wikimedia.org/T330337]

'''Problems'''
* Graph Extension update: Foundation developers have completed upgrading the visualization software to Vega5. Existing community graphs based on Vega2 are no longer compatible. Communities need to update local graphs and templates, and shared lua modules like &lt;bdi lang=&quot;de&quot; dir=&quot;ltr&quot;&gt;[[w:de:Modul:Graph]]&lt;/bdi&gt;. The [https://vega.github.io/vega/docs/porting-guide/ Vega Porting guide] provides the most comprehensive detail on migration from Vega2 and [https://www.mediawiki.org/w/index.php?title=Template:Graph:PageViews&amp;action=history here is an example migration]. Vega5 has currently just been enabled on mediawiki.org to provide a test environment for communities. [https://phabricator.wikimedia.org/T334940#8813922]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.8|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-05-09|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-05-10|en}}. It will be on all wikis from {{#time:j xg|2023-05-11|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Until now, all new OAuth apps went through manual review. Starting this week, apps using identification-only or basic authorizations will not require review. [https://phabricator.wikimedia.org/T67750]

'''Future changes'''
* During the next year, MediaWiki will stop using IP addresses to identify logged-out users, and will start automatically assigning unique temporary usernames. Read more at [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation/Updates|IP Editing: Privacy Enhancement and Abuse Mitigation/Updates]]. You can [[m:Talk:IP Editing: Privacy Enhancement and Abuse Mitigation#What should it look like?|join the discussion]] about the [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation/Updates#What will temporary usernames look like?|format of the temporary usernames]]. [https://phabricator.wikimedia.org/T332805]
* There will be an [[:w:en:A/B testing|A/B test]] on 10 Wikipedias where the Vector 2022 skin is the default skin. Half of logged-in desktop users will see an interface where the different parts of the page are more clearly separated. You can [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/2023-05 Zebra9 A/B test|read more]]. [https://phabricator.wikimedia.org/T333180][https://phabricator.wikimedia.org/T335972]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] &lt;code&gt;jquery.tipsy&lt;/code&gt; will be removed from the MediaWiki core. This will affect some user scripts. Many lines with &lt;code&gt;.tipsy(&lt;/code&gt; can be commented out. &lt;code&gt;OO.ui.PopupWidget&lt;/code&gt; can be used to keep things working like they are now. You can [[phab:T336019|read more]] and [[:mw:Help:Locating broken scripts|read about how to find broken scripts]]. [https://phabricator.wikimedia.org/T336019]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/19|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W19&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:36, 9 May 2023 (UTC)
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== Tech News: 2023-20 ==

&lt;section begin=&quot;technews-2023-W20&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/20|Translations]] are available.

'''Problems'''
* Citations that are automatically generated based on [[d:Q33057|ISBN]] are currently broken. This affects citations made with the [[mw:Special:MyLanguage/Help:VisualEditor/User_guide/Citations-Full#Automatic|VisualEditor Automatic tab]], and the use of the citoid API in gadgets and user scripts. Work is ongoing to restore this feature. [https://phabricator.wikimedia.org/T336298] 

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.9|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-05-16|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-05-17|en}}. It will be on all wikis from {{#time:j xg|2023-05-18|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* Starting on Wednesday, a new set of Wikipedias will get &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]&quot; ({{int:project-localized-name-gorwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hakwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hawwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hifwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hrwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hsbwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-htwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-iawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-iewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-igwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ilowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-inhwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-iowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-iswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-iuwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-jamwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-jvwiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T308134] 

'''Future changes'''
* There is a recently formed team at the Wikimedia Foundation which will be focusing on experimenting with new tools. Currently they are building [[m:Wikimedia_Foundation_Annual_Plan/2023-2024/Draft/Future_Audiences#FA2.2_Conversational_AI|a prototype ChatGPT plugin that allows information generated by ChatGPT to be properly attributed]] to the Wikimedia projects.
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Gadget and userscript developers should replace &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;jquery.cookie&lt;/code&gt;&lt;/bdi&gt; with &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;mediawiki.cookie&lt;/code&gt;&lt;/bdi&gt;. The &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;jquery.cookie&lt;/code&gt;&lt;/bdi&gt; library will be removed in ~1 month, and staff developers will run a script to replace any remaining uses at that time. [https://phabricator.wikimedia.org/T336018]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/20|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W20&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:45, 15 May 2023 (UTC)
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== Tech News: 2023-21 ==

&lt;section begin=&quot;technews-2023-W21&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/21|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] The &quot;recent edits&quot; time period for page watchers is now 30 days. It used to be 180 days. This was a [[m:Community Wishlist Survey 2023/Notifications, Watchlists and Talk Pages/Change information about the number of watchers on a page|Community Wishlist Survey proposal]]. [https://phabricator.wikimedia.org/T336250]

'''Changes later this week'''
* An [[mw:special:MyLanguage/Growth/Positive reinforcement#Impact|improved impact module]] will be available at Wikipedias. The impact module is a feature available to newcomers [[mw:Special:MyLanguage/Growth/Feature summary#Newcomer homepage|at their personal homepage]]. It will show their number of edits, how many readers their edited pages have, how many thanks they have received and similar things. It is also accessible by accessing Special:Impact. [https://phabricator.wikimedia.org/T336203]
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.10|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-05-23|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-05-24|en}}. It will be on all wikis from {{#time:j xg|2023-05-25|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/21|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W21&quot;/&gt;

16:55, 22 May 2023 (UTC)
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== Tech News: 2023-22 ==

&lt;section begin=&quot;technews-2023-W22&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/22|Translations]] are available.

'''Recent changes'''
* Citations can once again be added automatically from ISBNs, thanks to Zotero's ISBN searches. The current data sources are the Library of Congress (United States), the Bibliothèque nationale de France (French National Library), and K10plus ISBN (German repository). Additional data source searches can be [[mw:Citoid/Creating Zotero translators|proposed to Zotero]]. The ISBN labels in the [[mw:Special:MyLanguage/Help:VisualEditor/User_guide/Citations-Full#Automatic|VisualEditor Automatic tab]] will reappear later this week. [https://phabricator.wikimedia.org/T336298#8859917]
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] The page [[{{#special:EditWatchlist}}]] now has &quot;{{int:watchlistedit-normal-check-all}}&quot; options to select all the pages within a namespace. This feature request was [[m:Community Wishlist Survey 2023/Notifications, Watchlists and Talk Pages/Watchlist edit - &quot;check all&quot; checkbox|voted #161 in the 2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T334252]	

'''Problems'''
* For a few days earlier this month, the &quot;Add interlanguage link&quot; item in the Tools menu did not work properly. This has now been fixed. [https://phabricator.wikimedia.org/T337081]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.11|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-05-30|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-05-31|en}}. It will be on all wikis from {{#time:j xg|2023-06-01|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* VisualEditor will be switched to a new backend on [https://phabricator.wikimedia.org/source/mediawiki-config/browse/master/dblists/small.dblist small] and [https://phabricator.wikimedia.org/source/mediawiki-config/browse/master/dblists/medium.dblist medium] wikis this week. Large wikis will follow in the coming weeks. This is part of the effort to move Parsoid into MediaWiki core. The change should have no noticeable effect on users, but if you experience any slow loading or other strangeness when using VisualEditor, please report it on the phabricator ticket linked here. [https://phabricator.wikimedia.org/T320529]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/22|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W22&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:03, 29 May 2023 (UTC)
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== Tech News: 2023-23 ==

&lt;section begin=&quot;technews-2023-W23&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/23|Translations]] are available.

'''Recent changes'''
* The [[:mw:Special:MyLanguage/Help:Extension:RealMe|RealMe]] extension allows you to mark URLs on your user page as verified for Mastodon and similar software.
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] Citation and footnote editing can now be started from the reference list when using the visual editor. This feature request was [[m:Community Wishlist Survey 2023/Citations/Allow citations to be edited in the references section with VisualEditor|voted #2 in the 2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T54750]
* Previously, clicking on someone else's link to Recent Changes with filters applied within the URL could unintentionally change your preference for &quot;{{int:Rcfilters-group-results-by-page}}&quot;. This has now been fixed. [https://phabricator.wikimedia.org/T202916#8874081]

'''Problems'''
* For a few days last week, some tools and bots returned outdated information due to database replication problems, and may have been down entirely while it was being fixed. These issues have now been fixed. [https://phabricator.wikimedia.org/T337446]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.12|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-06-06|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-06-07|en}}. It will be on all wikis from {{#time:j xg|2023-06-08|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* Bots will no longer be prevented from making edits because of URLs that match the [[mw:Special:MyLanguage/Extension:SpamBlacklist|spam blacklist]]. [https://phabricator.wikimedia.org/T313107]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/23|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W23&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:52, 5 June 2023 (UTC)
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== Tech News: 2023-24 ==

&lt;section begin=&quot;technews-2023-W24&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/24|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] The content attribution tools [[mw:Special:MyLanguage/Who Wrote That?|Who Wrote That?]], [[xtools:authorship|XTools Authorship]], and [[xtools:blame|XTools Blame]] now support the Dutch, German, Hungarian, Indonesian, Japanese, Polish and Portuguese Wikipedias. This was the [[m:Community Wishlist Survey 2023/Reading/Extend &quot;Who Wrote That?&quot; tool to more wikis|#7 wish in the 2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T334891]
* The [[mw:Special:MyLanguage/Structured Data Across Wikimedia/Search Improvements#Search Preview panel|Search Preview panel]] has been deployed on four Wikipedias (Catalan, Dutch, Hungarian and Norwegian). The panel will show an image related to the article (if existing), the top sections of the article, related images (coming from MediaSearch on Commons), and eventually the sister projects associated with the article. [https://phabricator.wikimedia.org/T306341]
* The [[:mw:Special:MyLanguage/Help:Extension:RealMe#Verifying_a_link_on_non-user_pages|RealMe]] extension now allows administrators to verify URLs for any page, for Mastodon and similar software. [https://phabricator.wikimedia.org/T324937]
* The default project license [https://lists.wikimedia.org/hyperkitty/list/wikimediaannounce-l@lists.wikimedia.org/thread/7G6XPWZPQFLZ2JANN3ZX6RT4DVUI3HZQ/ has been officially upgraded] to CC BY-SA 4.0. The software interface messages have been updated. Communities should feel free to start updating any mentions of the old CC BY-SA 3.0 licensing within policies and related documentation pages. [https://phabricator.wikimedia.org/T319064]

'''Problems'''
* For three days last month, some Wikipedia pages edited with VisualEditor or DiscussionTools had an unintended &lt;code&gt;&lt;nowiki&gt;__TOC__&lt;/nowiki&gt;&lt;/code&gt; (or its localized form) added during an edit. There is [[mw:Parsoid/Deployments/T336101_followup|a listing of affected pages sorted by wiki]], that may still need to be fixed. [https://phabricator.wikimedia.org/T336101]
* Currently, the &quot;{{int:Visualeditor-dialog-meta-categories-defaultsort-label}}&quot; feature in VisualEditor is broken. Existing &lt;code&gt;&lt;nowiki&gt;{{DEFAULTSORT:...}}&lt;/nowiki&gt;&lt;/code&gt; keywords incorrectly appear as missing templates in VisualEditor. Developers are exploring how to fix this. In the meantime, those wishing to edit the default sortkey of a page are advised to switch to source editing. [https://phabricator.wikimedia.org/T337398]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Last week, an update to the delete form may have broken some gadgets or user scripts. If you need to manipulate (empty) the reason field, replace &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;#wpReason&lt;/code&gt;&lt;/bdi&gt; with &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot; style=&quot;white-space: nowrap;&quot;&gt;&lt;code&gt;#wpReason &gt; input&lt;/code&gt;&lt;/bdi&gt;. See [https://cs.wikipedia.org/w/index.php?title=MediaWiki%3AGadget-CleanDeleteReasons.js&amp;diff=22859956&amp;oldid=12794189 an example fix]. [https://phabricator.wikimedia.org/T337809]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.13|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-06-13|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-06-14|en}}. It will be on all wikis from {{#time:j xg|2023-06-15|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* VisualEditor will be switched to a new backend on English Wikipedia on Monday, and all other [https://phabricator.wikimedia.org/source/mediawiki-config/browse/master/dblists/large.dblist large] wikis on Thursday. The change should have no noticeable effect on users, but if you experience any slow loading or other strangeness when using VisualEditor, please report it on the phabricator ticket linked here. [https://phabricator.wikimedia.org/T320529]

'''Future changes'''
* From 5 June to 17 July, the Foundation's [[:mw:Wikimedia Security Team|Security team]] is holding a consultation with contributors regarding a draft policy to govern the use of third-party resources in volunteer-developed gadgets and scripts. Feedback and suggestions are warmly welcome at [[m:Special:MyLanguage/Third-party resources policy|Third-party resources policy]] on meta-wiki.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/24|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W24&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 14:51, 12 June 2023 (UTC)
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== Tech News: 2023-25 ==

&lt;section begin=&quot;technews-2023-W25&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/25|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Flame graphs are now available in WikimediaDebug. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/JXNQD3EHG5V5QW5UXFDPSHQG4MJ3FWJQ/][https://techblog.wikimedia.org/2023/06/08/flame-graphs-arrive-in-wikimediadebug/]

'''Changes later this week'''
* There is no new MediaWiki version this week.
* There is now a toolbar search popup in the visual editor. You can trigger it by typing &lt;code&gt;\&lt;/code&gt; or pressing &lt;code&gt;ctrl + shift + p&lt;/code&gt;. It can help you quickly access most tools in the editor. [https://commons.wikimedia.org/wiki/File:Visual_editor_toolbar_search_feature.png][https://phabricator.wikimedia.org/T66905]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/25|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W25&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:08, 19 June 2023 (UTC)
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== Tech News: 2023-26 ==

&lt;section begin=&quot;technews-2023-W26&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/26|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The Action API modules and Special:LinkSearch will now add a trailing &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;/&lt;/code&gt;&lt;/bdi&gt; to all &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;prop=extlinks&lt;/code&gt;&lt;/bdi&gt; responses for bare domains. This is part of the work to remove duplication in the &lt;code&gt;externallinks&lt;/code&gt; database table. [https://phabricator.wikimedia.org/T337994]

'''Problems'''
* Last week, search was broken on Commons and Wikidata for 23 hours. [https://phabricator.wikimedia.org/T339810][https://wikitech.wikimedia.org/wiki/Incidents/2023-06-18_search_broken_on_wikidata_and_commons]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.15|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-06-27|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-06-28|en}}. It will be on all wikis from {{#time:j xg|2023-06-29|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The Minerva skin now applies more predefined styles to the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;.mbox-text&lt;/code&gt;&lt;/bdi&gt; CSS class. This enables support for mbox templates that use divs instead of tables. Please make sure that the new styles won't affect other templates in your wiki. [https://gerrit.wikimedia.org/r/c/mediawiki/skins/MinervaNeue/+/930901/][https://phabricator.wikimedia.org/T339040]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Gadgets will now load on both desktop and mobile by default. Previously, gadgets loaded only on desktop by default. Changing this default using the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;|targets=&lt;/code&gt;&lt;/bdi&gt; parameter is also deprecated and should not be used. You should make gadgets work on mobile or disable them based on the skin (with the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;|skins=&lt;/code&gt;&lt;/bdi&gt; parameter in &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;MediaWiki:Gadgets-definition&lt;/bdi&gt;) rather than whether the user uses the mobile or the desktop website. Popular gadgets that create errors on mobile will be disabled by developers on the Minerva skin as a temporary solution. [https://phabricator.wikimedia.org/T127268]
* All namespace tabs now have the same browser [[m:Special:MyLanguage/Help:Keyboard_shortcuts|access key]] by default. Previously, custom and extension-defined namespaces would have to have their access keys set manually on-wiki, but that is no longer necessary. [https://phabricator.wikimedia.org/T22126]
* The review form of the Flagged Revisions extension now uses the standardized [[mw:Special:MyLanguage/Codex|user interface components]]. [https://phabricator.wikimedia.org/T191156]

'''Future changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] How media is structured in the parser's HTML output will change in the coming weeks at [[:wikitech:Deployments/Train#Thursday|group2 wikis]]. This change improves the accessibility of content. You may need to update your site-CSS, or userscripts and gadgets. There are [[mw:Special:MyLanguage/Parsoid/Parser_Unification/Media_structure/FAQ|details on what code to check, how to update the code, and where to report any related problems]]. [https://phabricator.wikimedia.org/T314318]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/26|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W26&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 16:18, 26 June 2023 (UTC)
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== Tech News: 2023-27 ==

&lt;section begin=&quot;technews-2023-W27&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/27|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] As part of the rolling out of the [[m:Community Wishlist Survey 2022/Multimedia and Commons/Audio links that play on click|audio links that play on click]] wishlist proposal, [https://noc.wikimedia.org/conf/highlight.php?file=dblists/small.dblist small wikis] will now be able to use the [[mw:Special:MyLanguage/Help:Extension:Phonos#Inline audio player mode|inline audio player]] that is implemented by the [[mw:Extension:Phonos|Phonos]] extension. [https://phabricator.wikimedia.org/T336763]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] From this week all gadgets automatically load on mobile and desktop sites. If you see any problems with gadgets on your wikis, please adjust the [[mw:Special:MyLanguage/Extension:Gadgets#Options|gadget options]] in your gadget definitions file. [https://phabricator.wikimedia.org/T328610]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.16|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-07-04|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-07-05|en}}. It will be on all wikis from {{#time:j xg|2023-07-06|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/27|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W27&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:51, 3 July 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25231546 --&gt;

== Tech News: 2023-28 ==

&lt;section begin=&quot;technews-2023-W28&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/28|Translations]] are available.

'''Recent changes'''
* The [[:mw:Special:MyLanguage/Structured Data Across Wikimedia/Section-level Image Suggestions|Section-level Image Suggestions feature]] has been deployed on seven Wikipedias (Portuguese, Russian, Indonesian, Catalan, Hungarian, Finnish and Norwegian Bokmål). The feature recommends images for articles on contributors' watchlists that are a good match for individual sections of those articles.
* [[:m:Special:MyLanguage/Global AbuseFilter|Global abuse filters]] have been enabled on all Wikimedia projects, except English and Japanese Wikipedias (who opted out). This change was made following a [[:m:Requests for comment/Make global abuse filters opt-out|global request for comments]]. [https://phabricator.wikimedia.org/T341159]
* [[{{#special:BlockedExternalDomains}}]] is a new tool for administrators to help fight spam. It provides a clearer interface for blocking plain domains (and their subdomains), is more easily searchable, and is faster for the software to process for each edit on the wiki. It does not support regex (for complex cases), nor URL path-matching, nor the [[MediaWiki:Spam-whitelist|MediaWiki:Spam-whitelist]], but otherwise it replaces most of the functionalities of the existing [[MediaWiki:Spam-blacklist|MediaWiki:Spam-blacklist]]. There is a Python script to help migrate all simple domains into this tool, and more feature details, within [[mw:Special:MyLanguage/Manual:BlockedExternalDomains|the tool's documentation]]. It is available at all wikis except for Meta-wiki, Commons, and Wikidata. [https://phabricator.wikimedia.org/T337431]
* The WikiEditor extension was updated. It includes some of the most frequently used features of wikitext editing. In the past, many of its messages could only be translated by administrators, but now all regular translators on translatewiki can translate them. Please check [https://translatewiki.net/wiki/Special:MessageGroupStats?group=ext-wikieditor&amp;messages=&amp;x=D#sortable:0=asc the state of WikiEditor localization into your language], and if the &quot;Completion&quot; for your language shows anything less than 100%, please complete the translation. See [https://lists.wikimedia.org/hyperkitty/list/wikitech-ambassadors@lists.wikimedia.org/thread/D4YELU2DXMZ75PGELUOKXXMFF3FH45XA/ a more detailed explanation].

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.17|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-07-11|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-07-12|en}}. It will be on all wikis from {{#time:j xg|2023-07-13|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* The default protocol of [[{{#special:LinkSearch}}]] and API counterparts has changed from http to both http and https. [https://phabricator.wikimedia.org/T14810]
* [[{{#special:LinkSearch}}]] and its API counterparts will now search for all of the URL provided in the query. It used to be only the first 60 characters. This feature was requested fifteen years ago. [https://phabricator.wikimedia.org/T17218]

'''Future changes'''
* There is an experiment with a [[:w:en:ChatGPT|ChatGPT]] plugin. This is to show users where the information is coming from when they read information from Wikipedia. It has been tested by Wikimedia Foundation staff and other Wikimedians. Soon all ChatGPT plugin users can use the Wikipedia plugin. This is the same plugin which was mentioned in [[m:Special:MyLanguage/Tech/News/2023/20|Tech News 2023/20]]. [https://meta.wikimedia.org/wiki/Wikimedia_Foundation_Annual_Plan/2023-2024/Draft/Future_Audiences#FA2.2_Conversational_AI]
* There is an ongoing discussion on a [[m:Special:MyLanguage/Third-party resources policy|proposed Third-party resources policy]]. The proposal will impact the use of third-party resources in gadgets and userscripts. Based on the ideas received so far, policy includes some of the risks related to user scripts and gadgets loading third-party resources, some best practices and exemption requirements such as code transparency and inspectability. Your feedback and suggestions are warmly welcome until July 17, 2023 on [[m:Talk:Third-party resources policy|on the policy talk page]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/28|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W28&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 19:54, 10 July 2023 (UTC)
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== Tech News: 2023-29 ==

&lt;section begin=&quot;technews-2023-W29&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/29|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] We are now serving 1% of all global user traffic from [[w:en:Kubernetes|Kubernetes]] (you can [[wikitech:MediaWiki On Kubernetes|read more technical details]]). We are planning to increment this percentage regularly. You can [[phab:T290536|follow the progress of this work]]. 

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.18|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-07-18|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-07-19|en}}. It will be on all wikis from {{#time:j xg|2023-07-20|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] MediaWiki [[mw:Special:MyLanguage/Help:System_message|system messages]] will now look for available local fallbacks, instead of always using the default fallback defined by software. This means wikis no longer need to override each language on the [[mw:Special:MyLanguage/Manual:Language#Fallback_languages|fallback chain]] separately. For example, English Wikipedia doesn't have to create &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;en-ca&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;en-gb&lt;/code&gt;&lt;/bdi&gt; subpages with a transclusion of the base pages anymore. This makes it easier to maintain local overrides. [https://phabricator.wikimedia.org/T229992]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;action=growthsetmentorstatus&lt;/code&gt;&lt;/bdi&gt; API will be deprecated with the new MediaWiki version. Bots or scripts calling that API should use the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;action=growthmanagementorlist&lt;/code&gt;&lt;/bdi&gt; API now. [https://phabricator.wikimedia.org/T321503]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/29|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W29&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:08, 17 July 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25289122 --&gt;

== Tech News: 2023-30 ==

&lt;section begin=&quot;technews-2023-W30&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/30|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]]   On July 18, the Wikimedia Foundation launched a survey about the [[:mw:Technical_decision_making|technical decision making process]] for people who do technical work that relies on software that is maintained by the Foundation or affiliates. If this applies to you, [https://wikimediafoundation.limesurvey.net/885471 please take part in the survey]. The survey will be open for three weeks, until August 7. You can find more information in [[listarchive:list/wikitech-l@lists.wikimedia.org/thread/Q7DUCFA75DXG3G2KHTO7CEWMLCYTSDB2/|the announcement e-mail on wikitech-l]].

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.19|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-07-25|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-07-26|en}}. It will be on all wikis from {{#time:j xg|2023-07-27|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/30|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W30&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 02:20, 25 July 2023 (UTC)
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== Tech News: 2023-31 ==

&lt;section begin=&quot;technews-2023-W31&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/31|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The [[mw:Synchronizer|Synchronizer]] tool is now available to keep Lua modules synced across Wikimedia wikis, along with [[mw:Multilingual Templates and Modules|updated documentation]] to develop global Lua modules and templates.
* The tag filter on [[{{#special:NewPages}}]] and revision history pages can now be inverted. For example, you can hide edits that were made using an automated tool. [https://phabricator.wikimedia.org/T334337][https://phabricator.wikimedia.org/T334338]
* The Wikipedia [[:w:en:ChatGPT|ChatGPT]] plugin experiment can now be used by ChatGPT users who can use plugins. You can participate in a [[:m:Talk:Wikimedia Foundation Annual Plan/2023-2024/Draft/Future Audiences#Announcing monthly Future Audiences open &quot;office hours&quot;|video call]] if you want to talk about this experiment or similar work. [https://meta.wikimedia.org/wiki/Wikimedia_Foundation_Annual_Plan/2023-2024/Draft/Future_Audiences#FA2.2_Conversational_AI]

'''Problems'''
* It was not possible to generate a PDF for pages with non-Latin characters in the title, for the last two weeks. This has now been fixed. [https://phabricator.wikimedia.org/T342442]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.20|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-08-01|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-08-02|en}}. It will be on all wikis from {{#time:j xg|2023-08-03|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* Starting on Tuesday, a new set of Wikipedias will get &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]&quot; ({{int:project-localized-name-kawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kaawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kabwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kbdwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kbpwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kiwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kkwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kmwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-knwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kshwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kuwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kwwiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T308135] 

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/31|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W31&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:54, 31 July 2023 (UTC)
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== Tech News: 2023-32 ==

&lt;section begin=&quot;technews-2023-W32&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/32|Translations]] are available.

'''Recent changes'''
* Mobile Web editors can now [[mw:Special:MyLanguage/Reading/Web/Advanced_mobile_contributions#August_1,_2023_-_Full-page_editing_added_on_mobile|edit a whole page at once]]. To use this feature, turn on &quot;{{int:Mobile-frontend-mobile-option-amc}}&quot; in your settings and use the &quot;{{int:Minerva-page-actions-editfull}}&quot; button in the &quot;{{int:Minerva-page-actions-overflow}}&quot; menu. [https://phabricator.wikimedia.org/T203151]

'''Changes later this week'''
* There is no new MediaWiki version this week.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/32|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W32&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:20, 7 August 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25420038 --&gt;

== Tech News: 2023-33 ==

&lt;section begin=&quot;technews-2023-W33&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/33|Translations]] are available.

'''Recent changes'''
* The Content translation system is no longer using Youdao's [[mw:Special:MyLanguage/Help:Content_translation/Translating/Initial_machine_translation|machine translation service]]. The service was in place for several years, but due to no usage, and availability of alternatives, it was deprecated to reduce maintenance overheads. Other services which cover the same languages are still available. [https://phabricator.wikimedia.org/T329137]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.22|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-08-15|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-08-16|en}}. It will be on all wikis from {{#time:j xg|2023-08-17|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* Starting on Wednesday, a new set of Wikipedias will get &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]&quot; ({{int:project-localized-name-lawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ladwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lbwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lbewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lezwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lfnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lgwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-liwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lijwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lmowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ltgwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lvwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-maiwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-map_bmswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mdfwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mgwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kywiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T308136] &lt;!-- TODO replace wiki codes --&gt;

'''Future changes'''
* A few gadgets/user scripts which add icons to the Minerva skin need to have their CSS updated. There are more details available including a [[phab:T344067|search for all existing instances and how to update them]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/33|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W33&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 05:59, 15 August 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25428668 --&gt;

== Tech News: 2023-34 ==

&lt;section begin=&quot;technews-2023-W34&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/34|Translations]] are available.

'''Recent changes'''
* The [https://gdrive-to-commons.toolforge.org/ GDrive to Commons Uploader] tool is now available. It enables [[m:Special:MyLanguage/GDrive to Commons Uploader|securely selecting and uploading files]] from your Google Drive directly to Wikimedia Commons. [https://phabricator.wikimedia.org/T267868]
* From now on, we will announce new Wikimedia wikis in Tech News, so you can update any tools or pages.
** Since the last edition, two new wikis have been created:
*** a Wiktionary in [[d:Q7121294|Pa'O]] ([[wikt:blk:|&lt;code&gt;wikt:blk:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T343540]
*** a Wikisource in [[d:Q34002|Sundanese]] ([[s:su:|&lt;code&gt;s:su:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T343539]
** To catch up, the next most recent six wikis are:
*** Wikifunctions ([[f:|&lt;code&gt;f:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T275945]
*** a Wiktionary in [[d:Q2891049|Mandailing]] ([[wikt:btm:|&lt;code&gt;wikt:btm:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T335216]
*** a Wikipedia in [[d:Q5555465|Ghanaian Pidgin]] ([[w:gpe:|&lt;code&gt;w:gpe:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T335969]
*** a Wikinews in [[d:Q3111668|Gungbe]] ([[n:guw:|&lt;code&gt;n:guw:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T334394]
*** a Wiktionary in [[d:Q33522|Kabardian]] ([[wikt:kbd:|&lt;code&gt;wikt:kbd:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T333266]
*** a Wikipedia in [[d:Q35570|Fante]] ([[w:fat:|&lt;code&gt;w:fat:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T335016]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.23|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-08-22|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-08-23|en}}. It will be on all wikis from {{#time:j xg|2023-08-24|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).

'''Future changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] There is an existing [[mw:Stable interface policy|stable interface policy]] for MediaWiki backend code. There is a [[mw:User:Jdlrobson/Stable interface policy/frontend|proposed stable interface policy for frontend code]]. This is relevant for anyone who works on gadgets or Wikimedia frontend code. You can read it, discuss it, and let the proposer know if there are any problems. [https://phabricator.wikimedia.org/T344079]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/34|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W34&quot;/&gt;

15:25, 21 August 2023 (UTC)
&lt;!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25497111 --&gt;

== Tech News: 2023-35 ==

&lt;section begin=&quot;technews-2023-W35&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/35|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] As part of the changes for the [[m:Community Wishlist Survey 2022/Better diff handling of paragraph splits|better diff handling of paragraph splits]], improved detection of splits is being rolled out. Over the last two weeks, we deployed this support to [[wikitech:Deployments/Train#Groups|group0]] and group1 wikis. This week it will be deployed to group2 wikis. [https://phabricator.wikimedia.org/T341754]
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] All [[{{#special:Contributions}}]] pages now show the user's local edit count and the account's creation date. [https://phabricator.wikimedia.org/T324166]
* Wikisource users can now use the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;prpbengalicurrency&lt;/code&gt;&lt;/bdi&gt; label to denote Bengali currency characters as page numbers inside the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;&lt;pagelist&gt;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; tag.  [https://phabricator.wikimedia.org/T268932]
* Two preferences have been relocated. The preference &quot;{{int:visualeditor-preference-visualeditor}}&quot; is now shown on the [[Special:Preferences#mw-prefsection-editing|&quot;{{int:prefs-editing}}&quot; tab]] at all wikis. Previously it was shown on the &quot;{{int:prefs-betafeatures}}&quot; tab at some wikis. The preference &quot;{{int:visualeditor-preference-newwikitexteditor-enable}}&quot; is now also shown on the &quot;{{int:prefs-editing}}&quot; tab at all wikis, instead of the &quot;{{int:prefs-betafeatures}}&quot; tab. [https://phabricator.wikimedia.org/T335056][https://phabricator.wikimedia.org/T344158]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.24|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-08-29|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-08-30|en}}. It will be on all wikis from {{#time:j xg|2023-08-31|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] New signups for a Wikimedia developer account will start being pushed towards &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[https://idm.wikimedia.org/ idm.wikimedia.org]&lt;/bdi&gt;, rather than going via Wikitech. [[wikitech:IDM|Further information about the new system is available]].
* All right-to-left language wikis, plus Korean, Armenian, Ukrainian, Russian, and Bulgarian Wikipedias, will have a link in the sidebar that provides a short URL of that page, using the [[m:Special:MyLanguage/Wikimedia URL Shortener|Wikimedia URL Shortener]]. This feature will come to more wikis in future weeks. [https://phabricator.wikimedia.org/T267921]

'''Future changes'''
* The removal of the [[mw:Special:MyLanguage/Extension:DoubleWiki|DoubleWiki extension]] is being discussed. This extension currently allows Wikisource users to view articles from multiple language versions side by side when the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;=&gt;&lt;/code&gt;&lt;/bdi&gt; symbol next to a specific language edition is selected. Comments on this are welcomed at [[phab:T344544|the phabricator task]].
* A proposal has been made to merge the second hidden-categories list (which appears below the wikitext editing form) with the main list of categories (which is further down the page). [[phab:T340606|More information is available on Phabricator]]; feedback is welcome!

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/35|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W35&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 14:00, 28 August 2023 (UTC)
&lt;!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25510866 --&gt;

== Tech News: 2023-36 ==

&lt;section begin=&quot;technews-2023-W36&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/36|Translations]] are available.

'''Recent changes'''
* [[m:Wikisource_EditInSequence|EditInSequence]], a feature that allows users to edit pages faster on Wikisource has been moved to a Beta Feature based on community feedback. To enable it, you can navigate to the [[Special:Preferences#mw-prefsection-betafeatures|beta features tab in Preferences]]. [https://phabricator.wikimedia.org/T308098]
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] As part of the changes for the [[m:Special:MyLanguage/Community Wishlist Survey 2022/Generate Audio for IPA|Generate Audio for IPA]] and [[m:Community Wishlist Survey 2022/Multimedia and Commons/Audio links that play on click|Audio links that play on click]] wishlist proposals, the [[mw:Special:MyLanguage/Help:Extension:Phonos#Inline_audio_player_mode|inline audio player mode]] of [[mw:Extension:Phonos|Phonos]] has been deployed to all projects. [https://phabricator.wikimedia.org/T336763]
* There is a new option for Administrators when they are changing the usergroups for a user, to add the user’s user page to their watchlist. This works both via [[{{#special:UserRights}}]] and via the API. [https://phabricator.wikimedia.org/T272294]
* One new wiki has been created:
** a {{int:project-localized-name-group-wikipedia}} in [[d:Q34318|Talysh]] ([[w:tly:|&lt;code&gt;w:tly:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T345166]
'''Problems'''
* The [[mw:Special:MyLanguage/Extension:LoginNotify|LoginNotify extension]] was not sending notifications since January. It has now been fixed, so going forward, you may see notifications for failed login attempts, and successful login attempts from a new device. [https://phabricator.wikimedia.org/T344785]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.25|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-09-05|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-09-06|en}}. It will be on all wikis from {{#time:j xg|2023-09-07|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* Starting on Wednesday, a new set of Wikipedias will get &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]&quot; ({{int:project-localized-name-mhrwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-miwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-minwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mkwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mlwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mrwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mrjwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mtwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mwlwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-myvwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mznwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nahwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-napwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ndswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nds_nlwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-newiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-newwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-novwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nqowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nrmwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nsowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nvwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ocwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-olowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-omwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-orwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-oswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pagwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pamwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-papwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pcdwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pdcwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pflwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pihwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pmswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pnbwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pntwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pswiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T308137][https://phabricator.wikimedia.org/T308138]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/36|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W36&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:33, 4 September 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25566983 --&gt;

== Tech News: 2023-37 ==

&lt;section begin=&quot;technews-2023-W37&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/37|Translations]] are available.

'''Recent changes'''
* [[mw:Special:MyLanguage/ORES|ORES]], the revision evaluation service, is now using a new open-source infrastructure on all wikis except for English Wikipedia and Wikidata. These two will follow this week. If you notice any unusual results from the Recent Changes filters that are related to ORES (for example, &quot;{{int:ores-rcfilters-damaging-title}}&quot; and &quot;{{int:ores-rcfilters-goodfaith-title}}&quot;), please [[mw:Talk:Machine Learning|report them]]. [https://phabricator.wikimedia.org/T342115]
* When you are logged in on one Wikimedia wiki and visit a different Wikimedia wiki, the system tries to log you in there automatically. This has been unreliable for a long time. You can now visit the login page to make the system try extra hard. If you feel that made logging in better or worse than it used to be, your feedback is appreciated. [https://phabricator.wikimedia.org/T326281]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.26|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-09-12|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-09-13|en}}. It will be on all wikis from {{#time:j xg|2023-09-14|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The [[mw:Special:MyLanguage/Technical decision making|Technical Decision-Making Forum Retrospective]] team invites anyone involved in the technical field of Wikimedia projects to signup to and join [[mw:Technical decision making/Listening Sessions|one of their listening sessions]] on 13 September. Another date will be scheduled later. The goal is to improve the technical decision-making processes.
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] As part of the changes for the [[m:Special:MyLanguage/Community Wishlist Survey 2022/Better diff handling of paragraph splits|Better diff handling of paragraph splits]] wishlist proposal, the inline switch widget in diff pages is being rolled out this week to all wikis. The inline switch will allow viewers to toggle between a unified inline or two-column diff wikitext format. [https://phabricator.wikimedia.org/T336716]

'''Future changes'''
* All wikis will be read-only for a few minutes on 20 September. [[m:Special:MyLanguage/Tech/Server switch|This is planned at 14:00 UTC.]] More information will be published in Tech News and will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T345263]
* The Enterprise API is launching a new feature called &quot;[http://breakingnews-beta.enterprise.wikimedia.com/ breaking news]&quot;. Currently in BETA, this attempts to identify likely &quot;newsworthy&quot; topics as they are currently being written about in any Wikipedia. Your help is requested to improve the accuracy of its detection model, especially on smaller language editions, by recommending templates or identifiable editing patterns. See more information at [[mw:Special:MyLanguage/Wikimedia Enterprise/Breaking news|the documentation page]] on MediaWiki or [[m:Special:MyLanguage/Wikimedia Enterprise/FAQ#What is Breaking News|the FAQ]] on Meta.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/37|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W37&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:07, 11 September 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25589064 --&gt;

== Tech News: 2023-38 ==

&lt;section begin=&quot;technews-2023-W38&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/38|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] MediaWiki now has a [[mw:Stable interface policy/frontend|stable interface policy for frontend code]] that more clearly defines how we deprecate MediaWiki code and wiki-based code (e.g. gadgets and user scripts). Thank you to everyone who contributed to the content and discussions. [https://phabricator.wikimedia.org/T346467][https://phabricator.wikimedia.org/T344079]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.27|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-09-19|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-09-20|en}}. It will be on all wikis from {{#time:j xg|2023-09-21|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* All wikis will be read-only for a few minutes on September 20. [[m:Special:MyLanguage/Tech/Server switch|This is planned at 14:00 UTC.]] [https://phabricator.wikimedia.org/T345263]
* All wikis will have a link in the sidebar that provides a short URL of that page, using the [[m:Special:MyLanguage/Wikimedia URL Shortener|Wikimedia URL Shortener]]. [https://phabricator.wikimedia.org/T267921]

'''Future changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The team investigating the Graph Extension posted [[mw:Extension:Graph/Plans#Proposal|a proposal for reenabling it]] and they need your input.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/38|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W38&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 19:19, 18 September 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25623533 --&gt;

== Tech News: 2023-39 ==

&lt;section begin=&quot;technews-2023-W39&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/39|Translations]] are available.

'''Recent changes'''
* The Vector 2022 skin will now remember the pinned/unpinned status for the Table of Contents for all logged-out users. [https://phabricator.wikimedia.org/T316060]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.28|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-09-26|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-09-27|en}}. It will be on all wikis from {{#time:j xg|2023-09-28|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The ResourceLoader &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;mediawiki.ui&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; modules are now deprecated as part of the move to Vue.js and Codex. There is a [[mw:Codex/Migrating_from_MediaWiki_UI|guide for migrating from MediaWiki UI to Codex]] for any tools that use it. More [[phab:T346468|details are available in the task]] and your questions are welcome there.
* Gadget definitions will have a [[mw:Special:MyLanguage/Extension:Gadgets#Options|new &quot;namespaces&quot; option]]. The option takes a list of namespace IDs. Gadgets that use this option will only load on pages in the given namespaces.

'''Future changes'''
* New variables will be added to [[mw:Special:MyLanguage/Extension:AbuseFilter|AbuseFilter]]: &lt;code&gt;&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;global_account_groups&lt;/bdi&gt;&lt;/code&gt; and &lt;code&gt;&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;global_account_editcount&lt;/bdi&gt;&lt;/code&gt;. They are available only when an account is being created. You can use them to prevent blocking automatic creation of accounts when users with many edits elsewhere visit your wiki for the first time. [https://phabricator.wikimedia.org/T345632][https://www.mediawiki.org/wiki/Special:MyLanguage/Extension:AbuseFilter/Rules_format]

'''Meetings'''
* You can join the next meeting with the Wikipedia mobile apps teams. During the meeting, we will discuss the current features and future roadmap. The meeting will be on [https://zonestamp.toolforge.org/1698426015 27 October at 17:00 (UTC)]. See [[mw:Special:MyLanguage/Wikimedia_Apps/Office_Hours#October_2023|details and how to join]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/39|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W39&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 16:51, 26 September 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25655264 --&gt;

== Tech News: 2023-40 ==

&lt;section begin=&quot;technews-2023-W40&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/40|Translations]] are available.

'''Recent changes'''
* There is a new [[Special:Preferences#mw-prefsection-rendering-advancedrendering|user preference]] for &quot;{{int:tog-forcesafemode}}&quot;. This setting will make pages load without including any on-wiki JavaScript or on-wiki stylesheet pages. It can be useful for debugging broken JavaScript gadgets. [https://phabricator.wikimedia.org/T342347]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Gadget definitions now have a [[mw:Special:MyLanguage/Extension:Gadgets#Options|new &quot;&lt;var&gt;contentModels&lt;/var&gt;&quot; option]]. The option takes a list of page content models, like &lt;code&gt;&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;wikitext&lt;/bdi&gt;&lt;/code&gt; or &lt;code&gt;&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;css&lt;/bdi&gt;&lt;/code&gt;. Gadgets that use this option will only load on pages with the given content models.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.29|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-10-03|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-10-04|en}}. It will be on all wikis from {{#time:j xg|2023-10-05|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).

'''Future changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The Vector 2022 skin will no longer use the custom styles and scripts of Vector legacy (2010). The change will be made later this year or in early 2024. See [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Features/Loading Vector 2010 scripts|how to adjust the CSS and JS pages on your wiki]]. [https://phabricator.wikimedia.org/T331679]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/40|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W40&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:26, 3 October 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25686930 --&gt;

== Tech News: 2023-41 ==

&lt;section begin=&quot;technews-2023-W41&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/41|Translations]] are available.

'''Recent changes'''
* One new wiki has been created: a {{int:project-localized-name-group-wikipedia}} in [[d:Q33291|Fon]] ([[w:fon:|&lt;code&gt;w:fon:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T347935]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.30|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-10-10|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-10-11|en}}. It will be on all wikis from {{#time:j xg|2023-10-12|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* Starting on Wednesday, a new set of Wikipedias will get &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]&quot; ({{int:project-localized-name-swwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-wawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-warwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-wowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-xalwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-xhwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-xmfwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-yiwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-yowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-zawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-zeawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-zh_min_nanwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-zuwiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T308139]
* At some wikis, newcomers are suggested images from Commons to add to articles without any images. Starting on Tuesday, newcomers at these wikis will be able to add images to unillustrated article sections. The specific wikis are listed under &quot;Images recommendations&quot; [[mw:Special:MyLanguage/Growth/Deployment table|at the Growth team deployment table]]. You can [[mw:Special:MyLanguage/Help:Growth/Tools/Add an image|learn more about this feature.]] [https://phabricator.wikimedia.org/T345940]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] In the mobile web skin (Minerva) the CSS ID &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;#page-actions&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; will be replaced with &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;#p-views&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. This change is to make it consistent with other skins and to improve support for gadgets and extensions in the mobile skin. A few gadgets may need to be updated; there are [https://phabricator.wikimedia.org/T348267 details and search-links in the task].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/41|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W41&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 14:39, 9 October 2023 (UTC)
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== Tech News: 2023-42 ==

&lt;section begin=&quot;technews-2023-W42&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/42|Translations]] are available.

'''Recent changes'''
* The [[m:Special:MyLanguage/Help:Unified login|Unified login]] system's edge login should now be fixed for some browsers (Chrome, Edge, Opera). This means that if you visit a new sister project wiki, you should be logged in automatically without the need to click &quot;Log in&quot; or reload the page. Feedback on whether it's working for you is welcome. [https://phabricator.wikimedia.org/T347889]
* [[mw:Special:MyLanguage/Manual:Interface/Edit_notice|Edit notices]] are now available within the MobileFrontend/Minerva skin. This feature was inspired by [[w:en:Wikipedia:EditNoticesOnMobile|the gadget on English Wikipedia]]. See more details in [[phab:T316178|T316178]].

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.1|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-10-17|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-10-18|en}}. It will be on all wikis from {{#time:j xg|2023-10-19|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).

'''Future changes'''
* In 3 weeks, in the Vector 2022 skin, code related to &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;addPortletLink&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;#p-namespaces&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; that was deprecated one year ago will be removed. If you notice tools that should appear next to the &quot;Discussion&quot; tab are then missing, please tell the gadget's maintainers to see [[phab:T347907|instructions in the Phabricator task]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/42|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W42&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:47, 16 October 2023 (UTC)
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== Tech News: 2023-43 ==

&lt;section begin=&quot;technews-2023-W43&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/43|Translations]] are available.

'''Recent changes'''
* There is a new [[mw:Special:MyLanguage/Wikimedia Language engineering/Newsletter/2023/October|Language and internationalization newsletter]], written quarterly. It contains updates on new feature development, improvements in various language-related technical projects, and related support work.
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Source map support has been enabled on all wikis. When you open the debugger in your browser's developer tools, you should be able to see the unminified JavaScript source code. [https://phabricator.wikimedia.org/T47514]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.2|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-10-24|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-10-25|en}}. It will be on all wikis from {{#time:j xg|2023-10-26|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/43|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W43&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:16, 23 October 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25782286 --&gt;

== Tech News: 2023-44 ==

&lt;section begin=&quot;technews-2023-W44&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/44|Translations]] are available.

'''Recent changes'''
* The Structured Content team, as part of its project of [[:commons:Commons:WMF support for Commons/Upload Wizard Improvements|improving UploadWizard on Commons]], made some UX improvements to the upload step of choosing own vs not own work ([[phab:T347590|T347590]]), as well as to the licensing step for own work ([[phab:T347756|T347756]]).
* The Design Systems team has released version 1.0.0 of [[wmdoc:codex/latest/|Codex]], the new design system for Wikimedia. See the [[mw:Special:MyLanguage/Design_Systems_Team/Announcing_Codex_1.0|full announcement about the release of Codex 1.0.0]].

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.3|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-10-31|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-11-01|en}}. It will be on all wikis from {{#time:j xg|2023-11-02|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]).
* Listings on category pages are sorted on each wiki for that language using a [[:w:en:International Components for Unicode|library]]. For a brief period on 2 November, changes to categories will not be sorted correctly for many languages. This is because the developers are upgrading to a new version of the library. They will then use a script to fix the existing categories. This will take a few hours or a few days depending on how big the wiki is. You can [[mw:Special:MyLanguage/Wikimedia Technical Operations/ICU announcement|read more]]. [https://phabricator.wikimedia.org/T345561][https://phabricator.wikimedia.org/T267145] 
* Starting November 1, the impact module (Special:Impact) will be upgraded by the Growth team. The new impact module shows newcomers more data regarding their impact on the wiki. It was tested by a few wikis during the last few months. [https://phabricator.wikimedia.org/T336203]

'''Future changes'''
* There is [[mw:Special:MyLanguage/Extension:Graph/Plans#Roadmap|a proposed plan]] for re-enabling the Graph Extension. You can help by reviewing this proposal and [[mw:Extension_talk:Graph/Plans#c-PPelberg_(WMF)-20231020221600-Update:_20_October|sharing what you think about it]].
* The WMF is working on making it possible for administrators to [[mw:Special:MyLanguage/Community_configuration_2.0|edit MediaWiki configuration directly]]. This is similar to previous work on Special:EditGrowthConfig. [[phab:T349757|A technical RfC is running until November 08, where you can provide feedback.]] 

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/44|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W44&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:21, 30 October 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25801989 --&gt;

== Tech News: 2023-45 ==

&lt;section begin=&quot;technews-2023-W45&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/45|Translations]] are available.

'''Recent changes'''
* In the Vector 2022 skin, the default font-size of a number of navigational elements (tagline, tools menu, navigational links, and more) has been increased slightly to match the font size used in page content. [https://phabricator.wikimedia.org/T346062]

'''Problems'''
* Last week, there was a problem displaying some recent edits on [https://noc.wikimedia.org/conf/highlight.php?file=dblists/s5.dblist a few wikis], for 1-6 hours. The edits were saved but not immediately shown. This was due to a database problem. [https://phabricator.wikimedia.org/T350443]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.4|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-11-07|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-11-08|en}}. It will be on all wikis from {{#time:j xg|2023-11-09|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]).
* The Growth team will reassign newcomers from former mentors to [[mw:Special:MyLanguage/Growth/Structured mentor list|the currently active mentors]]. They have also changed the notification language to be more user-friendly. [https://phabricator.wikimedia.org/T330071][https://phabricator.wikimedia.org/T327493]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/45|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W45&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:05, 6 November 2023 (UTC)
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== Tech News: 2023-46 ==

&lt;section begin=&quot;technews-2023-W46&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/46|Translations]] are available.

'''Recent changes'''
* Four new wikis have been created:
** a Wikipedia in [[d:Q7598268|Moroccan Amazigh]] ([[w:zgh:|&lt;code&gt;w:zgh:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T350216]
** a Wikipedia in [[d:Q35159|Dagaare]] ([[w:dga:|&lt;code&gt;w:dga:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T350218]
** a Wikipedia in [[d:Q33017|Toba Batak]] ([[w:bbc:|&lt;code&gt;w:bbc:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T350320]
** a Wikiquote in [[d:Q33151|Banjar]] ([[q:bjn:|&lt;code&gt;q:bjn:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T350217]

'''Problems'''
* Last week, users who previously visited Meta-Wiki or Wikimedia Commons and then became logged out on those wikis could not log in again. The problem is now resolved. [https://phabricator.wikimedia.org/T350695]
* Last week, some pop-up dialogs and menus were shown with the wrong font size. The problem is now resolved. [https://phabricator.wikimedia.org/T350544]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.5|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-11-14|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-11-15|en}}. It will be on all wikis from {{#time:j xg|2023-11-16|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]).

'''Future changes'''
* Reference Previews are coming to many wikis as a default feature. They are popups for references, similar to the [[mw:Special:MyLanguage/Page Previews|PagePreviews feature]]. [[m:WMDE Technical Wishes/ReferencePreviews#Opt-out feature|You can opt out]] of seeing them. If you are [[Special:Preferences#mw-prefsection-gadgets|using the gadgets]] Reference Tooltips or Navigation Popups, you won’t see Reference Previews. [[phab:T282999|Deployment]] is planned for November 22, 2023. 
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Canary (also known as heartbeat) events will be produced into [https://stream.wikimedia.org/?doc#/streams Wikimedia event streams] from December 11. Streams users are advised to filter out these events, by discarding all events where &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;meta.domain == &quot;canary&quot;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. Updates to [[mw:Special:MyLanguage/Manual:Pywikibot|Pywikibot]] or [https://github.com/ChlodAlejandro/wikimedia-streams wikimedia-streams] will discard these events by default. [https://phabricator.wikimedia.org/T266798]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/46|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W46&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:52, 13 November 2023 (UTC)
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== Tech News: 2023-47 ==

&lt;section begin=&quot;technews-2023-W47&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/47|Translations]] are available.

'''Changes later this week'''
* There is no new MediaWiki version this week. [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* Starting on Wednesday, a new set of Wikipedias will get &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]&quot; ({{int:project-localized-name-quwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-rmwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-rmywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-rnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-roa_rupwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-roa_tarawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ruewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-rwwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-sawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-sahwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-satwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-scwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-scnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-scowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-sdwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-sewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-sgwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-shwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-siwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-skwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-slwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-smwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-sowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-sqwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-srwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-srnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-sswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-stwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-stqwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-suwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-szlwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tcywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tetwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tgwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-thwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tkwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tlwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-towiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tpiwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-trwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ttwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-twwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tyvwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-udmwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ugwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-uzwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-vewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-vecwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-vepwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-vlswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-vowiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T308141][https://phabricator.wikimedia.org/T308142][https://phabricator.wikimedia.org/T308143] 
* The Vector 2022 skin will have some minor visual changes to drop-down menus, column widths, and more. These changes were added to four Wikipedias last week. If no issues are found, these changes will proceed to all wikis this week. These changes will make it possible to add new menus for readability and dark mode. [[mw:Special:MyLanguage/Reading/Web/Desktop_Improvements/Updates#November_2023:_Visual_changes,_more_deployments,_and_shifting_focus|Learn more]]. [https://phabricator.wikimedia.org/T347711]

'''Future changes'''
* There is [[mw:Extension talk:Graph/Plans#Update: 15 November|an update on re-enabling the Graph Extension]]. To speed up the process, Vega 2 will not be supported and only [https://phabricator.wikimedia.org/T335325 some protocols] will be available at launch. You can help by sharing what you think about the plan.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/47|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W47&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:55, 21 November 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25884616 --&gt;

== Tech News: 2023-48 ==

&lt;section begin=&quot;technews-2023-W48&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/48|Translations]] are available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.7|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-11-28|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-11-29|en}}. It will be on all wikis from {{#time:j xg|2023-11-30|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). There is no new MediaWiki version next week. [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] MediaWiki's JavaScript system will now allow &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;async&lt;/code&gt;/&lt;code&gt;await&lt;/code&gt;&lt;/bdi&gt; syntax in gadgets and user scripts. Gadget authors should remember that users' browsers may not support it, so it should be used appropriately. [https://phabricator.wikimedia.org/T343499]
* The deployment of &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add_a_link|Add a link]]&quot; announced [[m:Special:MyLanguage/Tech/News/2023/47|last week]] was postponed. It will resume this week.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/48|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W48&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:08, 27 November 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25906379 --&gt;

== Tech News: 2023-49 ==

&lt;section begin=&quot;technews-2023-W49&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/49|Translations]] are available.

'''Recent changes'''
* The spacing between paragraphs on Vector 2022 has been changed from 7px to 14px to match the size of the text. This will make it easier to distinguish paragraphs from sentences. [https://phabricator.wikimedia.org/T351754]
* The &quot;{{int:Visualeditor-dialog-meta-categories-defaultsort-label}}&quot; feature in VisualEditor is working again. You no longer need to switch to source editing to edit &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;{{DEFAULTSORT:...}}&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; keywords. [https://phabricator.wikimedia.org/T337398]

'''Changes later this week'''
* There is no new MediaWiki version this week. [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* On 6 December, people who have the enabled the preference for &quot;{{int:Discussiontools-preference-visualenhancements}}&quot; will notice the [[mw:Special:MyLanguage/Talk pages project/Usability|talk page usability improvements]] appear on pages that include the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;__NEWSECTIONLINK__&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; magic word. If you notice any issues, please [[phab:T352232|share them with the team on Phabricator]].

'''Future changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The Toolforge [[wikitech:News/Toolforge Grid Engine deprecation|Grid Engine shutdown process]] will start on December 14. Maintainers of [[toolforge:grid-deprecation|tools that still use this old system]] should plan to migrate to Kubernetes, or tell the team your plans on Phabricator in the task about your tool, before that date. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/VIWWQKMSQO2ED3TVUR7KPPWRTOBYBVOA/]
* Communities using [[mw:Special:MyLanguage/Structured_Discussions|Structured Discussions]] are being contacted regarding [[mw:Special:MyLanguage/Structured_Discussions/Deprecation|the upcoming deprecation of Structured Discussions]]. You can read more about this project, and share your comments, [[mw:Special:MyLanguage/Structured_Discussions/Deprecation|on the project's page]].

'''Events'''
* Registration &amp; Scholarship applications are now open for the [[mw:Special:MyLanguage/Wikimedia Hackathon 2024|Wikimedia Hackathon 2024]] that will take place from 3–5 May in Tallinn, Estonia. Scholarship applications are open until 5 January 2024.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/49|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W49&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:50, 4 December 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25914435 --&gt;

== Tech News: 2023-50 ==

&lt;section begin=&quot;technews-2023-W50&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/50|Translations]] are available.

'''Recent changes'''
* On Wikimedia Commons, there are some minor user-interface improvements for the &quot;choosing own vs not own work&quot; step in the UploadWizard. This is part of the Structured Content team's project of [[:commons:Commons:WMF support for Commons/Upload Wizard Improvements|improving UploadWizard on Commons]]. [https://phabricator.wikimedia.org/T352707][https://phabricator.wikimedia.org/T352709]

'''Problems'''
* There was a problem showing the [[mw:Special:MyLanguage/Growth/Personalized first day/Newcomer homepage|Newcomer homepage]] feature with the &quot;impact module&quot; and their page-view graphs, for a few days in early December. This has now been fixed. [https://phabricator.wikimedia.org/T352352][https://phabricator.wikimedia.org/T352349]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.9|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-12-12|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-12-13|en}}. It will be on all wikis from {{#time:j xg|2023-12-14|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

'''Future changes'''
* [[File:Octicons-tools.svg|15px|link=]] The [https://wikimediafoundation.limesurvey.net/796964 2023 Developer Satisfaction Survey] is seeking the opinions of the Wikimedia developer community. Please take the survey if you have any role in developing software for the Wikimedia ecosystem. The survey is open until 5 January 2024, and has an associated [[foundation:Legal:December_2023_Developer_Satisfaction_Survey|privacy statement]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/50|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W50&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 02:12, 12 December 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25945501 --&gt;

== Tech News: 2023-51 ==

&lt;section begin=&quot;technews-2023-W51&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/51|Translations]] are available.

'''Tech News'''
* The next issue of Tech News will be sent out on 8 January 2024 because of [[w:en:Christmas and holiday season|the holidays]].

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.10|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-12-19|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-12-20|en}}. It will be on all wikis from {{#time:j xg|2023-12-21|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). There is no new MediaWiki version next week. [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* Starting December 18, it won't be possible to activate Structured Discussions on a user's own talk page using the Beta feature. The Beta feature option remains available for users who want to deactivate Structured Discussions. This is part of [[mw:Structured Discussions/Deprecation|Structured Discussions' deprecation work]]. [https://phabricator.wikimedia.org/T248309] 
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] There will be full support for redirects in the Module namespace. The &quot;Move Page&quot; feature will leave an appropriate redirect behind, and such redirects will be appropriately recognized by the software (e.g. hidden from [[{{#special:UnconnectedPages}}]]). There will also be support for [[mw:Special:MyLanguage/Extension:Scribunto/Lua reference manual#Renaming or moving modules|manual redirects]]. [https://phabricator.wikimedia.org/T120794]

'''Future changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The MediaWiki JavaScript documentation is moving to a new format. During the move, you can read the old docs using [https://doc.wikimedia.org/mediawiki-core/REL1_41/js/ version 1.41]. Feedback about [https://doc.wikimedia.org/mediawiki-core/master/js/ the new site] is welcome on the [[mw:Talk:JSDoc_WMF_theme|project talk page]].
* The Wishathon is a new initiative that encourages collaboration across the Wikimedia community to develop solutions for wishes collected through the [[m:Special:MyLanguage/Community Wishlist Survey|Community Wishlist Survey]]. The first community Wishathon will take place from 15–17 March. If you are interested in a project proposal as a user, developer, designer, or product lead, you can [[m:Special:MyLanguage/Event:WishathonMarch2024|register for the event and read more]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/51|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W51&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 16:17, 18 December 2023 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25959059 --&gt;

== Tech News: 2024-02 ==

&lt;section begin=&quot;technews-2024-W02&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/02|Translations]] are available.

'''Recent changes'''
* [https://mediawiki2latex.wmflabs.org/ mediawiki2latex] is a tool that converts wiki content into the formats of LaTeX, PDF, ODT, and EPUB. The code now runs many times faster due to recent improvements. There is also an optional Docker container you can [[b:de:Benutzer:Dirk_Hünniger/wb2pdf/install#Using_Docker|install]] on your local machine.
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The way that Random pages are selected has been updated. This will slowly reduce the problem of some pages having a lower chance of appearing. [https://phabricator.wikimedia.org/T309477]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.13|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-01-09|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-01-10|en}}. It will be on all wikis from {{#time:j xg|2024-01-11|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/02|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W02&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:19, 9 January 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26026251 --&gt;

== Tech News: 2024-03 ==

&lt;section begin=&quot;technews-2024-W03&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/03|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Pages that use the JSON [[mw:Special:MyLanguage/Manual:ContentHandler|contentmodel]] will now use tabs instead of spaces for auto-indentation. This will significantly reduce the page size. [https://phabricator.wikimedia.org/T326065]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] [[mw:Special:MyLanguage/Extension:Gadgets|Gadgets]] and personal user scripts may now use JavaScript syntax introduced in ES6 (also known as &quot;ES2015&quot;) and ES7 (&quot;ES2016&quot;). MediaWiki validates the source code to protect other site functionality from syntax errors, and to ensure scripts are valid in all [[mw:Special:MyLanguage/Compatibility#Browsers|supported browsers]]. Previously, Gadgets could use the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;requiresES6&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; option. This option is no longer needed and will be removed in the future. [https://phabricator.wikimedia.org/T75714]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] [[mw:Special:MyLanguage/Manual:Bot passwords|Bot passwords]] and [[mw:Special:MyLanguage/OAuth/Owner-only consumers|owner-only OAuth consumers]] can now be restricted to allow editing only specific pages. [https://phabricator.wikimedia.org/T349957]
* You can now [[mw:Special:MyLanguage/Extension:Thanks|thank]] edits made by bots. [https://phabricator.wikimedia.org/T341388]
* An update on the status of the Community Wishlist Survey for 2024 [[m:Special:MyLanguage/Community Wishlist Survey/Future Of The Wishlist/January 4, 2024 Update|has been published]]. Please read and give your feedback.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.14|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-01-16|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-01-17|en}}. It will be on all wikis from {{#time:j xg|2024-01-18|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* Starting on January 17, it will not be possible to login to Wikimedia wikis from some specific old versions of the Chrome browser (versions 51–66, released between 2016 and 2018). Additionally, users of iOS 12, or Safari on Mac OS 10.14, may need to login to each wiki separately. [https://phabricator.wikimedia.org/T344791]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;jquery.cookie&lt;/code&gt;&lt;/bdi&gt; module was deprecated and replaced with the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;mediawiki.cookie&lt;/code&gt;&lt;/bdi&gt; module last year. A script has now been run to replace any remaining uses, and this week the temporary alias will be removed. [https://phabricator.wikimedia.org/T354966]

'''Future changes'''
* Wikimedia Deutschland is working to [[m:WMDE Technical Wishes/Reusing references|make reusing references easier]]. They are looking for people who are interested in participating in [https://wikimedia.sslsurvey.de/User-research-into-Reusing-References-Sign-up-Form-2024/en/ individual video calls for user research in January and February].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/03|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W03&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:13, 16 January 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26074460 --&gt;

== Tech News: 2024-04 ==

&lt;section begin=&quot;technews-2024-W04&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/04|Translations]] are available.

'''Problems'''
* A bug in UploadWizard prevented linking to the userpage of the uploader when uploading. It has now been fixed. [https://phabricator.wikimedia.org/T354529]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.15|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-01-23|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-01-24|en}}. It will be on all wikis from {{#time:j xg|2024-01-25|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/04|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W04&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:03, 23 January 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26096197 --&gt;

== Tech News: 2024-05 ==

&lt;section begin=&quot;technews-2024-W05&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/05|Translations]] are available.

'''Recent changes'''
* Starting Monday January 29, all talk pages messages' timestamps will become a link. This link is a permanent link to the comment. It allows users to find the comment they are looking for, even if this comment was moved elsewhere. This will affect all wikis except for the English Wikipedia. You can read more about this change [https://diff.wikimedia.org/2024/01/29/talk-page-permalinks-dont-lose-your-threads/ on Diff] or [[mw:Special:MyLanguage/Help:DiscussionTools#Talk_pages_permalinking|on Mediawiki.org]].&lt;!-- The Diff post will be published on Monday morning UTC--&gt; [https://phabricator.wikimedia.org/T302011]
* There are some improvements to the CAPTCHA to make it harder for spam bots and scripts to bypass it. If you have feedback on this change, please comment on [[phab:T141490|the task]]. Staff are monitoring metrics related to the CAPTCHA, as well as secondary metrics such as account creations and edit counts.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.16|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-01-30|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-01-31|en}}. It will be on all wikis from {{#time:j xg|2024-02-01|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] On February 1, a link will be added to the &quot;Tools&quot; menu to download a [[w:en:QR code|QR code]] that links to the page you are viewing. There will also be a new [[{{#special:QrCode}}]] page to create QR codes for any Wikimedia URL. This addresses the [[m:Community Wishlist Survey 2023/Mobile and apps/Add ability to share QR code for a page in any Wikimedia project|#19 most-voted wish]] from the [[m:Community Wishlist Survey 2023/Results|2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T329973]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] [[mw:Special:MyLanguage/Extension:Gadgets|Gadgets]] which only work in some skins have sometimes used the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;targets&lt;/code&gt;&lt;/bdi&gt; option to limit where you can use them. This will stop working this week. You should use the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;skins&lt;/code&gt;&lt;/bdi&gt; option instead. [https://phabricator.wikimedia.org/T328497]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/05|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W05&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 19:31, 29 January 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26137870 --&gt;

== Tech News: 2024-06 ==

&lt;section begin=&quot;technews-2024-W06&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/06|Translations]] are available.

'''Recent changes'''
*The mobile site history pages now use the same HTML as the desktop history pages. If you hear of any problems relating to mobile history usage please point them to [[phab:T353388|the phabricator task]].
*On most wikis, admins can now block users from making specific actions. These actions are: uploading files, creating new pages, moving (renaming) pages, and sending thanks. The goal of this feature is to allow admins to apply blocks that are adequate to the blocked users' activity. [[m:Special:MyLanguage/Community health initiative/Partial blocks#action-blocks|Learn more about &quot;action blocks&quot;]]. [https://phabricator.wikimedia.org/T242541][https://phabricator.wikimedia.org/T280531]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.17|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-02-06|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-02-07|en}}. It will be on all wikis from {{#time:j xg|2024-02-08|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* Talk pages permalinks that included diacritics and non-Latin script were malfunctioning. This issue is fixed. [https://phabricator.wikimedia.org/T356199]

'''Future changes'''
* [[m:WMDE Technical Wishes/ReferencePreviews#24WPs|24 Wikipedias]] with [[mw:Special:MyLanguage/Reference_Tooltips|Reference Tooltips]] as a default gadget are encouraged to remove that default flag. This would make [[mw:Special:MyLanguage/Help:Reference_Previews|Reference Previews]] the new default for reference popups, leading to a more consistent experience across wikis. For [[m:WMDE Technical Wishes/ReferencePreviews#46WPs|46 Wikipedias]] with less than 4 interface admins, the change is already scheduled for mid-February, [[m:Talk:WMDE Technical Wishes/ReferencePreviews#Reference Previews to become the default for previewing references on more wikis.|unless there are concerns]]. The older Reference Tooltips gadget will still remain usable and will override this feature, if it is available on your wiki and you have enabled it in your settings. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/ReferencePreviews#Reference_Previews_to_become_the_default_for_previewing_references_on_more_wikis][https://phabricator.wikimedia.org/T355312]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/06|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W06&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 19:22, 5 February 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26180971 --&gt;

== Tech News: 2024-07 ==

&lt;section begin=&quot;technews-2024-W07&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/07|Translations]] are available.

'''Recent changes'''
*  The [[d:Wikidata:SPARQL query service/WDQS graph split|WDQS Graph Split experiment]] is working and loaded onto 3 test servers. The team in charge is testing the split's impact and requires feedback from WDQS users through the UI or programmatically in different channels. [https://www.wikidata.org/wiki/Wikidata_talk:SPARQL_query_service/WDQS_graph_split][https://phabricator.wikimedia.org/T356773][https://www.wikidata.org/wiki/User:Sannita_(WMF)] Users' feedback will validate the impact of various use cases and workflows around the Wikidata Query service. [https://www.wikidata.org/wiki/Wikidata:SPARQL_query_service/WDQS_backend_update/October_2023_scaling_update][https://www.mediawiki.org/wiki/Wikidata_Query_Service/User_Manual#Federation]

'''Problems'''
*There was a bug that affected the appearance of visited links when using mobile device to access wiki sites. It made the links appear black; [[phab:T356928|this issue]] is fixed.  

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.18|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-02-13|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-02-14|en}}. It will be on all wikis from {{#time:j xg|2024-02-15|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] As work continues on the grid engine deprecation,[https://wikitech.wikimedia.org/wiki/News/Toolforge_Grid_Engine_deprecation] tools on the grid engine will be stopped starting on February 14th, 2024. If you have tools actively migrating you can ask for an extension so they are not stopped. [https://wikitech.wikimedia.org/wiki/Portal:Toolforge/About_Toolforge#Communication_and_support] 

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/07|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W07&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 05:48, 13 February 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26223994 --&gt;

== Tech News: 2024-08 ==

&lt;section begin=&quot;technews-2024-W08&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/08|Translations]] are available.

'''Recent changes'''
* If you have the &quot;{{int:Tog-enotifwatchlistpages}}&quot; option enabled, edits by bot accounts no longer trigger notification emails. Previously, only minor edits would not trigger the notification emails. [https://phabricator.wikimedia.org/T356984]
* There are changes to how user and site scripts load for [[mw:Special:MyLanguage/Skin:Vector/2022| Vector 2022]] on specific wikis. The changes impacted the following Wikis: all projects with [[mw:Special:MyLanguage/Skin:Vector|Vector legacy]] as the default skin, Wikivoyage, and Wikibooks. Other wikis will be affected over the course of the next three months. Gadgets are not impacted. If you have been affected or want to minimize the impact on your project, see [[Phab:T357580| this ticket]]. Please coordinate and take action proactively.
*Newly auto-created accounts (the accounts you get when you visit a new wiki) now have the same local notification preferences as users who freshly register on that wiki. It is effected in four notification types listed in the [[phab:T353225|task's description]].
*The maximum file size when using [[c:Special:MyLanguage/Commons:Upload_Wizard|Upload Wizard]] is now 5 GiB. [https://phabricator.wikimedia.org/T191804]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.19|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-02-20|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-02-21|en}}. It will be on all wikis from {{#time:j xg|2024-02-22|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Selected tools on the grid engine have been [[wikitech:News/Toolforge_Grid_Engine_deprecation|stopped]] as we prepare to shut down the grid on March 14th, 2024. The tool's code and data have not been deleted. If you are a maintainer and you want your tool re-enabled reach out to the [[wikitech:Portal:Toolforge/About_Toolforge#Communication_and_support|team]]. Only tools that have asked for extension are still running on the grid.
* The CSS &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;[https://developer.mozilla.org/en-US/docs/Web/CSS/filter filter]&lt;/code&gt;&lt;/bdi&gt; property can now be used in HTML &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;style&lt;/code&gt;&lt;/bdi&gt; attributes in wikitext. [https://phabricator.wikimedia.org/T308160]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/08|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W08&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 15:36, 19 February 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26254282 --&gt;

== Tech News: 2024-09 ==

&lt;section begin=&quot;technews-2024-W09&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/09|Translations]] are available.

'''Recent changes'''
* The [[mw:Special:MyLanguage/VisualEditor_on_mobile|mobile visual editor]] is now the default editor for users who never edited before, at a small group of wikis. [[mw:Special:MyLanguage/VisualEditor_on_mobile/VE_mobile_default#A/B_test_results| Research ]] shows that users using this editor are slightly more successful publishing the edits they started, and slightly less successful publishing non-reverted edits. Users who defined the wikitext editor as their default on desktop will get the wikitext editor on mobile for their first edit on mobile as well. [https://phabricator.wikimedia.org/T352127]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The [[mw:Special:MyLanguage/ResourceLoader/Core modules#mw.config|mw.config]] value &lt;code&gt;wgGlobalGroups&lt;/code&gt; now only contains groups that are active in the wiki. Scripts no longer have to check whether the group is active on the wiki via an API request. A code example of the above is: &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;if (/globalgroupname/.test(mw.config.get(&quot;wgGlobalGroups&quot;)))&lt;/code&gt;&lt;/bdi&gt;. [https://phabricator.wikimedia.org/T356008]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.20|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-02-27|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-02-28|en}}. It will be on all wikis from {{#time:j xg|2024-02-29|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

'''Future changes'''
* The right to change [[mw:Special:MyLanguage/Manual:Tags|edit tags]] (&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;changetags&lt;/code&gt;&lt;/bdi&gt;) will be removed from users in Wikimedia sites, keeping it by default for admins and bots only. Your community can ask to retain the old configuration on your wiki before this change happens. Please indicate in [[phab:T355639|this ticket]] to keep it for your community before the end of March 2024.
'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/09|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W09&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 19:23, 26 February 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26294125 --&gt;

== Tech News: 2024-10 ==

&lt;section begin=&quot;technews-2024-W10&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/10|Translations]] are available.

'''Recent changes'''
* The &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;Special:Book&lt;/code&gt;&lt;/bdi&gt; page (as well as the associated &quot;Create a book&quot; functionality) provided by the old [[mw:Special:MyLanguage/Extension:Collection|Collection extension]] has been removed from all Wikisource wikis, as it was broken. This does not affect the ability to download normal books, which is provided by the [[mw:Special:MyLanguage/Extension:Wikisource|Wikisource extension]]. [https://phabricator.wikimedia.org/T358437]
* [[m:Wikitech|Wikitech]] now uses the next-generation [[mw:Special:MyLanguage/Parsoid|Parsoid]] wikitext parser by default to generate all pages in the Talk namespace.  Report any problems on the [[mw:Talk:Parsoid/Parser_Unification/Known_Issues|Known Issues discussion page]]. You can use the [[mw:Special:MyLanguage/Extension:ParserMigration|ParserMigration]] extension to control the use of Parsoid; see the [[mw:Special:MyLanguage/Help:Extension:ParserMigration|ParserMigration help documentation]] for more details.
* Maintenance on [https://etherpad.wikimedia.org etherpad] is completed. If you encounter any issues, please indicate in [[phab:T316421|this ticket]].
* [[File:Octicons-tools.svg|12px|link=|alt=| Advanced item]] [[mw:Special:MyLanguage/Extension:Gadgets|Gadgets]] allow interface admins to create custom features with CSS and JavaScript. The &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;Gadget&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;Gadget_definition&lt;/code&gt;&lt;/bdi&gt; namespaces and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;gadgets-definition-edit&lt;/code&gt;&lt;/bdi&gt; user right were reserved for an experiment in 2015, but were never used. These were visible on Special:Search and Special:ListGroupRights. The unused namespaces and user rights are now removed. No pages are moved, and no changes need to be made. [https://phabricator.wikimedia.org/T31272]
* A usability improvement to the &quot;Add a citation&quot; in Wikipedia workflow has been made, the insert button was moved to the popup header. [https://phabricator.wikimedia.org/T354847]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.21|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-03-05|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-03-06|en}}. It will be on all wikis from {{#time:j xg|2024-03-07|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

'''Future changes'''
*  All wikis will be read-only for a few minutes on March 20. This is planned at 14:00 UTC. More information will be published in Tech News and will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T358233]
* The HTML markup of headings and section edit links will be changed later this year to improve accessibility. See [[mw:Special:MyLanguage/Heading_HTML_changes|Heading HTML changes]] for details. The new markup will be the same as in the new Parsoid wikitext parser. You can test your gadget or stylesheet with the new markup if you add &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;?useparsoid=1&lt;/code&gt;&lt;/bdi&gt; to your URL ([[mw:Special:MyLanguage/Help:Extension:ParserMigration#Selecting_a_parser_using_a_URL_query_string|more info]]) or turn on Parsoid read views in your user options ([[mw:Special:MyLanguage/Help:Extension:ParserMigration#Enabling_via_user_preference|more info]]).
*

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/10|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W10&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 19:47, 4 March 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26329807 --&gt;

== Tech News: 2024-11 ==

&lt;section begin=&quot;technews-2024-W11&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/11|Translations]] are available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.22|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-03-12|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-03-13|en}}. It will be on all wikis from {{#time:j xg|2024-03-14|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* After consulting with various communities, the line height of the text on the [[mw:Special:MyLanguage/Skin:Minerva Neue|Minerva skin]] will be increased to its previous value of 1.65. Different options for typography can also be set using the options in the menu, as needed. [https://phabricator.wikimedia.org/T358498]
*The active link color in [[mw:Special:MyLanguage/Skin:Minerva Neue|Minerva]] will be changed to provide more consistency with our other platforms and best practices. [https://phabricator.wikimedia.org/T358516]
* [[c:Special:MyLanguage/Commons:Structured data|Structured data on Commons]] will no longer ask whether you want to leave the page without saving. This will prevent the “information you’ve entered may not be saved” popups from appearing when no information have been entered. It will also make file pages on Commons load faster in certain cases. However, the popups will be hidden even if information has indeed been entered. If you accidentally close the page before saving the structured data you entered, that data will be lost. [https://phabricator.wikimedia.org/T312315]

'''Future changes'''
* All wikis will be read-only for a few minutes on March 20. This is planned at 14:00 UTC. More information will be published in Tech News and will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T358233][https://meta.wikimedia.org/wiki/Special:MyLanguage/Tech/Server_switch]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/11|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W11&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:04, 11 March 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26374013 --&gt;

== Tech News: 2024-12 ==

&lt;section begin=&quot;technews-2024-W12&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/12|Translations]] are available.

'''Recent changes'''
* The notice &quot;Language links are at the top of the page&quot; that appears in the [[mw:Special:MyLanguage/Skin:Vector/2022|Vector 2022 skin]] main menu has been removed now that users have learned the new location of the Language switcher. [https://phabricator.wikimedia.org/T353619]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] [[m:Special:MyLanguage/IP_Editing:_Privacy_Enhancement_and_Abuse_Mitigation/IP_Info_feature|IP info feature]] displays data from Spur, an IP addresses database. Previously, the only data source for this feature was MaxMind. Now, IP info is more useful for patrollers. [https://phabricator.wikimedia.org/T341395]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The Toolforge Grid Engine services have been shut down after the final migration process from Grid Engine to Kubernetes. [https://wikitech.wikimedia.org/wiki/Obsolete:Toolforge/Grid][https://wikitech.wikimedia.org/wiki/News/Toolforge_Grid_Engine_deprecation][https://techblog.wikimedia.org/2022/03/14/toolforge-and-grid-engine/]
* Communities can now customize the default reasons for undeleting a page by creating [[MediaWiki:Undelete-comment-dropdown]]. [https://phabricator.wikimedia.org/T326746]

'''Problems'''
* [[m:Special:MyLanguage/WMDE_Technical_Wishes/RevisionSlider|RevisionSlider]] is an interface to interactively browse a page's history. Users in [[mw:Special:MyLanguage/Extension:RevisionSlider/Developing_a_RTL-accessible_feature_in_MediaWiki_-_what_we%27ve_learned_while_creating_the_RevisionSlider|right-to-left]] languages reported RevisionSlider reacting wrong to mouse clicks. This should be fixed now. [https://phabricator.wikimedia.org/T352169]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.23|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-03-19|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-03-20|en}}. It will be on all wikis from {{#time:j xg|2024-03-21|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* All wikis will be read-only for a few minutes on March 20. This is planned at [https://zonestamp.toolforge.org/1710943200 14:00 UTC]. [https://phabricator.wikimedia.org/T358233][https://meta.wikimedia.org/wiki/Special:MyLanguage/Tech/Server_switch]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/12|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W12&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 17:39, 18 March 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26410165 --&gt;

== Tech News: 2024-13 ==

&lt;section begin=&quot;technews-2024-W13&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/13|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] An update was made on March 18th 2024 to how various projects load site, user JavaScript and CSS in [[mw:Special:MyLanguage/Skin:Vector/2022|Vector 2022 skin]]. A [[phab:T360384|checklist]] is provided for site admins to follow.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.24|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-03-26|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-03-27|en}}. It will be on all wikis from {{#time:j xg|2024-03-28|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/13|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W13&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 18:56, 25 March 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26446209 --&gt;

== Tech News: 2024-14 ==

&lt;section begin=&quot;technews-2024-W14&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/14|Translations]] are available.

'''Recent changes'''
* Users of the [[mw:Special:MyLanguage/Reading/Web/Accessibility_for_reading|reading accessibility]] beta feature will notice that the default line height for the standard and large text options has changed. [https://phabricator.wikimedia.org/T359030]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.25|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-04-02|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-04-03|en}}. It will be on all wikis from {{#time:j xg|2024-04-04|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

'''Future changes'''
* The Wikimedia Foundation has an annual plan. The annual plan decides what the Wikimedia Foundation will work on. You can now read [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2024-2025/Product &amp; Technology OKRs#Draft Key Results|the draft key results]] for the Product and Technology department. They are suggestions for what results the Foundation wants from big technical changes from July 2024 to June 2025. You can [[m:Talk:Wikimedia Foundation Annual Plan/2024-2025/Product &amp; Technology OKRs|comment on the talk page]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/14|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W14&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 03:36, 2 April 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26462933 --&gt;

== Tech News: 2024-15 ==

&lt;section begin=&quot;technews-2024-W15&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/15|Translations]] are available.

'''Recent changes'''
* Web browsers can use tools called [[:w:en:Browser extension|extensions]]. There is now a Chrome extension called [[m:Future Audiences/Experiment:Citation Needed|Citation Needed]] which you can use to see if an online statement is supported by a Wikipedia article. This is a small experiment to see if Wikipedia can be used this way. Because it is a small experiment, it can only be used in Chrome in English.
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] A new [[mw:Special:MyLanguage/Help:Edit Recovery|Edit Recovery]] feature has been added to all wikis, available as a [[Special:Preferences#mw-prefsection-editing|user preference]]. Once you enable it, your in-progress edits will be stored in your web browser, and if you accidentally close an editing window or your browser or computer crashes, you will be prompted to recover the unpublished text. Please leave any feedback on the [[m:Special:MyLanguage/Talk:Community Wishlist Survey 2023/Edit-recovery feature|project talk page]]. This was the #8 wish in the 2023 Community Wishlist Survey.
* Initial results of [[mw:Special:MyLanguage/Edit check|Edit check]] experiments [[mw:Special:MyLanguage/Edit_check#4_April_2024|have been published]]. Edit Check is now deployed as a default feature at [[phab:T342930#9538364|the wikis that tested it]]. [[mw:Talk:Edit check|Let us know]] if you want your wiki to be part of the next deployment of Edit check. [https://phabricator.wikimedia.org/T342930][https://phabricator.wikimedia.org/T361727]
* Readers using the [[mw:Special:MyLanguage/Skin:Minerva Neue|Minerva skin]] on mobile will notice there has been an improvement in the line height across all typography settings. [https://phabricator.wikimedia.org/T359029]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.26|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-04-09|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-04-10|en}}. It will be on all wikis from {{#time:j xg|2024-04-11|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* New accounts and logged-out users will get the [[mw:Special:MyLanguage/VisualEditor|visual editor]] as their default editor on mobile. This deployment is made at all wikis except for the English Wikipedia. [https://phabricator.wikimedia.org/T361134]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/15|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W15&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:37, 8 April 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26564838 --&gt;

== Tech News: 2024-16 ==

&lt;section begin=&quot;technews-2024-W16&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/16|Translations]] are available.

'''Problems'''
* Between 2 April and 8 April, on wikis using [[mw:Special:MyLanguage/Extension:FlaggedRevs|Flagged Revisions]], the &quot;{{Int:tag-mw-reverted}}&quot; tag was not applied to undone edits. In addition, page moves, protections and imports were not autoreviewed. This problem is now fixed. [https://phabricator.wikimedia.org/T361918][https://phabricator.wikimedia.org/T361940]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.1|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-04-16|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-04-17|en}}. It will be on all wikis from {{#time:j xg|2024-04-18|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* [[mw:Special:MyLanguage/Help:Magic words#DEFAULTSORT|Default category sort keys]] will now affect categories added by templates placed in [[mw:Special:MyLanguage/Help:Cite|footnotes]]. Previously footnotes used the page title as the default sort key even if a different default sort key was specified (category-specific sort keys already worked). [https://phabricator.wikimedia.org/T40435]
* A new variable &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;page_last_edit_age&lt;/code&gt;&lt;/bdi&gt; will be added to [[Special:AbuseFilter|abuse filters]]. It tells how many seconds ago the last edit to a page was made. [https://phabricator.wikimedia.org/T269769]

'''Future changes'''
* Volunteer developers are kindly asked to update the code of their tools and features to handle [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|temporary accounts]]. [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/For developers/2024-04 CTA|Learn more]].
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Four database fields will be removed from database replicas (including [[quarry:|Quarry]]). This affects only the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;abuse_filter&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;abuse_filter_history&lt;/code&gt;&lt;/bdi&gt; tables. Some queries might need to be updated. [https://phabricator.wikimedia.org/T361996]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/16|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W16&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:29, 15 April 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26564838 --&gt;

== Tech News: 2024-17 ==

&lt;section begin=&quot;technews-2024-W17&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/17|Translations]] are available.

'''Recent changes'''
* Starting this week, newcomers editing Wikipedia [[mw:Special:MyLanguage/Growth/Positive reinforcement#Leveling up 3|will be encouraged]] to try structured tasks. [[mw:Special:MyLanguage/Growth/Feature summary#Newcomer tasks|Structured tasks]] have been shown to [[mw:Special:MyLanguage/Growth/Personalized first day/Structured tasks/Add a link/Experiment analysis, December 2021|improve newcomer activation and retention]]. [https://phabricator.wikimedia.org/T348086]
* You can [[m:Special:MyLanguage/Coolest Tool Award|nominate your favorite tools]] for the fifth edition of the Coolest Tool Award. Nominations will be open until May 10. 

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.2|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-04-23|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-04-24|en}}. It will be on all wikis from {{#time:j xg|2024-04-25|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

'''Future changes'''
* This is the last warning that by the end of May 2024 the Vector 2022 skin will no longer share site and user scripts/styles with old Vector. For user-scripts that you want to keep using on Vector 2022, copy the contents of [[{{#special:MyPage}}/vector.js]] to [[{{#special:MyPage}}/vector-2022.js]]. There are [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Features/Loading Vector 2010 scripts|more technical details]] available. Interface administrators who foresee this leading to lots of technical support questions may wish to send a mass message to your community, as was done on French Wikipedia. [https://phabricator.wikimedia.org/T362701]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/17|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W17&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:28, 22 April 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26647188 --&gt;

== Tech News: 2024-18 ==

&lt;section begin=&quot;technews-2024-W18&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/18|Translations]] are available.

'''Recent changes'''

[[File:Talk_pages_default_look_(April_2023).jpg|thumb|alt=Screenshot of the visual improvements made on talk pages|Example of a talk page with the new design, in French.]]
* The appearance of talk pages changed for the following wikis: {{int:project-localized-name-azwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-dewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-fawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hiwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-idwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nlwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ptwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-rowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-thwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-trwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ukwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-viwiki/en}}. These wikis participated to a test, where 50% of users got the new design, for one year. As this test [[Mw:Special:MyLanguage/Talk pages project/Usability/Analysis|gave positive results]], the new design is deployed on these wikis as the default design. It is possible to opt-out these changes [[Special:Preferences#mw-prefsection-editing|in user preferences]] (&quot;{{int:discussiontools-preference-visualenhancements}}&quot;). The deployment will happen at all wikis in the coming weeks. [https://phabricator.wikimedia.org/T341491]
* Seven new wikis have been created: 
** a {{int:project-localized-name-group-wikipedia}} in [[d:Q33014|Betawi]] ([[w:bew:|&lt;code&gt;w:bew:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T357866]
** a {{int:project-localized-name-group-wikipedia}} in [[d:Q35708|Kusaal]] ([[w:kus:|&lt;code&gt;w:kus:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T359757]
** a {{int:project-localized-name-group-wikipedia}} in [[d:Q35513|Igala]] ([[w:igl:|&lt;code&gt;w:igl:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T361644]
** a {{int:project-localized-name-group-wiktionary}} in [[d:Q33541|Karakalpak]] ([[wikt:kaa:|&lt;code&gt;wikt:kaa:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T362135]
** a {{int:project-localized-name-group-wikisource}} in [[d:Q9228|Burmese]] ([[s:my:|&lt;code&gt;s:my:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T361085]
** a {{int:project-localized-name-group-wikisource}} in [[d:Q9237|Malay]] ([[s:ms:|&lt;code&gt;s:ms:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T363039]
** a {{int:project-localized-name-group-wikisource}} in [[d:Q8108|Georgian]] ([[s:ka:|&lt;code&gt;s:ka:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T363085]
* You can now [https://translatewiki.net/wiki/Support#Early_access:_Watch_Message_Groups_on_Translatewiki.net watch message groups/projects] on [[m:Special:MyLanguage/translatewiki.net|Translatewiki.net]]. Initially, this feature will notify you of added or deleted messages in these groups. [https://phabricator.wikimedia.org/T348501]
* Dark mode is now available on all wikis, on mobile web for logged-in users who opt into the [[Special:MobileOptions|advanced mode]]. This is the early release of the feature. Technical editors are invited to [https://night-mode-checker.wmcloud.org/ check for accessibility issues on wikis]. See [[mw:Special:MyLanguage/Reading/Web/Accessibility for reading/Updates/2024-04|more detailed guidelines]].

'''Problems'''
* [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] maps can use an alternative visual style without labels, by using &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;mapstyle=&quot;osm&quot;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. This wasn't working in previews, creating the wrong impression that it wasn't supported. This has now been fixed. [https://phabricator.wikimedia.org/T362531]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.3|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-04-30|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-05-01|en}}. It will be on all wikis from {{#time:j xg|2024-05-02|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/18|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W18&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 03:33, 30 April 2024 (UTC)
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== Tech News: 2024-19 ==

&lt;section begin=&quot;technews-2024-W19&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/19|Translations]] are available.

'''Recent changes'''
[[File:Talk_pages_default_look_(April_2023).jpg|thumb|alt=Screenshot of the visual improvements made on talk pages|Example of a talk page with the new design, in French.]]
* The appearance of talk pages changed for all wikis, except for Commons, Wikidata and most Wikipedias ([[m:Special:MyLanguage/Tech/News/2024/18|a few]] have already received this design change). You can read the detail of the changes [[diffblog:2024/05/02/making-talk-pages-better-for-everyone/|on ''Diff'']]. It is possible to opt-out these changes [[Special:Preferences#mw-prefsection-editing|in user preferences]] (&quot;{{int:discussiontools-preference-visualenhancements}}&quot;). The deployment will happen at remaining wikis in the coming weeks. [https://phabricator.wikimedia.org/T352087][https://phabricator.wikimedia.org/T319146]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Interface admins now have greater control over the styling of article components on mobile with the introduction of the &lt;code&gt;SiteAdminHelper&lt;/code&gt;. More information on how styles can be disabled can be found [[mw:Special:MyLanguage/Extension:WikimediaMessages#Site_admin_helper|at the extension's page]]. [https://phabricator.wikimedia.org/T363932]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] [[m:Special:MyLanguage/Wikimedia Enterprise|Wikimedia Enterprise]] has added article body sections in JSON format and a curated short description field to the existing parsed Infobox. This expansion to the API is also available via Wikimedia Cloud Services. [https://enterprise.wikimedia.com/blog/article-sections-and-description/]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.4|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-05-07|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-05-08|en}}. It will be on all wikis from {{#time:j xg|2024-05-09|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* When you look at the Special:Log page, the first view is labelled &quot;All public logs&quot;, but it only shows some logs. This label will now say &quot;Main public logs&quot;. [https://phabricator.wikimedia.org/T237729]

'''Future changes'''
* A new service will be built to replace [[mw:Special:MyLanguage/Extension:Graph|Extension:Graph]]. Details can be found in [[mw:Special:MyLanguage/Extension:Graph/Plans|the latest update]] regarding this extension.
* Starting May 21, English Wikipedia and German Wikipedia will get the possibility to activate &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]&quot;. This is part of the [[phab:T304110|progressive deployment of this tool to all Wikipedias]]. These communities can [[mw:Special:MyLanguage/Growth/Community configuration|activate and configure the feature locally]]. [https://phabricator.wikimedia.org/T308144]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/19|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W19&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 16:44, 6 May 2024 (UTC)
&lt;!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26729363 --&gt;

== Tech News: 2024-20 ==

&lt;section begin=&quot;technews-2024-W20&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/20|Translations]] are available.

'''Recent changes'''
* On Wikisource there is a special page listing pages of works without corresponding scan images. Now you can use the new magic word &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;__EXPECTWITHOUTSCANS__&lt;/code&gt;&lt;/bdi&gt; to exclude certain pages (list of editions or translations of works) from that list. [https://phabricator.wikimedia.org/T344214]
* If you use the [[Special:Preferences#mw-prefsection-editing|user-preference]] &quot;{{int:tog-uselivepreview}}&quot;, then the template-page feature &quot;{{int:Templatesandbox-editform-legend}}&quot; will now also work without reloading the page. [https://phabricator.wikimedia.org/T136907]
* [[mw:Special:Mylanguage/Extension:Kartographer|Kartographer]] maps can now specify an alternative text via the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;alt=&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; attribute. This is identical in usage to the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;alt=&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; attribute in the [[mw:Special:MyLanguage/Help:Images#Syntax|image and gallery syntax]]. An exception for this feature is wikis like Wikivoyage where the miniature maps are interactive. [https://phabricator.wikimedia.org/T328137]
* The old [[mw:Special:MyLanguage/Extension:GuidedTour|Guided Tour]] for the &quot;[[mw:Special:MyLanguage/Edit Review Improvements/New filters for edit review|New Filters for Edit Review]]&quot; feature has been removed. It was created in 2017 to show people with older accounts how the interface had changed, and has now been seen by most of the intended people. [https://phabricator.wikimedia.org/T217451]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.5|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-05-14|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-05-15|en}}. It will be on all wikis from {{#time:j xg|2024-05-16|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The [[{{#special:search}}]] results page will now use CSS flex attributes, for better accessibility, instead of a table. If you have a gadget or script that adjusts search results, you should update your script to the new HTML structure. [https://phabricator.wikimedia.org/T320295]

'''Future changes'''
* In the Vector 2022 skin, main pages will be displayed at full width (like special pages). The goal is to keep the number of characters per line large enough. This is related to the coming changes to typography in Vector 2022. [[mw:Special:MyLanguage/Reading/Web/Accessibility for reading/Updates|Learn more]]. [https://phabricator.wikimedia.org/T357706]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Two columns of the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;[[mw:Special:MyLanguage/Manual:pagelinks table|pagelinks]]&lt;/code&gt;&lt;/bdi&gt; database table (&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;pl_namespace&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;pl_title&lt;/code&gt;&lt;/bdi&gt;) are being dropped soon. Users must use two columns of the new &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;[[mw:special:MyLanguage/Manual:linktarget table|linktarget]]&lt;/code&gt;&lt;/bdi&gt; table instead (&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;lt_namespace&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;lt_title&lt;/code&gt;&lt;/bdi&gt;). In your existing SQL queries:
*# Replace &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;JOIN pagelinks&lt;/code&gt;&lt;/bdi&gt; with &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;JOIN linktarget&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;pl_&lt;/code&gt;&lt;/bdi&gt; with &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;lt_&lt;/code&gt;&lt;/bdi&gt; in the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;ON&lt;/code&gt;&lt;/bdi&gt; statement
*# Below that add &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;JOIN pagelinks ON lt_id = pl_target_id&lt;/code&gt;&lt;/bdi&gt;
** See &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[phab:T222224]]&lt;/bdi&gt; for technical reasoning. [https://phabricator.wikimedia.org/T222224][https://phabricator.wikimedia.org/T299947]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/20|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W20&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:58, 13 May 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26762074 --&gt;

== Tech News: 2024-21 ==

&lt;section begin=&quot;technews-2024-W21&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/21|Translations]] are available.

'''Recent changes'''
* The [[mw:Special:MyLanguage/Extension:Nuke|Nuke]] feature, which enables administrators to mass delete pages, will now correctly delete pages which were moved to another title. [https://phabricator.wikimedia.org/T43351]
* New changes have been made to the UploadWizard in Wikimedia Commons: the overall layout has been improved, by following new styling and spacing for the form and its fields; the headers and helper text for each of the fields was changed; the Caption field is now a required field, and there is an option for users to copy their caption into the media description. [https://commons.wikimedia.org/wiki/Commons:WMF_support_for_Commons/Upload_Wizard_Improvements#Changes_to_%22Describe%22_workflow][https://phabricator.wikimedia.org/T361049]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.6|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-05-21|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-05-22|en}}. It will be on all wikis from {{#time:j xg|2024-05-23|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]]  The HTML used to render all headings [[mw:Heading_HTML_changes|is being changed to improve accessibility]]. It will change on 22 May in some skins (Timeless, Modern, CologneBlue, Nostalgia, and Monobook). Please test gadgets on your wiki on these skins and [[phab:T13555|report any related problems]] so that they can be resolved before this change is made in all other skins. The developers are also considering the introduction of a [[phab:T337286|Gadget API for adding buttons to section titles]] if that would be helpful to tool creators, and would appreciate any input you have on that.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/21|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W21&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:04, 20 May 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26786311 --&gt;

== Tech News: 2024-22 ==

&lt;section begin=&quot;technews-2024-W22&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/22|Translations]] are available.

'''Recent changes'''
* Several bugs related to the latest updates to the UploadWizard on Wikimedia Commons have been fixed. For more information, see [[:phab:T365107|T365107]] and [[:phab:T365119|T365119]].
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] In March 2024 a new [[mw:ResourceLoader/Core_modules#addPortlet|addPortlet]] API was added to allow gadgets to create new portlets (menus) in the skin. In certain skins this can be used to create dropdowns. Gadget developers are invited to try it and [[phab:T361661|give feedback]].
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Some CSS in the Minerva skin has been removed to enable easier community configuration. Interface editors should check the rendering on mobile devices for aspects related to the classes: &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;.collapsible&lt;/code&gt;&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;.multicol&lt;/code&gt;&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;.reflist&lt;/code&gt;&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;.coordinates&lt;/code&gt;&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;.topicon&lt;/code&gt;&lt;/bdi&gt;. [[phab:T361659|Further details are available on replacement CSS]] if it is needed.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.7|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-05-28|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-05-29|en}}. It will be on all wikis from {{#time:j xg|2024-05-30|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* When you visit a wiki where you don't yet have a local account, local rules such as edit filters can sometimes prevent your account from being created. Starting this week, MediaWiki takes your global rights into account when evaluating whether you can override such local rules. [https://phabricator.wikimedia.org/T316303]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/22|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W22&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:15, 28 May 2024 (UTC)
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== Tech News: 2024-23 ==

&lt;section begin=&quot;technews-2024-W23&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/23|Translations]] are available.

'''Recent changes'''
* It is now possible for local administrators to add new links to the bottom of the site Tools menu without JavaScript. [[mw:Manual:Interface/Sidebar#Add or remove toolbox sections|Documentation is available]]. [https://phabricator.wikimedia.org/T6086]
* The message name for the definition of the tracking category of WikiHiero has changed from &quot;&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;MediaWiki:Wikhiero-usage-tracking-category&lt;/code&gt;&lt;/bdi&gt;&quot; to &quot;&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;MediaWiki:Wikihiero-usage-tracking-category&lt;/code&gt;&lt;/bdi&gt;&quot;. [https://gerrit.wikimedia.org/r/c/mediawiki/extensions/wikihiero/+/1035855]
* One new wiki has been created: a {{int:project-localized-name-group-wikipedia}} in [[d:Q5317225|Kadazandusun]] ([[w:dtp:|&lt;code&gt;w:dtp:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T365220]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.8|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-06-04|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-06-05|en}}. It will be on all wikis from {{#time:j xg|2024-06-06|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

'''Future changes'''
* Next week, on wikis with the Vector 2022 skin as the default, logged-out desktop users will be able to choose between different font sizes. The default font size will also be increased for them. This is to make Wikimedia projects easier to read. [[mw:Special:MyLanguage/Reading/Web/Accessibility for reading/Updates/2024-06 deployments|Learn more]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/23|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W23&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:35, 3 June 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26844397 --&gt;

== Tech News: 2024-24 ==

&lt;section begin=&quot;technews-2024-W24&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/24|Translations]] are available.

'''Recent changes'''
* The software used to render SVG files has been updated to a new version, fixing many longstanding bugs in SVG rendering. [https://phabricator.wikimedia.org/T265549]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The HTML used to render all headings [[mw:Heading HTML changes|is being changed to improve accessibility]]. It was changed last week in some skins (Vector legacy and Minerva). Please test gadgets on your wiki on these skins and [[phab:T13555|report any related problems]] so that they can be resolved before this change is made in Vector-2022. The developers are still considering the introduction of a [[phab:T337286|Gadget API for adding buttons to section titles]] if that would be helpful to tool creators, and would appreciate any input you have on that.
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The HTML markup used for citations by [[mw:Special:MyLanguage/Parsoid|Parsoid]] changed last week. In places where Parsoid previously added the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;mw-reference-text&lt;/code&gt;&lt;/bdi&gt; class, Parsoid now also adds the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;reference-text&lt;/code&gt;&lt;/bdi&gt; class for better compatibility with the legacy parser. [[mw:Specs/HTML/2.8.0/Extensions/Cite/Announcement|More details are available]]. [https://gerrit.wikimedia.org/r/1036705]

'''Problems'''
* There was a bug with the Content Translation interface that caused the tools menus to appear in the wrong location. This has now been fixed. [https://phabricator.wikimedia.org/T366374]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.9|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-06-11|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-06-12|en}}. It will be on all wikis from {{#time:j xg|2024-06-13|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The new version of MediaWiki includes another change to the HTML markup used for citations: [[mw:Special:MyLanguage/Parsoid|Parsoid]] will now generate a &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;&lt;span class=&quot;mw-cite-backlink&quot;&gt;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; wrapper for both named and unnamed references for better compatibility with the legacy parser. Interface administrators should verify that gadgets that interact with citations are compatible with the new markup. [[mw:Specs/HTML/2.8.0/Extensions/Cite/Announcement|More details are available]]. [https://gerrit.wikimedia.org/r/1035809] 
* On multilingual wikis that use the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;&lt;translate&gt;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; system, there is a feature that shows potentially-outdated translations with a pink background until they are updated or confirmed. From this week, confirming translations will be logged, and there is a new user-right that can be required for confirming translations if the community [[m:Special:MyLanguage/Requesting wiki configuration changes|requests it]]. [https://phabricator.wikimedia.org/T49177]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/24|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W24&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:20, 10 June 2024 (UTC)
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== Tech News: 2024-25 ==

&lt;section begin=&quot;technews-2024-W25&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/25|Translations]] are available.

'''Recent changes'''
* People who attempt to add an external link in the visual editor will now receive immediate feedback if they attempt to link to a domain that a project has decided to block. Please see [[mw:Special:MyLanguage/Edit_check#11_June_2024|Edit check]] for more details. [https://phabricator.wikimedia.org/T366751]
* The new [[mw:Special:MyLanguage/Extension:CommunityConfiguration|Community Configuration extension]] is available [[testwiki:Special:CommunityConfiguration|on Test Wikipedia]]. This extension allows communities to customize specific features to meet their local needs. Currently only Growth features are configurable, but the extension will support other [[mw:Special:MyLanguage/Community_configuration#Use_cases|Community Configuration use cases]] in the future. [https://phabricator.wikimedia.org/T323811][https://phabricator.wikimedia.org/T360954]
* The dark mode [[Special:Preferences#mw-prefsection-betafeatures|beta feature]] is now available on category and help pages, as well as more special pages. There may be contrast issues. Please report bugs on the [[mw:Talk:Reading/Web/Accessibility_for_reading|project talk page]]. [https://phabricator.wikimedia.org/T366370]

'''Problems'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Cloud Services tools were not available for 25 minutes last week. This was caused by a faulty hardware cable in the data center. [https://wikitech.wikimedia.org/wiki/Incidents/2024-06-11_WMCS_Ceph]
* Last week, styling updates were made to the Vector 2022 skin. This caused unforeseen issues with templates, hatnotes, and images. Changes to templates and hatnotes were reverted. Most issues with images were fixed. If you still see any, [[phab:T367463|report them here]]. [https://phabricator.wikimedia.org/T367480]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.10|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-06-18|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-06-19|en}}. It will be on all wikis from {{#time:j xg|2024-06-20|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* Starting June 18, the [[mw:Special:MyLanguage/Help:Edit check#ref|Reference Edit Check]] will be deployed to [[phab:T361843|a new set of Wikipedias]]. This feature is intended to help newcomers and to assist edit-patrollers by inviting people who are adding new content to a Wikipedia article to add a citation when they do not do so themselves. During [[mw:Special:MyLanguage/Edit_check#Reference_Check_A/B_Test|a test at 11 wikis]], the number of citations added [https://diff.wikimedia.org/?p=127553 more than doubled] when Reference Check was shown to people. Reference Check is [[mw:Special:MyLanguage/Edit check/Configuration|community configurable]]. [https://phabricator.wikimedia.org/T361843]&lt;!-- NOTE: THE DIFF BLOG WILL BE PUBLISHED ON MONDAY --&gt;
* [[m:Special:MyLanguage/Mailing_lists|Mailing lists]] will be unavailable for roughly two hours on Tuesday 10:00–12:00 UTC. This is to enable migration to a new server and upgrade its software. [https://phabricator.wikimedia.org/T367521]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/25|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W25&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:48, 17 June 2024 (UTC)
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== Tech News: 2024-26 ==

&lt;section begin=&quot;technews-2024-W26&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/26|Translations]] are available.

'''Recent changes'''
* Editors will notice that there have been some changes to the background color of text in the diff view, and the color of the byte-change numbers, last week. These changes are intended to make text more readable in both light mode and dark mode, and are part of a larger effort to increase accessibility. You can share your comments or questions [[mw:Talk:Reading/Web/Accessibility for reading|on the project talkpage]]. [https://phabricator.wikimedia.org/T361717]
* The text colors that are used for visited-links, hovered-links, and active-links, were also slightly changed last week to improve their accessibility in both light mode and dark mode. [https://phabricator.wikimedia.org/T366515]

'''Problems'''
* You can [[mw:Special:MyLanguage/Help:DiscussionTools#Talk pages permalinking|copy permanent links to talk page comments]] by clicking on a comment's timestamp. [[mw:Talk pages project/Permalinks|This feature]] did not always work when the topic title was very long and the link was used as a wikitext link. This has been fixed. Thanks to Lofhi for submitting the bug. [https://phabricator.wikimedia.org/T356196]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.11|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-06-25|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-06-26|en}}. It will be on all wikis from {{#time:j xg|2024-06-27|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* Starting 26 June, all talk pages messages' timestamps will become a link at English Wikipedia, making this feature available for you to use at all wikis. This link is a permanent link to the comment. It allows users to find the comment they were linked to, even if this comment has since been moved elsewhere. You can read more about this feature [[DiffBlog:/2024/01/29/talk-page-permalinks-dont-lose-your-threads/|on Diff]] or [[mw:Special:MyLanguage/Help:DiscussionTools#Talk pages permalinking|on Mediawiki.org]]. [https://phabricator.wikimedia.org/T365974]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/26|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W26&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:32, 24 June 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26989424 --&gt;

== Tech News: 2024-27 ==

&lt;section begin=&quot;technews-2024-W27&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/27|Translations]] are available.

'''Recent changes'''
* Over the next three weeks, dark mode will become available for all users, both logged-in and logged-out, starting with the mobile web version. This fulfils one of the [[m:Special:MyLanguage/Community_Wishlist_Survey_2023/Reading/Dark_mode|top-requested community wishes]], and improves low-contrast reading and usage in low-light settings. As part of these changes, dark mode will also work on User-pages and Portals. There is more information in [[mw:Special:MyLanguage/Reading/Web/Accessibility_for_reading/Updates#June_2024:_Typography_and_dark_mode_deployments,_new_global_preferences|the latest Web team update]]. [https://phabricator.wikimedia.org/T366364]
* Logged-in users can now set [[m:Special:GlobalPreferences#mw-prefsection-rendering-skin-skin-prefs|global preferences for the text-size and dark-mode]], thanks to a combined effort across Foundation teams. This allows Wikimedians using multiple wikis to set up a consistent reading experience easily, for example by switching between light and dark mode only once for all wikis. [https://phabricator.wikimedia.org/T341278] 
* If you use a very old web browser some features might not work on the Wikimedia wikis. This affects Internet Explorer 11 and versions of Chrome, Firefox and Safari older than 2016. This change makes it possible to use new [[d:Q46441|CSS]] features and to send less code to all readers. [https://phabricator.wikimedia.org/T288287][https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:How_to_make_a_MediaWiki_skin#Using_CSS_variables_for_supporting_different_themes_e.g._dark_mode]
* Wikipedia Admins can customize local wiki configuration options easily using [[mw:Special:MyLanguage/Community Configuration|Community Configuration]]. Community Configuration was created to allow communities to customize how some features work, because each language wiki has unique needs. At the moment, admins can configure [[mw:Special:MyLanguage/Growth/Feature_summary|Growth features]] on their home wikis, in order to better recruit and retain new editors. More options will be provided in the coming months. [https://phabricator.wikimedia.org/T366458]
* Editors interested in language issues that are related to [[w:en:Unicode|Unicode standards]], can now discuss those topics at [[mw:Talk:WMF membership with Unicode Consortium|a new conversation space in MediaWiki.org]]. The Wikimedia Foundation is now a [[mw:Special:MyLanguage/WMF membership with Unicode Consortium|member of the Unicode Consortium]], and the coordination group can collaboratively review the issues discussed and, where appropriate, bring them to the attention of the Unicode Consortium.
* One new wiki has been created: a {{int:project-localized-name-group-wikipedia}} in [[d:Q2891049|Mandailing]] ([[w:btm:|&lt;code&gt;w:btm:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T368038] 

'''Problems'''
* Editors can once again click on links within the visual editor's citation-preview, thanks to a bug fix by the Editing Team. [https://phabricator.wikimedia.org/T368119]

'''Future changes'''
* Please [https://wikimediafoundation.limesurvey.net/758713?lang=en help us to improve Tech News by taking this short survey]. The goal is to better meet the needs of the various types of people who read Tech News. The survey will be open for 2 weeks. The survey is covered by [https://foundation.wikimedia.org/wiki/Legal:Tech_News_Survey_2024_Privacy_Statement this privacy statement]. Some translations are available.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/27|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W27&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:59, 1 July 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27038456 --&gt;

== Tech News: 2024-28 ==

&lt;section begin=&quot;technews-2024-W28&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/28|Translations]] are available.

'''Recent changes'''
* At the Wikimedia Foundation a new task force was formed to replace the disabled Graph with [[mw:Special:MyLanguage/Extension:Chart/Project|more secure, easy to use, and extensible Chart]]. You can [[mw:Special:MyLanguage/Newsletter:Chart Project|subscribe to the newsletter]] to get notified about new project updates and other news about Chart.
* The [[m:Special:MyLanguage/CampaignEvents|CampaignEvents]] extension is now available on Meta-wiki, Igbo Wikipedia, and Swahili Wikipedia, and can be requested on your wiki. This extension helps in managing and making events more visible, giving Event organizers the ability to use tools like the Event registration tool. To learn more about the deployment status and how to request this extension for your wiki, visit the [[m:Special:MyLanguage/CampaignEvents/Deployment_status|CampaignEvents page on Meta-wiki]]. 
* Editors using the iOS Wikipedia app who have more than 50 edits can now use the [[mw:Special:MyLanguage/Wikimedia Apps/iOS Suggested edits#Add an image|Add an Image]] feature. This feature presents opportunities for small but useful contributions to Wikipedia.
* Thank you to [[mw:MediaWiki Product Insights/Contributor retention and growth/Celebration|all of the authors]] who have contributed to MediaWiki Core. As a result of these contributions, the [[mw:MediaWiki Product Insights/Contributor retention and growth|percentage of authors contributing more than 5 patches has increased by 25% since last year]], which helps ensure the sustainability of the platform for the Wikimedia projects.

'''Problems'''
* A problem with the color of the talkpage tabs always showing as blue, even for non-existent pages which should have been red, affecting the Vector 2022 skin, [[phab:T367982|has been fixed]]. 

'''Future changes'''
* The Trust and Safety Product team wants to introduce [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|temporary accounts]] with as little disruption to tools and workflows as possible. Volunteer developers, including gadget and user-script maintainers, are kindly asked to update the code of their tools and features to handle temporary accounts. The team has [[mw:Trust and Safety Product/Temporary Accounts/For developers|created documentation]] explaining how to do the update. [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/For developers/2024-04 CTA|Learn more]].

'''Tech News survey'''
* Please [https://wikimediafoundation.limesurvey.net/758713?lang=en help us to improve Tech News by taking this short survey]. The goal is to better meet the needs of the various types of people who read Tech News. The survey will be open for 1 more week. The survey is covered by [https://foundation.wikimedia.org/wiki/Legal:Tech_News_Survey_2024_Privacy_Statement this privacy statement]. Some translations are available.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/28|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W28&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:31, 8 July 2024 (UTC)
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== Tech News: 2024-29 ==

&lt;section begin=&quot;technews-2024-W29&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/29|Translations]] are available.

'''Tech News survey'''
* Please [https://wikimediafoundation.limesurvey.net/758713?lang=en help us to improve Tech News by taking this short survey]. The goal is to better meet the needs of the various types of people who read Tech News. The survey will be open for 3 more days. The survey is covered by [https://foundation.wikimedia.org/wiki/Legal:Tech_News_Survey_2024_Privacy_Statement this privacy statement]. Some translations are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Wikimedia developers can now officially continue to use both [[mw:Special:MyLanguage/Gerrit|Gerrit]] and [[mw:Special:MyLanguage/GitLab|GitLab]], due to a June 24 decision by the Wikimedia Foundation to support software development on both platforms. Gerrit and GitLab are both code repositories used by developers to write, review, and deploy the software code that supports the MediaWiki software that the wiki projects are built on, as well as the tools used by editors to create and improve content. This decision will safeguard the productivity of our developers and prevent problems in code review from affecting our users. More details are available in the [[mw:GitLab/Migration status|Migration status]] page.
* The Wikimedia Foundation seeks applicants for the [[m:Special:MyLanguage/Product and Technology Advisory Council/Proposal|Product and Technology Advisory Council]] (PTAC). This group will bring technical contributors and Wikimedia Foundation together to co-define a more resilient, future-proof technological platform. Council members will evaluate and consult on the movement's product and technical activities, so that we develop multi-generational projects. We are looking for a range of technical contributors across the globe, from a variety of Wikimedia projects. [[m:Special:MyLanguage/Product and Technology Advisory Council/Proposal#Joining the PTAC as a technical volunteer|Please apply here by August 10]].
* Editors with rollback user-rights who use the Wikipedia App for Android can use the new [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android/Anti Vandalism|Edit Patrol]] features. These features include a new feed of Recent Changes, related links such as Undo and Rollback, and the ability to create and save a personal library of user talk messages to use while patrolling. If your wiki wants to make these features available to users who do not have rollback rights but have reached a certain edit threshold, [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android#Contact us|you can contact the team]]. You can [[diffblog:2024/07/10/ِaddressing-vandalism-with-a-tap-the-journey-of-introducing-the-patrolling-feature-in-the-mobile-app/|read more about this project on Diff blog]].
* Editors who have access to [[m:Special:MyLanguage/The_Wikipedia_Library|The Wikipedia Library]] can once again use non-open access content in SpringerLinks, after the Foundation [[phab:T368865|contacted]] them to restore access. You can read more about [[m:Tech/News/Recently_resolved_community_tasks|this and 21 other community-submitted tasks that were completed last week]]. 

'''Changes later this week'''
* This week, [[mw:Special:MyLanguage/Reading/Web/Accessibility for reading/Updates/2024-07 deployments|dark mode will be available on a number of Wikipedias]], both desktop and mobile, for logged-in and logged-out users. Interface admins and user script maintainers are encouraged to check gadgets and user scripts in the dark mode, to find any hard-coded colors and fix them. There are some [[mw:Special:MyLanguage/Recommendations for night mode compatibility on Wikimedia wikis|recommendations for dark mode compatibility]] to help.

'''Future changes'''
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Next week, functionaries, volunteers maintaining tools, and software development teams are invited to test the [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|temporary accounts]] feature on testwiki. Temporary accounts is a feature that will help improve privacy on the wikis. No further temporary account deployments are scheduled yet. Please [[mw:Talk:Trust and Safety Product/Temporary Accounts|share your opinions and questions on the project talk page]]. [https://phabricator.wikimedia.org/T348895]
* Editors who upload files cross-wiki, or teach other people how to do so, may wish to join a Wikimedia Commons discussion. The Commons community is discussing limiting who can upload files through the cross-wiki upload/Upload dialog feature to users auto-confirmed on Wikimedia Commons. This is due to the large amount of copyright violations uploaded this way. There is a short summary at [[c:Special:MyLanguage/Commons:Cross-wiki upload|Commons:Cross-wiki upload]] and [[c:Commons:Village pump/Proposals#Deactivate cross-wiki uploads for new users|discussion at Commons:Village Pump]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/29|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' You can also get other news from the [[m:Special:MyLanguage/Wikimedia Foundation Bulletin|Wikimedia Foundation Bulletin]].
&lt;/div&gt;&lt;section end=&quot;technews-2024-W29&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:31, 16 July 2024 (UTC)
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== Tech News: 2024-30 ==

&lt;section begin=&quot;technews-2024-W30&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/30|Translations]] are available.

'''Feature News'''
* Stewards can now [[:m:Special:MyLanguage/Global_blocks|globally block]] accounts. Before [[phab:T17294|the change]] only IP addresses and IP ranges could be blocked globally. Global account blocks are useful when the blocked user should not be logged out. [[:m:Special:MyLanguage/Global_locks|Global locks]] (a similar tool logging the user out of their account) are unaffected by this change. The new global account block feature is related to the [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|Temporary Accounts]] project, which is a new type of user account that replaces IP addresses of unregistered editors that are no longer made public.
* Later this week, Wikimedia site users will notice that the Interface of [[mw:Special:MyLanguage/Extension:FlaggedRevs|FlaggedRevs]] (also known as &quot;Pending Changes&quot;) is improved and consistent with the rest of the MediaWiki interface and [[mw:Special:MyLanguage/Codex|Wikimedia's design system]]. The FlaggedRevs interface experience on mobile and [[mw:Special:MyLanguage/Skin:MinervaNeue|Minerva skin]] was inconsistent before it was fixed and ported to [[mw:Special:MyLanguage/Codex|Codex]] by the WMF Growth team and some volunteers. [https://phabricator.wikimedia.org/T191156]
*  Wikimedia site users can now submit account vanishing requests via [[m:Special:GlobalVanishRequest|GlobalVanishRequest]]. This feature is used when a contributor wishes to stop editing forever. It helps you hide your past association and edit to protect your privacy. Once processed, the account will be locked and renamed. [https://phabricator.wikimedia.org/T367329]
* Have you tried monitoring and addressing vandalism in Wikipedia using your phone? [https://diff.wikimedia.org/2024/07/10/%d9%90addressing-vandalism-with-a-tap-the-journey-of-introducing-the-patrolling-feature-in-the-mobile-app/ A Diff blog post on Patrolling features in the Mobile App] highlights some of the new capabilities of the feature, including swiping through a feed of recent changes and a personal library of user talk messages for use when patrolling from your phone. 
* Wikimedia contributors and GLAM (galleries, libraries, archives, and museums) organisations can now learn and measure the impact Wikimedia Commons is having towards creating quality encyclopedic content using the [https://doc.wikimedia.org/generated-data-platform/aqs/analytics-api/reference/commons.html Commons Impact Metrics] analytics dashboard. The dashboard offers organizations analytics on things like monthly edits in a category, the most viewed files, and which Wikimedia articles are using Commons images. As a result of these new data dumps, GLAM organisation can more reliably measure their return on investment for programs bringing content into the digital Commons. [https://diff.wikimedia.org/2024/07/19/commons-impact-metrics-now-available-via-data-dumps-and-api/]

'''Project Updates'''
*  Come share your ideas for improving the wikis on the newly reopened [[m:Special:MyLanguage/Community Wishlist|Community Wishlist]]. The Community Wishlist is Wikimedia’s forum for volunteers to share ideas (called wishes) to improve how the wikis work. The new version of the wishlist is always open, works with both wikitext and Visual Editor, and allows wishes in any language.

'''Learn more'''
* Have you ever wondered how Wikimedia software works across over 300 languages? This is 253 languages more than the Google Chrome interface, and it's no accident. The Language and Product Localization Team at the Wikimedia Foundation supports your work by adapting all the tools and interfaces in the MediaWiki software so that contributors in our movement who translate pages and strings can translate them and have the sites in all languages. Read more about the team and their upcoming work on [https://diff.wikimedia.org/2024/07/17/building-towards-a-robust-multilingual-knowledge-ecosystem-for-the-wikimedia-movement/ Diff]. 
* How can Wikimedia build innovative and experimental products while maintaining such heavily used websites? A recent [https://diff.wikimedia.org/2024/07/09/on-the-value-of-experimentation/ blog post] by WMF staff Johan Jönsson highlights the work of the [[m:Future Audiences#Objectives and Key Results|WMF Future Audience initiative]], where the goal is not to build polished products but test out new ideas, such as a [[m:Future_Audiences/Experiments: conversational/generative AI|ChatGPT plugin]] and [[m:Future_Audiences/Experiment:Add a Fact|Add a Fact]], to help take Wikimedia into the future. 

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/30|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' You can also get other news from the [[m:Special:MyLanguage/Wikimedia Foundation Bulletin|Wikimedia Foundation Bulletin]].
&lt;/div&gt;&lt;section end=&quot;technews-2024-W30&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:04, 23 July 2024 (UTC)
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== Tech News: 2024-31 ==

&lt;section begin=&quot;technews-2024-W31&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/31|Translations]] are available.

'''Feature news'''
* Editors using the Visual Editor in languages that use non-Latin characters for numbers, such as Hindi, Manipuri and Eastern Arabic, may notice some changes in the formatting of reference numbers. This is a side effect of preparing a new sub-referencing feature, and will also allow fixing some general numbering issues in Visual Editor. If you notice any related problems on your wiki, please share details at the [[m:Talk:WMDE Technical Wishes/Sub-referencing|project talkpage]].

'''Bugs status'''
* Some logged-in editors were briefly unable to edit or load pages last week. [[phab:T370304|These errors]] were mainly due to the addition of new [[mw:Special:MyLanguage/Help:Extension:Linter|linter]] rules which led to caching problems. Fixes have been applied and investigations are continuing.
* Editors can use the [[mw:Special:MyLanguage/Trust and Safety Product/IP Info|IP Information tool]] to get information about IP addresses. This tool is available as a Beta Feature in your preferences. The tool was not available for a few days last week, but is now working again. Thank you to Shizhao for filing the bug report. You can read about that, and [[m:Tech/News/Recently resolved community tasks#2024-07-25|28 other community-submitted tasks]] that were resolved last week.

'''Project updates'''
* There are new features and improvements to Phabricator from the Release Engineering and Collaboration Services teams, and some volunteers, including: the search systems, the new task creation system, the login systems, the translation setup which has resulted in support for more languages (thanks to Pppery), and fixes for many edge-case errors. You can [[phab:phame/post/view/316/iterative_improvements/|read details about these and other improvements in this summary]].
* There is an [[mw:Special:MyLanguage/Extension:Chart/Project/Updates|update on the Charts project]]. The team has decided which visualization library to use, which chart types to start focusing on, and where to store chart definitions.
* One new wiki has been created: a {{int:project-localized-name-group-wikivoyage}} in [[d:Q9056|Czech]] ([[voy:cs:|&lt;code&gt;voy:cs:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T370905] 

'''Learn more'''
* There is a [[diffblog:2024/07/26/the-journey-to-open-our-first-data-center-in-south-america/|new Wikimedia Foundation data center]] in São Paulo, Brazil which helps to reduce load times.
* There is new [[diffblog:2024/07/22/the-perplexing-process-of-uploading-images-to-wikipedia/|user research]] on problems with the process of uploading images.
* Commons Impact Metrics are [[diffblog:2024/07/19/commons-impact-metrics-now-available-via-data-dumps-and-api/|now available]] via data dumps and API.
* The latest quarterly [[mw:Technical Community Newsletter/2024/July|Technical Community Newsletter]] is now available.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/31|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W31&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:10, 29 July 2024 (UTC)
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== Tech News: 2024-32 ==

&lt;section begin=&quot;technews-2024-W32&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/32|Translations]] are available.

'''Feature news'''
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Two new parser functions will be available this week: &lt;code&gt;&lt;nowiki&gt;{{&lt;/nowiki&gt;[[mw:Special:MyLanguage/Help:Magic_words#dir|#dir]]&lt;nowiki&gt;}}&lt;/nowiki&gt;&lt;/code&gt; and &lt;code&gt;&lt;nowiki&gt;{{&lt;/nowiki&gt;[[mw:Special:MyLanguage/Help:Magic_words#bcp47|#bcp47]]&lt;nowiki&gt;}}&lt;/nowiki&gt;&lt;/code&gt;. These will reduce the need for &lt;code&gt;Template:Dir&lt;/code&gt; and &lt;code&gt;Template:BCP47&lt;/code&gt; on Commons and allow us to [[phab:T343131|drop 100 million rows]] from the &quot;what links here&quot; database. Editors at any wiki that use these templates, can help by replacing the templates with these new functions. The templates at Commons will be updated during the Hackathon at Wikimania. [https://phabricator.wikimedia.org/T359761][https://phabricator.wikimedia.org/T366623]
* Communities can request the activation of the visual editor on entire namespaces where discussions sometimes happen (for instance ''Wikipedia:'' or ''Wikisource:'' namespaces) if they understand the [[mw:Special:MyLanguage/Help:VisualEditor/FAQ#WPNS|known limitations]]. For discussions, users can already use [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] in these namespaces.
* The tracking category &quot;Pages using Timeline&quot; has been renamed to &quot;Pages using the EasyTimeline extension&quot; [https://translatewiki.net/wiki/Special:Translations?message=MediaWiki%3ATimeline-tracking-category&amp;namespace=8 in TranslateWiki]. Wikis that have created the category locally should rename their local creation to match.

'''Project updates'''
* Editors who help to organize WikiProjects and similar on-wiki collaborations, are invited to share ideas and examples of successful collaborations with the Campaigns and Programs teams. You can fill out [[m:Special:MyLanguage/Campaigns/WikiProjects|a brief survey]] or share your thoughts [[m:Talk:Campaigns/WikiProjects|on the talkpage]]. The teams are particularly looking for details about successful collaborations on non-English wikis.
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] The new parser is being rolled out on {{int:project-localized-name-group-wikivoyage}} wikis over the next few months. The {{int:project-localized-name-enwikivoyage}} and {{int:project-localized-name-hewikivoyage}} were [[phab:T365367|switched]] to Parsoid last week. For more information, see [[mw:Parsoid/Parser_Unification|Parsoid/Parser Unification]].

'''Learn more'''
* There will be more than 200 sessions at Wikimania this week. Here is a summary of some of the [[diffblog:2024/08/05/interested-in-product-and-tech-here-are-some-wikimania-sessions-you-dont-want-to-miss/|key sessions related to the product and technology area]].
* The latest [[m:Special:MyLanguage/Wikimedia Foundation Bulletin/2024/07-02|Wikimedia Foundation Bulletin]] is available.
* The latest quarterly [[mw:Special:MyLanguage/Wikimedia Language and Product Localization/Newsletter/2024/July|Language and Internationalization newsletter]] is available. It includes: New design previews for Translatable pages; Updates about MinT for Wiki Readers; the release of Translation dumps; and more.
* The latest quarterly [[mw:Special:MyLanguage/Growth/Newsletters/31|Growth newsletter]] is available.
* The latest monthly [[mw:Special:MyLanguage/MediaWiki Product Insights/Reports/July 2024|MediaWiki Product Insights newsletter]] is available.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/32|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W32&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:43, 5 August 2024 (UTC)
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== Tech News: 2024-33 ==

&lt;section begin=&quot;technews-2024-W33&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/33|Translations]] are available.

'''Feature news'''
* [[mw:Special:MyLanguage/Extension:AbuseFilter|AbuseFilter]] editors and maintainers can now [[mw:Special:MyLanguage/Extension:AbuseFilter/Actions#Show a CAPTCHA|make a CAPTCHA show if a filter matches an edit]]. This allows communities to quickly respond to spamming by automated bots. [https://phabricator.wikimedia.org/T20110]
* [[m:Special:MyLanguage/Stewards|Stewards]] can now specify if global blocks should prevent account creation. Before [[phab:T17273|this change]] by the [[mw:Special:MyLanguage/Trust and Safety Product|Trust and Safety Product]] Team, all global blocks would prevent account creation. This will allow stewards to reduce the unintended side-effects of global blocks on IP addresses.

'''Project updates'''
* [[wikitech:Help talk:Toolforge/Toolforge standards committee#August_2024_committee_nominations|Nominations are open on Wikitech]] for new members to refresh the [[wikitech:Help:Toolforge/Toolforge standards committee|Toolforge standards committee]]. The committee oversees the Toolforge [[wikitech:Help:Toolforge/Right to fork policy|Right to fork policy]] and [[wikitech:Help:Toolforge/Abandoned tool policy|Abandoned tool policy]] among other duties. Nominations will remain open until at least 2024-08-26.
* One new wiki has been created: a {{int:project-localized-name-group-wikipedia}} in [[d:Q2880037|West Coast Bajau]] ([[w:bdr:|&lt;code&gt;w:bdr:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T371757]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/33|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W33&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:21, 12 August 2024 (UTC)
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== Tech News: 2024-34 ==

&lt;section begin=&quot;technews-2024-W34&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/34|Translations]] are available.

'''Feature news'''
* Editors who want to re-use references but with different details such as page numbers, will be able to do so by the end of 2024, using a new [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing#Sub-referencing in a nutshell|sub-referencing]] feature. You can read more [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing|about the project]] and [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing#Test|how to test the prototype]].
* Editors using tracking categories to identify which pages use specific extensions may notice that six of the categories have been renamed to make them more easily understood and consistent. These categories are automatically added to pages that use specialized MediaWiki extensions. The affected names are for: [https://translatewiki.net/wiki/Special:Translations?message=MediaWiki%3Aintersection-category&amp;namespace=8 DynamicPageList], [https://translatewiki.net/wiki/Special:Translations?message=MediaWiki%3Akartographer-tracking-category&amp;namespace=8 Kartographer], [https://translatewiki.net/wiki/Special:Translations?message=MediaWiki%3Aphonos-tracking-category&amp;namespace=8 Phonos], [https://translatewiki.net/wiki/Special:Translations?message=MediaWiki%3Arss-tracking-category&amp;namespace=8 RSS], [https://translatewiki.net/wiki/Special:Translations?message=MediaWiki%3Ascore-use-category&amp;namespace=8 Score], [https://translatewiki.net/wiki/Special:Translations?message=MediaWiki%3Awikihiero-usage-tracking-category&amp;namespace=8 WikiHiero]. Wikis that have created the category locally should rename their local creation to match. Thanks to Pppery for these improvements. [https://phabricator.wikimedia.org/T347324]
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Technical volunteers who edit modules and want to get a list of the categories used on a page, can now do so using the &lt;code&gt;&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;categories&lt;/bdi&gt;&lt;/code&gt; property of &lt;code&gt;&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;[[mediawikiwiki:Special:MyLanguage/Extension:Scribunto/Lua reference manual#Title objects|mw.title objects]]&lt;/bdi&gt;&lt;/code&gt;. This enables wikis to configure workflows such as category-specific edit notices. Thanks to SD001 for these improvements. [https://phabricator.wikimedia.org/T50175][https://phabricator.wikimedia.org/T85372]

'''Bugs status'''
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Your help is needed to check if any pages need to be moved or deleted. A maintenance script was run to clean up unreachable pages (due to Unicode issues or introduction of new namespaces/namespace aliases). The script tried to find appropriate names for the pages (e.g. by following the Unicode changes or by moving pages whose titles on Wikipedia start with &lt;code&gt;Talk:WP:&lt;/code&gt; so that their titles start with &lt;code&gt;Wikipedia talk:&lt;/code&gt;), but it may have failed for some pages, and moved them to &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;[[Special:PrefixIndex/T195546/]]&lt;/bdi&gt; instead. Your community should check if any pages are listed there, and move them to the correct titles, or delete them if they are no longer needed. A full log (including pages for which appropriate names could be found) is available in [[phab:P67388]].
* Editors who volunteer as [[mw:Special:MyLanguage/Help:Growth/Mentorship|mentors]] to newcomers on their wiki are once again able to access lists of potential mentees who they can connect with to offer help and guidance. This functionality was restored thanks to [[phab:T372164|a bug fix]]. Thank you to Mbch331 for filing the bug report. You can read about that, and 18 other community-submitted tasks that were [[m:Tech/News/Recently resolved community tasks|resolved last week]].

'''Project updates'''
* The application deadline for the [[m:Special:MyLanguage/Product and Technology Advisory Council/Proposal|Product &amp; Technology Advisory Council]] (PTAC) has been extended to September 16. Members will help by providing advice to Foundation Product and Technology leadership on short and long term plans, on complex strategic problems, and help to get feedback from more contributors and technical communities. Selected members should expect to spend roughly 5 hours per month for the Council, during the one year pilot. Please consider applying, and spread the word to volunteers you think would make a positive contribution to the committee.

'''Learn more'''
* The [[m:Special:MyLanguage/Coolest Tool Award#2024 Winners|2024 Coolest Tool Awards]] were awarded at Wikimania, in seven categories. For example, one award went to the ISA Tool, used for adding structured data to files on Commons, which was recently improved during the [[m:Event:Wiki Mentor Africa ISA Hackathon 2024|Wiki Mentor Africa Hackathon]]. You can see video demonstrations of each tool at the awards page. Congratulations to this year's recipients, and thank you to all tool creators and maintainers.
* The latest [[m:Special:MyLanguage/Wikimedia Foundation Bulletin/2024/08-01|Wikimedia Foundation Bulletin]] is available, and includes some highlights from Wikimania, an upcoming Language community meeting, and other news from the movement.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/34|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W34&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:54, 20 August 2024 (UTC)
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== Tech News: 2024-35 ==

&lt;section begin=&quot;technews-2024-W35&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/35|Translations]] are available.

'''Feature news'''
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Administrators can now test the [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|temporary accounts]] feature on test2wiki. This was done to allow cross-wiki testing of temporary accounts, for when temporary accounts switch between projects. The feature was enabled on testwiki a few weeks ago. No further temporary account deployments are scheduled yet. Temporary Accounts is a project to create a new type of user account that replaces IP addresses of unregistered editors which are no longer made public. Please [[mw:Talk:Trust and Safety Product/Temporary Accounts|share your opinions and questions on the project talk page]].
* Later this week, editors at wikis that use [[mw:Special:MyLanguage/Extension:FlaggedRevs|FlaggedRevs]] (also known as &quot;Pending Changes&quot;) may notice that the indicators at the top of articles have changed. This change makes the system more consistent with the rest of the MediaWiki interface. [https://phabricator.wikimedia.org/T191156]

'''Bugs status'''
* Editors who use the 2010 wikitext editor, and use the Character Insert buttons, will [[phab:T361465|no longer]] experience problems with the buttons adding content into the edit-summary instead of the edit-window. You can read more about that, and 26 other community-submitted tasks that were [[m:Tech/News/Recently resolved community tasks|resolved last week]].

'''Project updates'''
* [[File:Octicons-gift.svg|12px|link=|class=skin-invert|Wishlist item]] Please review and vote on [[m:Special:MyLanguage/Community Wishlist/Focus areas|Focus Areas]], which are groups of wishes that share a problem. Focus Areas were created for the newly reopened Community Wishlist, which is now open year-round for submissions. The first batch of focus areas are specific to moderator workflows, around welcoming newcomers, minimizing repetitive tasks, and prioritizing tasks. Once volunteers have reviewed and voted on focus areas, the Foundation will then review and select focus areas for prioritization.
* Do you have a project and are willing to provide a three (3) month mentorship for an intern? [[mw:Special:MyLanguage/Outreachy|Outreachy]] is a twice a year program for people to participate in a paid internship that will start in December 2024 and end in early March 2025, and they need mentors and projects to work on. Projects can be focused on coding or non-coding (design, documentation, translation, research). See the Outreachy page for more details, and a list of past projects since 2013.

'''Learn more'''
* If you're curious about the product and technology improvements made by the Wikimedia Foundation last year, read [[diffblog:2024/08/21/wikimedia-foundation-product-technology-improving-the-user-experience/|this recent highlights summary on Diff]].
* To learn more about the technology behind the Wikimedia projects, you can now watch sessions from the technology track at Wikimania 2024 on Commons. This week, check out:
** [[c:File:Wikimania 2024 - Ohrid - Day 2 - Community Configuration - Shaping On-Wiki Functionality Together.webm|Community Configuration - Shaping On-Wiki Functionality Together]] (55 mins) - about the [[mw:Special:MyLanguage/Community Configuration|Community Configuration]] project.
** [[c:File:Wikimania 2024 - Belgrade - Day 1 - Future of MediaWiki. A sustainable platform to support a collaborative user base and billions of page views.webm|Future of MediaWiki. A sustainable platform to support a collaborative user base and billions of page views]] (30 mins) - an overview for both technical and non technical audiences, covering some of the challenges and open questions, related to the [[mw:MediaWiki Product Insights|platform evolution, stewardship and developer experiences]] research.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/35|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W35&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:33, 26 August 2024 (UTC)
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== Tech News: 2024-36 ==

&lt;section begin=&quot;technews-2024-W36&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/36|Translations]] are available.

'''Weekly highlight'''
* Editors and volunteer developers interested in data visualisation can now test the new software for charts. Its early version is available on beta Commons and beta Wikipedia. This is an important milestone before making charts available on regular wikis. You can [[mw:Special:MyLanguage/Extension:Chart/Project/Updates|read more about this project update]] and help to test the charts.

'''Feature news'''
* Editors who use the [[{{#special:Unusedtemplates}}]] page can now filter out pages which are expected to be there permanently, such as sandboxes, test-cases, and templates that are always substituted. Editors can add the new magic word [[mw:Special:MyLanguage/Help:Magic words#EXPECTUNUSEDTEMPLATE|&lt;code dir=&quot;ltr&quot;&gt;&lt;nowiki&gt;__EXPECTUNUSEDTEMPLATE__&lt;/nowiki&gt;&lt;/code&gt;]] to a template page to hide it from the listing. Thanks to Sophivorus and DannyS712 for these improvements. [https://phabricator.wikimedia.org/T184633]
* Editors who use the New Topic tool on discussion pages, will [[phab:T334163|now be reminded]] to add a section header, which should help reduce the quantity of newcomers who add sections without a header. You can read more about that, and {{formatnum:28}} other community-submitted tasks that were [[m:Tech/News/Recently resolved community tasks|resolved last week]].
* Last week, some Toolforge tools had occasional connection problems. The cause is still being investigated, but the problems have been resolved for now. [https://phabricator.wikimedia.org/T373243]
* Translation administrators at multilingual wikis, when editing multiple translation units, can now easily mark which changes require updates to the translation. This is possible with the [[phab:T298852#10087288|new dropdown menu]].

'''Project updates'''
* A new draft text of a policy discussing the use of Wikimedia's APIs [[m:Special:MyLanguage/API Policy Update 2024|has been published on Meta-Wiki]]. The draft text does not reflect a change in policy around the APIs; instead, it is an attempt to codify existing API rules. Comments, questions, and suggestions are welcome on [[m:Talk:API Policy Update 2024|the proposed update’s talk page]] until September 13 or until those discussions have concluded.

'''Learn more'''
* To learn more about the technology behind the Wikimedia projects, you can now watch sessions from the technology track at Wikimania 2024 on Commons. This week, check out:
** [[c:File:Wikimania 2024 - Ohrid - Day 2 - Charts, the successor of Graphs - A secure and extensible tool for data visualization.webm|Charts, the successor of Graphs - A secure and extensible tool for data visualization]] (25 mins) – about the above-mentioned Charts project.
** [[c:File:Wikimania 2024 - Ohrid - Day 3 - State of Language Technology and Onboarding at Wikimedia.webm|State of Language Technology and Onboarding at Wikimedia]] (90 mins) – about some of the language tools that support Wikimedia sites, such as [[mw:Special:MyLanguage/Content translation|Content]]/[[mw:Special:MyLanguage/Content translation/Section translation|Section Translation]], [[mw:Special:MyLanguage/MinT|MinT]], and LanguageConverter; also the current state and future of languages onboarding. [https://phabricator.wikimedia.org/T368772]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/36|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W36&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:07, 3 September 2024 (UTC)
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== Tech News: 2024-37 ==

&lt;section begin=&quot;technews-2024-W37&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/37|Translations]] are available.

'''Feature news'''
* Starting this week, the standard [[mw:Special:MyLanguage/Extension:CodeMirror|syntax highlighter]] will receive new colors that make them compatible in dark mode. This is the first of many changes to come as part of a major upgrade to syntax highlighting. You can learn more about what's to come on the [[mw:Special:MyLanguage/Help:Extension:CodeMirror|help page]]. [https://phabricator.wikimedia.org/T365311][https://phabricator.wikimedia.org/T259059]
* Editors of wikis using Wikidata will now be notified of only relevant Wikidata changes in their watchlist. This is because the Lua functions &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;entity:getSitelink()&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;mw.wikibase.getSitelink(qid)&lt;/code&gt;&lt;/bdi&gt; will have their logic unified for tracking different aspects of sitelinks to reduce junk notifications from [[m:Wikidata For Wikimedia Projects/Projects/Watchlist Wikidata Sitelinks Tracking|inconsistent sitelinks tracking]]. [https://phabricator.wikimedia.org/T295356]

'''Project updates'''
*  Users of all Wikis will have access to Wikimedia sites as read-only for a few minutes on September 25, starting at 15:00 UTC. This is a planned datacenter switchover for maintenance purposes. More information will be published in Tech News and will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T370962]
* Contributors of [[phab:T363538#10123348|11 Wikipedias]], including English will have a new &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;MOS&lt;/code&gt;&lt;/bdi&gt; namespace added to their Wikipedias. This improvement ensures that links beginning with &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;MOS:&lt;/code&gt;&lt;/bdi&gt; (usually shortcuts to the [[w:en:Wikipedia:Manual of Style|Manual of Style]]) are not broken by [[w:en:Mooré|Mooré]] Wikipedia (language code &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;mos&lt;/code&gt;&lt;/bdi&gt;). [https://phabricator.wikimedia.org/T363538]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/37|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W37&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 18:52, 9 September 2024 (UTC)
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== Tech News: 2024-38 ==

&lt;section begin=&quot;technews-2024-W38&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/38|Translations]] are available.

'''Improvements and Maintenance'''
* [[File:Octicons-gift.svg|12px|link=|class=skin-invert|Wishlist item]] Editors interested in templates can help by reading the latest Wishlist focus area, [[m:Special:MyLanguage/Community Wishlist/Focus areas/Template recall and discovery|Template recall and discovery]], and share your feedback on the talkpage. This input helps the Community Tech team to decide the right technical approach to build. Everyone is also encouraged to continue adding [[m:Special:MyLanguage/Community Wishlist|new wishes]].
* The new automated [[{{#special:NamespaceInfo}}]] page helps editors understand which [[mw:Special:MyLanguage/Help:Namespaces|namespaces]] exist on each wiki, and some details about how they are configured. Thanks to DannyS712 for these improvements. [https://phabricator.wikimedia.org/T263513]
* [[mw:Special:MyLanguage/Help:Edit check#Reference check|References Check]] is a feature that encourages editors to add a citation when they add a new paragraph to a Wikipedia article. For a short time, the corresponding tag &quot;Edit Check (references) activated&quot; was erroneously being applied to some edits outside of the main namespace. This has been fixed. [https://phabricator.wikimedia.org/T373692]
* It is now possible for a wiki community to change the order in which a page’s categories are displayed on their wiki. By default, categories are displayed in the order they appear in the wikitext. Now, wikis with a consensus to do so can [[m:Special:MyLanguage/Requesting wiki configuration changes|request]] a configuration change to display them in alphabetical order. [https://phabricator.wikimedia.org/T373480]
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Tool authors can now access ToolsDB's [[wikitech:Portal:Data Services#ToolsDB|public databases]] from both [[m:Special:MyLanguage/Research:Quarry|Quarry]] and [[wikitech:Superset|Superset]]. Those databases have always been accessible to every [[wikitech:Portal:Toolforge|Toolforge]] user, but they are now more broadly accessible, as Quarry can be accessed by anyone with a Wikimedia account. In addition, Quarry's internal database can now be [[m:Special:MyLanguage/Research:Quarry#Querying Quarry's own database|queried from Quarry itself]]. This database contains information about all queries that are being run and starred by users in Quarry. This information was already public through the web interface, but you can now query it using SQL. You can read more about that, and {{formatnum:20}} other community-submitted tasks that were [[m:Tech/News/Recently resolved community tasks|resolved last week]].
* Any pages or tools that still use the very old CSS classes &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;mw-message-box&lt;/code&gt;&lt;/bdi&gt; need to be updated. These old classes will be removed next week or soon afterwards. Editors can use a [https://global-search.toolforge.org/?q=mw-message-box&amp;regex=1&amp;namespaces=&amp;title= global-search] to determine what needs to be changed. It is possible to use the newer &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;cdx-message&lt;/code&gt;&lt;/bdi&gt; group of classes as a replacement (see [https://doc.wikimedia.org/codex/latest/components/demos/message.html#css-only-version the relevant Codex documentation], and [https://meta.wikimedia.org/w/index.php?title=Tech/Header&amp;diff=prev&amp;oldid=27449042 an example update]), but using locally defined onwiki classes would be best. [https://phabricator.wikimedia.org/T374499]

'''Technical project updates'''
* Next week, all Wikimedia wikis will be read-only for a few minutes. This will start on September 25 at [https://zonestamp.toolforge.org/1727276400 15:00 UTC]. This is a planned datacenter switchover for maintenance purposes. [[m:Special:MyLanguage/Tech/Server switch|This maintenance process also targets other services.]] The previous switchover took 3 minutes, and the Site Reliability Engineering teams use many tools to make sure that this essential maintenance work happens as quickly as possible. [https://phabricator.wikimedia.org/T370962]

'''Tech in depth'''
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] The latest monthly [[mw:Special:MyLanguage/MediaWiki Product Insights/Reports/August 2024|MediaWiki Product Insights newsletter]] is available. This edition includes details about: research about [[mw:Special:MyLanguage/Manual:Hooks|hook]] handlers to help simplify development, research about performance improvements, work to improve the REST API for end-users, and more.
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] To learn more about the technology behind the Wikimedia projects, you can now watch sessions from the technology track at Wikimania 2024 on Commons. This week, check out: 
** [[c:File:Wikimania 2024 - Auditorium Kyiv - Day 4 - Hackathon Showcase.webm|Hackathon Showcase]] (45 mins) - 19 short presentations by some of the Hackathon participants, describing some of the projects they worked on, such as automated testing of maintenance scripts, a video-cutting command line tool, and interface improvements for various tools. There are [[phab:T369234|more details and links available]] in the Phabricator task.
** [[c:File:Co-Creating a Sustainable Future for the Toolforge Ecosystem.webm|Co-Creating a Sustainable Future for the Toolforge Ecosystem]] (40 mins) - a roundtable discussion for tool-maintainers, users, and supporters of Toolforge about how to make the platform sustainable and how to evaluate the tools available there.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/38|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W38&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:02, 17 September 2024 (UTC)
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== Tech News: 2024-39 ==

&lt;section begin=&quot;technews-2024-W39&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/39|Translations]] are available.

'''Weekly highlight'''
* All wikis will be [[m:Special:MyLanguage/Tech/Server switch|read-only]] for a few minutes on Wednesday September 25 at [https://zonestamp.toolforge.org/1727276400 15:00 UTC]. Reading the wikis will not be interrupted, but editing will be paused. These twice-yearly processes allow WMF's site reliability engineering teams to remain prepared to keep the wikis functioning even in the event of a major interruption to one of our data centers.

'''Updates for editors'''
[[File:Add alt text from a halfsheet, with the article behind.png|thumb|A screenshot of the interface for the Alt Text suggested-edit feature]]
* Editors who use the iOS Wikipedia app in Spanish, Portuguese, French, or Chinese, may see the [[mw:Special:MyLanguage/Wikimedia Apps/iOS Suggested edits project/Alt Text Experiment|Alt Text suggested-edit experiment]] after editing an article, or completing a suggested edit using &quot;[[mw:Special:MyLanguage/Wikimedia Apps/iOS Suggested edits project#Hypothesis 2 Add an Image Suggested Edit|Add an image]]&quot;. Alt-text helps people with visual impairments to read Wikipedia articles. The team aims to learn if adding alt-text to images is a task that editors can be successful with. Please share any feedback on [[mw:Talk:Wikimedia Apps/iOS Suggested edits project/Alt Text Experiment|the discussion page]].
* The Codex color palette has been updated with new and revised colors for the MediaWiki user interfaces. The [[mw:Special:MyLanguage/Design System Team/Color/Design documentation#Updates|most noticeable changes]] for editors include updates for: dark mode colors for Links and for quiet Buttons (progressive and destructive), visited Link colors for both light and dark modes, and background colors for system-messages in both light and dark modes.
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] It is now possible to include clickable wikilinks and external links inside code blocks. This includes links that are used within &lt;code&gt;&lt;nowiki&gt;&lt;syntaxhighlight&gt;&lt;/nowiki&gt;&lt;/code&gt; tags and on code pages (JavaScript, CSS, Scribunto and Sanitized CSS). Uses of template syntax &lt;code&gt;&lt;nowiki&gt;{{…}}&lt;/nowiki&gt;&lt;/code&gt; are also linked to the template page. Thanks to SD0001 for these improvements. [https://phabricator.wikimedia.org/T368166]
* Two bugs were fixed in the [[m:Special:MyLanguage/Account vanishing|GlobalVanishRequest]] system by improving the logging and by removing an incorrect placeholder message. [https://phabricator.wikimedia.org/T370595][https://phabricator.wikimedia.org/T372223]
* View all {{formatnum:25}} community-submitted {{PLURAL:25|task|tasks}} that were [[m:Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] From [[m:Special:MyLanguage/Wikimedia Enterprise|Wikimedia Enterprise]]: 
** The API now enables 5,000 on-demand API requests per month and twice-monthly HTML snapshots freely (gratis and libre). More information on the updates and also improvements to the software development kits (SDK) are explained on [https://enterprise.wikimedia.com/blog/enhanced-free-api/ the project's blog post]. While Wikimedia Enterprise APIs are designed for high-volume commercial reusers, this change enables many more community use-cases to be built on the service too.
** The Snapshot API (html dumps) have added beta Structured Contents endpoints ([https://enterprise.wikimedia.com/blog/structured-contents-snapshot-api/ blog post on that]) as well as released two beta datasets (English and French Wikipedia) from that endpoint to Hugging Face for public use and feedback ([https://enterprise.wikimedia.com/blog/hugging-face-dataset/ blog post on that]). These pre-parsed data sets enable new options for researchers, developers, and data scientists to use and study the content.

'''In depth'''
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] The Wikidata Query Service (WDQS) is used to get answers to questions using the Wikidata data set. As Wikidata grows, we had to make a major architectural change so that WDQS could remain performant. As part of the [[d:Special:MyLanguage/Wikidata:SPARQL query service/WDQS graph split|WDQS Graph Split project]], we have new SPARQL endpoints available for serving the &quot;[https://query-scholarly.wikidata.org scholarly]&quot; and &quot;[https://query-main.wikidata.org main]&quot; subgraphs of Wikidata. The [http://query.wikidata.org query.wikidata.org endpoint] will continue to serve the full Wikidata graph until March 2025. After this date, it will only serve the main graph. For more information, please see [[d:Special:MyLanguage/Wikidata:SPARQL query service/WDQS backend update/September 2024 scaling update|the announcement on Wikidata]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/39|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W39&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:36, 23 September 2024 (UTC)
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== Tech News: 2024-40 ==

&lt;section begin=&quot;technews-2024-W40&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/40|Translations]] are available.

'''Updates for editors'''
* Readers of [[phab:T375401|42 more wikis]] can now use Dark Mode. If the option is not yet available for logged-out users of your wiki, this is likely because many templates do not yet display well in Dark Mode. Please use the [https://night-mode-checker.wmcloud.org/ night-mode-checker tool] if you are interested in helping to reduce the number of issues. The [[mw:Special:MyLanguage/Recommendations for night mode compatibility on Wikimedia wikis|recommendations page]] provides guidance on this. Dark Mode is enabled on additional wikis once per month.
* Editors using the 2010 wikitext editor as their default can access features from the 2017 wikitext editor by adding &lt;code dir=ltr&gt;?veaction=editsource&lt;/code&gt; to the URL. If you would like to enable the 2017 wikitext editor as your default, it can be set in [[Special:Preferences#mw-input-wpvisualeditor-newwikitext|your preferences]]. [https://phabricator.wikimedia.org/T239796]
* For logged-out readers using the Vector 2022 skin, the &quot;donate&quot; link has been moved from a collapsible menu next to the content area into a more prominent top menu, next to &quot;Create an account&quot;. This restores the link to the level of prominence it had in the Vector 2010 skin. [[mw:Readers/2024 Reader and Donor Experiences#Donor Experiences (Key Result WE 3.2 and the related hypotheses)|Learn more]] about the changes related to donor experiences. [https://phabricator.wikimedia.org/T373585]
* The CampaignEvents extension provides tools for organizers to more easily manage events, communicate with participants, and promote their events on the wikis. The extension has been [[m:Special:MyLanguage/CampaignEvents/Deployment status|enabled]] on Arabic Wikipedia, Igbo Wikipedia, Swahili Wikipedia, and Meta-Wiki. [[w:zh:Wikipedia:互助客栈/其他#引進CampaignEvents擴充功能|Chinese Wikipedia has decided]] to enable the extension, and discussions on the extension are in progress [[w:es:Wikipedia:Votaciones/2024/Sobre la política de Organizadores de Eventos|on Spanish Wikipedia]] and [[d:Wikidata:Project chat#Enabling the CampaignEvents Extention on Wikidata|on Wikidata]]. To learn how to enable the extension on your wiki, you can visit [[m:Special:MyLanguage/CampaignEvents|the CampaignEvents page on Meta-Wiki]].
* View all {{formatnum:22}} community-submitted {{PLURAL:22|task|tasks}} that were [[m:Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* Developers with an account on Wikitech-wiki should [[wikitech:Wikitech/SUL-migration|check if any action is required]] for their accounts. The wiki is being changed to use the single-user-login (SUL) system, and other configuration changes. This change will help reduce the overall complexity for the weekly software updates across all our wikis.

'''In depth'''
* The [[m:Special:MyLanguage/Tech/Server switch|server switch]] was completed successfully last week with a read-only time of [[wikitech:Switch Datacenter#Past Switches|only 2 minutes 46 seconds]]. This periodic process makes sure that engineers can switch data centers and keep all of the wikis available for readers, even if there are major technical issues. It also gives engineers a chance to do maintenance and upgrades on systems that normally run 24 hours a day, and often helps to reveal weaknesses in the infrastructure. The process involves dozens of software services and hundreds of hardware servers, and requires multiple teams working together. Work over the past few years has reduced the time from 17 minutes down to 2–3 minutes. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/66ZW7B2MG63AESQVTXDIFQBDBS766JGW/]

'''Meetings and events'''
* October 4–6: [[m:Special:MyLanguage/WikiIndaba conference 2024|WikiIndaba Conference's Hackathon]] in Johannesburg, South Africa
* November 4–6: [[mw:Special:MyLanguage/MediaWiki Users and Developers Conference Fall 2024|MediaWiki Users and Developers Conference Fall 2024]] in Vienna, Austria

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/40|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W40&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:20, 30 September 2024 (UTC)
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== Tech News: 2024-41 ==

&lt;section begin=&quot;technews-2024-W41&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/41|Translations]] are available.

'''Weekly highlight'''
* Communities can now request installation of [[mw:Special:MyLanguage/Moderator Tools/Automoderator|Automoderator]] on their wiki. Automoderator is an automated anti-vandalism tool that reverts bad edits based on scores from the new &quot;Revert Risk&quot; machine learning model. You can [[mw:Special:MyLanguage/Extension:AutoModerator/Deploying|read details about the necessary steps]] for installation and configuration. [https://phabricator.wikimedia.org/T336934]

'''Updates for editors'''
* Translators in wikis where [[mw:Special:MyLanguage/Content translation/Section translation#Try the tool|the mobile experience of Content Translation is available]], can now customize their articles suggestion list from 41 filtering options when using the tool. This topic-based article suggestion feature makes it easy for translators to self-discover relevant articles based on their area of interest and translate them. You can [https://test.wikipedia.org/w/index.php?title=Special:ContentTranslation&amp;active-list=suggestions try it with your mobile device]. [https://phabricator.wikimedia.org/T368422]
* View all {{formatnum:12}} community-submitted {{PLURAL:12|task|tasks}} that were [[m:Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* It is now possible for &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;&lt;syntaxhighlight&gt;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; code blocks to offer readers a &quot;Copy&quot; button if the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;copy=1&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; attribute is [[mw:Special:MyLanguage/Extension:SyntaxHighlight#copy|set on the tag]]. Thanks to SD0001 for these improvements. [https://phabricator.wikimedia.org/T40932]
* Customized copyright footer messages on all wikis will be updated. The new versions will use wikitext markup instead of requiring editing raw HTML. [https://phabricator.wikimedia.org/T375789]
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Later this month, [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|temporary accounts]] will be rolled out on several pilot wikis. The final list of the wikis will be published in the second half of the month. If you maintain any tools, bots, or gadgets on [[phab:T376499|these 11 wikis]], and your software is using data about IP addresses or is available for logged-out users, please check if it needs to be updated to work with temporary accounts. [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/For developers|Guidance on how to update the code is available]].
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Rate limiting has been enabled for the code review tools [[Wikitech:Gerrit|Gerrit]] and [[Wikitech:GitLab|GitLab]] to address ongoing issues caused by malicious traffic and scraping. Clients that open too many concurrent connections will be restricted for a few minutes. This rate limiting is managed through [[Wikitech:nftables|nftables]] firewall rules. For more details, see Wikitech's pages on [[Wikitech:Firewall#Throttling with nftables|Firewall]], [[Wikitech:GitLab/Abuse and rate limiting|GitLab limits]] and [[Wikitech:Gerrit/Operations#Throttling IPs|Gerrit operations]].
* Five new wikis have been created: 
** a {{int:project-localized-name-group-wikipedia}} in [[d:Q49224|Komering]] ([[w:kge:|&lt;code&gt;w:kge:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T374813]
** a {{int:project-localized-name-group-wikipedia}} in [[d:Q36096|Mooré]] ([[m:mos:|&lt;code&gt;m:mos:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T374641]
** a {{int:project-localized-name-group-wiktionary}} in [[d:Q36213|Madurese]] ([[wikt:mad:|&lt;code&gt;wikt:mad:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T374968]
** a {{int:project-localized-name-group-wikiquote}} in [[d:Q2501174|Gorontalo]] ([[q:gor:|&lt;code&gt;q:gor:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T375088]
** a {{int:project-localized-name-group-wikinews}} in [[d:Q56482|Shan]] ([[n:shn:|&lt;code&gt;n:shn:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T375430]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/41|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W41&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:42, 7 October 2024 (UTC)
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== Tech News: 2024-42 ==

&lt;section begin=&quot;technews-2024-W42&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/42|Translations]] are available.

'''Updates for editors'''
* The Structured Discussion extension (also known as Flow) is starting to be removed. This extension is unmaintained and causes issues. It will be replaced by [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]], which is used on any regular talk page. [[mw:Special:MyLanguage/Structured Discussions/Deprecation#Deprecation timeline|A first set of wikis]] are being contacted. These wikis are invited to stop using Flow, and to move all Flow boards to sub-pages, as archives. At these wikis, a script will move all Flow pages that aren't a sub-page to a sub-page automatically, starting on 22 October 2024. On 28 October 2024, all Flow boards at these wikis will be set in read-only mode. [https://www.mediawiki.org/wiki/Structured_Discussions/Deprecation][https://phabricator.wikimedia.org/T370722]
* WMF's Search Platform team is working on making it easier for readers to perform text searches in their language. A [[phab:T332342|change last week]] on over 30 languages makes it easier to find words with accents and other diacritics. This applies to both full-text search and to types of advanced search such as the &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;''hastemplate''&lt;/bdi&gt; and &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;''incategory''&lt;/bdi&gt; keywords. More technical details (including a few other minor search upgrades) are available. [https://www.mediawiki.org/wiki/User:TJones_%28WMF%29/Notes/Language_Analyzer_Harmonization_Notes#ASCII-folding/ICU-folding_%28T332342%29]
* View all {{formatnum:20}} community-submitted {{PLURAL:20|task|tasks}} that were [[m:Tech/News/Recently resolved community tasks|resolved last week]]. For example, [[mw:Special:MyLanguage/Help:Edit check|EditCheck]] was installed at Russian Wikipedia, and fixes were made for some missing user interface styles.

'''Updates for technical contributors'''
* Editors who use the Toolforge tool [[toolforge:copyvios|Earwig's Copyright Violation Detector]] will now be required to log in with their Wikimedia account before running checks using the &quot;search engine&quot; option. This change is needed to help prevent external bots from misusing the system. Thanks to Chlod for these improvements. [https://en.wikipedia.org/wiki/Wikipedia_talk:New_pages_patrol/Reviewers#Authentication_is_now_required_for_search_engine_checks_on_Earwig's_Copyvio_Tool]
* [[m:Special:MyLanguage/Phabricator|Phabricator]] users can create tickets and add comments on existing tickets via Email again. [[mw:Special:MyLanguage/Phabricator/Help#Using email|Sending email to Phabricator]] has been fixed. [https://phabricator.wikimedia.org/T356077]
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Some HTML elements in the interface are now wrapped with a &lt;code&gt;&lt;nowiki&gt;&lt;bdi&gt;&lt;/nowiki&gt;&lt;/code&gt; element, to make our HTML output more aligned with Web standards. More changes like this will be coming in future weeks. This change might break some tools that rely on the previous HTML structure of the interface. Note that relying on the HTML structure of the interface is [[mw:Special:MyLanguage/Stable interface policy/Frontend#What is not stable?|not recommended]] and might break at any time. [https://phabricator.wikimedia.org/T375975]

'''In depth'''
* The latest monthly [[mw:Special:MyLanguage/MediaWiki Product Insights/Reports/September 2024|MediaWiki Product Insights newsletter]] is available. This edition includes: updates on Wikimedia's authentication system, research to simplify feature development in the MediaWiki platform, updates on Parser Unification and MathML rollout, and more.
* The latest quarterly [[mw:Technical Community Newsletter/2024/October|Technical Community Newsletter]] is now available. This edition include: research about improving topic suggestions related to countries, improvements to PHPUnit tests, and more.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/42|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W42&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:21, 14 October 2024 (UTC)
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== Tech News: 2024-43 ==

&lt;section begin=&quot;technews-2024-W43&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/43|Translations]] are available.

'''Weekly highlight'''
* The Mobile Apps team has released an [[mw:Special:MyLanguage/Wikimedia Apps/Team/iOS/Navigation Refresh#Phase 1: Creating a user Profile Menu (T373714)|update]] to the iOS app's navigation, and it is now available in the latest App store version. The team added a new Profile menu that allows for easy access to editor features like Notifications and Watchlist from the Article view, and brings the &quot;Donate&quot; button into a more accessible place for users who are reading an article. This is the first phase of a larger planned [[mw:Special:MyLanguage/Wikimedia Apps/Team/iOS/Navigation Refresh|navigation refresh]] to help the iOS app transition from a primarily reader-focused app, to an app that fully supports reading and editing. The Wikimedia Foundation has added more editing features and support for on-wiki communication based on volunteer requests in recent years.
[[File:IOS App Navigation refresh first phase 05.png|thumb|iOS Wikipedia App's profile menu and contents]]

'''Updates for editors'''
* Wikipedia readers can now download a browser extension to experiment with some early ideas on potential features that recommend articles for further reading, automatically summarize articles, and improve search functionality. For more details and to stay updated, check out the Web team's [[mw:Special:MyLanguage/Reading/Web/Content Discovery Experiments|Content Discovery Experiments page]] and [[mw:Special:MyLanguage/Newsletter:Web team's projects|subscribe to their newsletter]].
* Later this month, logged-out editors of [[phab:T376499|these 12 wikis]] will start to have [[mw:Special:Mylanguage/Trust and Safety Product/Temporary Accounts|temporary accounts]] created. The list may slightly change - some wikis may be removed but none will be added. Temporary account is a new [[mw:Special:MyLanguage/User account types|type of user account]]. It enhances the logged-out editors' privacy and makes it easier for community members to communicate with them. If you maintain any tools, bots, or gadgets on these 12 wikis, and your software is using data about IP addresses or is available for logged-out users, please check if it needs to be updated to work with temporary accounts. [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/For developers|Guidance on how to update the code is available]]. Read more about the [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/Updates|deployment plan across all wikis]].
* View all {{formatnum:33}} community-submitted {{PLURAL:33|task|tasks}} that were [[m:Tech/News/Recently resolved community tasks|resolved last week]]. For example, the [[w:nr:Main Page|South Ndebele]], [[w:rsk:Главни бок|Pannonian Rusyn]], [[w:ann:Uwu|Obolo]], [[w:iba:Lambar Keterubah|Iban]] and [[w:tdd:ᥞᥨᥝᥴ ᥘᥣᥲ ᥖᥥᥰ|Tai Nüa]] Wikipedia languages were created last week. [https://www.wikidata.org/wiki/Q36785][https://www.wikidata.org/wiki/Q35660][https://www.wikidata.org/wiki/Q36614][https://www.wikidata.org/wiki/Q33424][https://www.wikidata.org/wiki/Q36556]
* It is now possible to create functions on Wikifunctions using Wikidata lexemes, through the new [[f:Z6005|Wikidata lexeme type]] launched last week. When you go to one of these functions, the user interface provides a lexeme selector that helps you pick a lexeme from Wikidata that matches the word you type. After hitting run, your selected lexeme is retrieved from Wikidata, transformed into a Wikidata lexeme type, and passed into the selected function. Read more about this in [[f:Special:MyLanguage/Wikifunctions:Status updates/2024-10-17#Function of the Week: select representation from lexeme|the latest Wikifunctions newsletter]].

'''Updates for technical contributors'''
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Users of the Wikimedia sites can now format dates more easily in different languages with the new &lt;code dir=&quot;ltr&quot;&gt;{{[[mw:Special:MyLanguage/Help:Extension:ParserFunctions##timef|#timef]]:…}}&lt;/code&gt; parser function. For example, &lt;code dir=&quot;ltr&quot;&gt;&lt;nowiki&gt;{{#timef:now|date|en}}&lt;/nowiki&gt;&lt;/code&gt; will show as &quot;&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;{{#timef:now|date|en}}&lt;/bdi&gt;&quot;. Previously, &lt;code dir=&quot;ltr&quot;&gt;&lt;nowiki&gt;{{#time:…}}&lt;/nowiki&gt;&lt;/code&gt; could be used to format dates, but this required knowledge of the order of the time and date components and their intervening punctuation. &lt;code dir=&quot;ltr&quot;&gt;#timef&lt;/code&gt; (or &lt;code dir=&quot;ltr&quot;&gt;#timefl&lt;/code&gt; for local time) provides access to the standard date formats that MediaWiki uses in its user interface. This may help to simplify some templates on multi-lingual wikis like Commons and Meta. [https://phabricator.wikimedia.org/T223772][https://www.mediawiki.org/wiki/Special:MyLanguage/Help:Extension:ParserFunctions##timef]
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Commons and Meta users can now efficiently [[mw:Special:MyLanguage/Help:Magic words#Localization|retrieve the user's language]] using &lt;code dir=&quot;ltr&quot;&gt;&lt;nowiki&gt;{{USERLANGUAGE}}&lt;/nowiki&gt;&lt;/code&gt; instead of using &lt;code dir=&quot;ltr&quot;&gt;&lt;nowiki&gt;{{int:lang}}&lt;/nowiki&gt;&lt;/code&gt;. [https://phabricator.wikimedia.org/T4085]
* The [[m:Special:MyLanguage/Product and Technology Advisory Council|Product and Tech Advisory Council]] (PTAC) now has its pilot members with representation across Africa, Asia, Europe, North America and South America. They will work to address the [[Special:MyLanguage/Movement Strategy/Initiatives/Technology Council|Movement Strategy's Technology Council]] initiative of having a co-defined and more resilient technological platform. [https://meta.wikimedia.org/wiki/Movement_Strategy/Initiatives/Technology_Council]

'''In depth'''
* The latest quarterly [[mw:Special:MyLanguage/Growth/Newsletters/32|Growth newsletter]] is available. It includes: an upcoming Newcomer Homepage Community Updates module, new Community Configuration options, and details on new projects.
* The Wikimedia Foundation is [[mw:Special:MyLanguage/Wikimedia Security Team#CNA Partnership|now an official partner of the CVE program]], which is an international effort to catalog publicly disclosed cybersecurity vulnerabilities. This partnership will allow the Security Team to instantly publish [[w:en:Common Vulnerabilities and Exposures|common vulnerabilities and exposures]] (CVE) records that are affecting MediaWiki core, extensions, and skins, along with any other code the Foundation is a steward of.
* The [[m:Special:MyLanguage/Community Wishlist|Community Wishlist]] is now [[m:Community Wishlist/Updates#October 16, 2024: Conversations Made Easier: Machine-Translated Wishes Are Here!|testing machine translations]] for Wishlist content. Volunteers can now read machine-translated versions of wishes and dive into discussions even before translators arrive to translate content.

'''Meetings and events'''
* 24 October - Wiki Education Speaker Series Webinar - [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/N4XTB4G55BUY3M3PNGUAKQWJ7A4UOPAK/ Open Source Tech: Building the Wiki Education Dashboard], featuring Wikimedia interns and a Web developer in the panel.
* 20–22 December 2024 - [[m:Special:MyLanguage/Indic Wikimedia Hackathon Bhubaneswar 2024|Indic Wikimedia Hackathon Bhubaneswar 2024]] in Odisha, India. A hackathon for community members, including developers, designers and content editors, to build technical solutions that improve contributors' experiences.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/43|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W43&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:52, 21 October 2024 (UTC)
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== Tech News: 2024-44 ==

&lt;section begin=&quot;technews-2024-W44&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/44|Translations]] are available.

'''Updates for editors'''
* Later in November, the Charts extension will be deployed to the test wikis in order to help identify and fix any issue. A security review is underway to then enable deployment to pilot wikis for broader testing. You can read [[mw:Special:MyLanguage/Extension:Chart/Project/Updates#October 2024: Working towards production deployment|the October project update]] and see the [https://en.wikipedia.beta.wmflabs.org/wiki/Charts latest documentation and examples on Beta Wikipedia].
* View all {{formatnum:32}} community-submitted {{PLURAL:32|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, [[w:en:PediaPress|Pediapress.com]], an external service that creates books from Wikipedia, can now use [[mw:Special:MyLanguage/Wikimedia Maps|Wikimedia Maps]] to include existing pre-rendered infobox map images in their printed books on Wikipedia. [https://phabricator.wikimedia.org/T375761]

'''Updates for technical contributors'''
* Wikis can use [[:mw:Special:MyLanguage/Extension:GuidedTour|the Guided Tour extension]] to help newcomers understand how to edit. The Guided Tours extension now works with [[mw:Special:MyLanguage/Manual:Dark mode|dark mode]]. Guided Tour maintainers can check their tours to see that nothing looks odd. They can also set &lt;code&gt;emitTransitionOnStep&lt;/code&gt; to &lt;code&gt;true&lt;/code&gt; to fix an old bug. They can use the new flag &lt;code&gt;allowAutomaticBack&lt;/code&gt; to avoid back-buttons they don't want. [https://phabricator.wikimedia.org/T73927#10241528]
*  Administrators in the Wikimedia projects who use the [[mw:Special:MyLanguage/Help:Extension:Nuke|Nuke Extension]] will notice that mass deletions done with this tool have the &quot;Nuke&quot; tag. This change will make reviewing and analyzing deletions performed with the tool easier. [https://phabricator.wikimedia.org/T366068]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/44|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W44&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:56, 28 October 2024 (UTC)
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== Tech News: 2024-45 ==

&lt;section begin=&quot;technews-2024-W45&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/45|Translations]] are available.

'''Updates for editors'''
* Stewards can now make [[m:Special:MyLanguage/Global blocks|global account blocks]] cause global [[mw:Special:MyLanguage/Autoblock|autoblocks]]. This will assist stewards in preventing abuse from users who have been globally blocked. This includes preventing globally blocked temporary accounts from exiting their session or switching browsers to make subsequent edits for 24 hours. Previously, temporary accounts could exit their current session or switch browsers to continue editing. This is an anti-abuse tool improvement for the [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|Temporary Accounts]] project. You can read more about the [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/Updates|progress on key features for temporary accounts]]. [https://phabricator.wikimedia.org/T368949]
* Wikis that have the [[m:Special:MyLanguage/CampaignEvents/Deployment status|CampaignEvents extension enabled]] can now use the [[m:Special:MyLanguage/Campaigns/Foundation Product Team/Event list#October 29, 2024: Collaboration List launched|Collaboration List]] feature. This list provides a new, easy way for contributors to learn about WikiProjects on their wikis. Thanks to the Campaign team for this work that is part of [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2024-2025/Product %26 Technology OKRs#WE KRs|the 2024/25 annual plan]]. If you are interested in bringing the CampaignEvents extension to your wiki, you can [[m:Special:MyLanguage/CampaignEvents/Deployment status#How to Request the CampaignEvents Extension for your wiki|follow these steps]] or you can reach out to User:Udehb-WMF for help.
* The text color for red links will be slightly changed later this week to improve their contrast in light mode. [https://phabricator.wikimedia.org/T370446]
* View all {{formatnum:32}} community-submitted {{PLURAL:32|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, on multilingual wikis, users [[phab:T216368|can now]] hide translations from the WhatLinksHere special page.

'''Updates for technical contributors'''
* XML [[m:Special:MyLanguage/Data dumps|data dumps]] have been temporarily paused whilst a bug is investigated. [https://lists.wikimedia.org/hyperkitty/list/xmldatadumps-l@lists.wikimedia.org/message/BXWJDPO5QI2QMBCY7HO36ELDCRO6HRM4/]

'''In depth'''
* Temporary Accounts have been deployed to six wikis; thanks to the Trust and Safety Product team for [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|this work]], you can read about [[phab:T340001|the deployment plans]]. Beginning next week, Temporary Accounts will also be enabled on [[phab:T378336|seven other projects]]. If you are active on these wikis and need help migrating your tools, please reach out to [[m:User:Udehb-WMF|User:Udehb-WMF]] for assistance.
*  The latest quarterly [[mw:Special:MyLanguage/Wikimedia Language and Product Localization/Newsletter/2024/October|Language and Internationalization newsletter]] is available.  It includes: New languages supported in translatewiki or in MediaWiki; New keyboard input methods for some languages; details about recent and upcoming meetings, and more.

'''Meetings and events'''
* [[mw:Special:MyLanguage/MediaWiki Users and Developers Conference Fall 2024|MediaWiki Users and Developers Conference Fall 2024]] is happening in Vienna, Austria and online from 4 to 6 November 2024. The conference will feature discussions around the usage of MediaWiki software by and within companies in different industries and will inspire and onboard new users.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/45|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W45&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:50, 4 November 2024 (UTC)
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== Tech News: 2024-46 ==

&lt;section begin=&quot;technews-2024-W46&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/46|Translations]] are available.

'''Updates for editors'''
* On wikis with the [[mw:Special:MyLanguage/Help:Extension:Translate|Translate extension]] enabled, users will notice that the FuzzyBot will now automatically create translated versions of categories used on translated pages. [https://phabricator.wikimedia.org/T285463]
* View all {{formatnum:29}} community-submitted {{PLURAL:29|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the submitted task to use the [[mw:Special:MyLanguage/Extension:SecurePoll|SecurePoll extension]] for English Wikipedia's special [[w:en:Wikipedia:Administrator elections|administrator election]] was resolved on time. [https://phabricator.wikimedia.org/T371454] 

'''Updates for technical contributors'''
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] In &lt;code dir=&quot;ltr&quot;&gt;[[mw:MediaWiki_1.44/wmf.2|1.44.0-wmf-2]]&lt;/code&gt;, the logic of Wikibase function &lt;code&gt;getAllStatements&lt;/code&gt; changed to behave like &lt;code&gt;getBestStatements&lt;/code&gt;. Invoking the function now returns a copy of values which are immutable. [https://phabricator.wikimedia.org/T270851]
* [https://en.wikipedia.org/api/rest_v1/ Wikimedia REST API] users, such as bot operators and tool maintainers, may be affected by ongoing upgrades. The API will be rerouting some page content endpoints from RESTbase to the newer [[mw:Special:MyLanguage/API:REST API|MediaWiki REST API]] endpoints. The [[phab:T374683|impacted endpoints]] include getting page/revision metadata and rendered HTML content. These changes will be available on testwiki later this week, with other projects to follow. This change should not affect existing functionality, but active users of the impacted endpoints should verify behavior on testwiki, and raise any concerns on the related [[phab:T374683|Phabricator ticket]].

'''In depth'''
* Admins and users of the Wikimedia projects [[mw:Special:MyLanguage/Moderator_Tools/Automoderator#Usage|where Automoderator is enabled]] can now monitor and evaluate important metrics related to Automoderator's actions. [https://superset.wmcloud.org/superset/dashboard/unified-automoderator-activity-dashboard/ This Superset dashboard] calculates and aggregates metrics about Automoderator's behaviour on the projects in which it is deployed. Thanks to the Moderator Tools team for this Dashboard; you can visit [[mw:Special:MyLanguage/Moderator Tools/Automoderator/Unified Activity Dashboard|the documentation page]] for more information about this work. [https://phabricator.wikimedia.org/T369488]

'''Meetings and events'''
* 21 November 2024 ([[m:Special:MyLanguage/Event:Commons community discussion - 21 November 2024 8:00 UTC|8:00 UTC]] &amp; [[m:Special:MyLanguage/Event:Commons community discussion - 21 November 2024 16:00 UTC|16:00 UTC]]) - [[c:Commons:WMF support for Commons/Commons community calls|Community call]] with Wikimedia Commons volunteers and stakeholders to help prioritize support efforts for 2025-2026 Fiscal Year. The theme of this call is how content should be organised on Wikimedia Commons.
'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/46|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W46&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:07, 12 November 2024 (UTC)
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== Tech News: 2024-47 ==

&lt;section begin=&quot;technews-2024-W47&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/47|Translations]] are available.

'''Updates for editors'''
* Users of Wikimedia sites will now be warned when they create a [[mw:Special:MyLanguage/Help:Redirects|redirect]] to a page that doesn't exist. This will reduce the number of broken redirects to red links in our projects. [https://phabricator.wikimedia.org/T326057]
* View all {{formatnum:42}} community-submitted {{PLURAL:42|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, [[mw:Special:MyLanguage/Manual:Pywikibot/Overview|Pywikibot]], which automates work on MediaWiki sites, was upgraded to 9.5.0 on Toolforge. [https://phabricator.wikimedia.org/T378676]

'''Updates for technical contributors'''
* On wikis that use the [[mw:Special:MyLanguage/Extension:FlaggedRevs|FlaggedRevs extension]], pages created or moved by users with the appropriate permissions are marked as flagged automatically. This feature has not been working recently, and changes fixing it should be deployed this week. Thanks to Daniel and Wargo for working on this. [https://phabricator.wikimedia.org/T379218][https://phabricator.wikimedia.org/T368380]

'''In depth'''
* There is a new [https://diff.wikimedia.org/2024/11/05/say-hi-to-temporary-accounts-easier-collaboration-with-logged-out-editors-with-better-privacy-protection Diff post] about Temporary Accounts, available in more than 15 languages. Read it to learn about what Temporary Accounts are, their impact on different groups of users, and the plan to introduce the change on all wikis.

'''Meetings and events'''
* Technical volunteers can now register for the [[mw:Special:MyLanguage/Wikimedia Hackathon 2025|2025 Wikimedia Hackathon]], which will take place in Istanbul, Turkey. [https://pretix.eu/wikimedia/hackathon2025/ Application for travel and accommodation scholarships] is open from '''November 12 to December 10 2024'''. The registration for the event will close in mid-April 2025. The Wikimedia Hackathon is an annual gathering that unites the global technical community to collaborate on existing projects and explore new ideas.
* Join the [[C:Special:MyLanguage/Commons:WMF%20support%20for%20Commons/Commons%20community%20calls|Wikimedia Commons community calls]] this week to help prioritize support for Commons which will be planned for 2025–2026. The theme will be how content should be organised on Wikimedia Commons. This is an opportunity for volunteers who work on different things to come together and talk about what matters for the future of the project. The calls will take place '''November 21, 2024, [[m:Special:MyLanguage/Event:Commons community discussion - 21 November 2024 8:00 UTC|8:00 UTC]] and [[m:Special:MyLanguage/Event:Commons community discussion - 21 November 2024 16:00 UTC|16:00 UTC]]'''.
* A [[mw:Special:MyLanguage/Wikimedia_Language_and_Product_Localization/Community meetings#29 November 2024|Language community meeting]] will take place '''November 29, 16:00 UTC''' to discuss updates and technical problem-solving.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/47|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W47&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 02:00, 19 November 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27806858 --&gt;

== Tech News: 2024-48 ==

&lt;section begin=&quot;technews-2024-W48&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/48|Translations]] are available.

'''Updates for editors'''
* [[File:Octicons-gift.svg|12px|link=|class=skin-invert|Wishlist item]] A new version of the standard wikitext editor-mode [[mw:Special:MyLanguage/Extension:CodeMirror|syntax highlighter]] will be available as a [[Special:Preferences#mw-prefsection-betafeatures|beta feature]] later this week. This brings many new features and bug fixes, including right-to-left support, [[mw:Special:MyLanguage/Help:Extension:CodeMirror#Template folding|template folding]], [[mw:Special:MyLanguage/Help:Extension:CodeMirror#Autocompletion|autocompletion]], and an improved search panel. You can learn more on the [[mw:Special:MyLanguage/Help:Extension:CodeMirror|help page]].
* The 2010 wikitext editor now supports common keyboard shortcuts such &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;Ctrl&lt;/code&gt;+&lt;code&gt;B&lt;/code&gt;&lt;/bdi&gt; for bold and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;Ctrl&lt;/code&gt;+&lt;code&gt;I&lt;/code&gt;&lt;/bdi&gt; for italics. A full [[mw:Help:Extension:WikiEditor#Keyboard shortcuts|list of all six shortcuts]] is available. Thanks to SD0001 for this improvement. [https://phabricator.wikimedia.org/T62928]
* Starting November 28, Flow/Structured Discussions pages will be automatically archived and set to read-only at the following wikis: &lt;bdi&gt;bswiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;elwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;euwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;fawiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;fiwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;frwikiquote&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;frwikisource&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;frwikiversity&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;frwikivoyage&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;idwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;lvwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;plwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;ptwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;urwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;viwikisource&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;zhwikisource&lt;/bdi&gt;. This is done as part of [[mw:Special:MyLanguage/Structured_Discussions/Deprecation|StructuredDiscussions deprecation work]]. If you need any assistance to archive your page in advance, please contact [[m:User:Trizek (WMF)|Trizek (WMF)]]. 
* View all {{formatnum:25}} community-submitted {{PLURAL:25|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, a user creating a new AbuseFilter can now only set the filter to &quot;protected&quot; [[phab:T377765|if it includes a protected variable]].

'''Updates for technical contributors'''
* The [[mw:Special:MyLanguage/Extension:CodeEditor|CodeEditor]], which can be used in JavaScript, CSS, JSON, and Lua pages, [[phab:T377663|now offers]] live autocompletion. Thanks to SD0001 for this improvement. The feature can be temporarily disabled on a page by pressing &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;Ctrl&lt;/code&gt;+&lt;code&gt;,&lt;/code&gt;&lt;/bdi&gt; and un-selecting &quot;&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;Live Autocompletion&lt;/bdi&gt;&quot;.
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Tool-maintainers who use the Graphite system for tracking metrics, need to migrate to the newer Prometheus system. They can check [https://grafana.wikimedia.org/d/K6DEOo5Ik/grafana-graphite-datasource-utilization?orgId=1 this dashboard] and the list in the Description of the [[phab:T350592|task T350592]] to see if their tools are listed, and they should claim metrics and dashboards connected to their tools. They can then disable or migrate all existing metrics by following the instructions in the task. The Graphite service will become read-only in April. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/KLUV4IOLRYXPQFWD6WKKJUHMWE77BMSZ/]
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] The [[mw:Special:MyLanguage/NewPP parser report|New PreProcessor parser performance report]] has been fixed to give an accurate count for the number of Wikibase entities accessed. It had previously been resetting after 400 entities. [https://phabricator.wikimedia.org/T279069]

'''Meetings and events'''
* A [[mw:Special:MyLanguage/Wikimedia_Language_and_Product_Localization/Community meetings#29 November 2024|Language community meeting]] will take place November 29 at [https://zonestamp.toolforge.org/1732896000 16:00 UTC]. There will be presentations on topics like developing language keyboards, the creation of the Mooré Wikipedia, the language support track at [[m:Wiki Indaba|Wiki Indaba]], and a report from the Wayuunaiki community on their experiences with the Incubator and as a new community over the last 3 years. This meeting will be in English and will also have Spanish interpretation.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/48|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W48&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:42, 25 November 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27847039 --&gt;

== Tech News: 2024-49 ==

&lt;section begin=&quot;technews-2024-W49&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/49|Translations]] are available.

'''Updates for editors'''
* Two new parser functions were added this week. The &lt;code dir=&quot;ltr&quot;&gt;&lt;nowiki&gt;{{&lt;/nowiki&gt;[[mw:Special:MyLanguage/Help:Magic words#interwikilink|#interwikilink]]&lt;nowiki&gt;}}&lt;/nowiki&gt;&lt;/code&gt; function adds an [[mw:Special:MyLanguage/Help:Links#Interwiki links|interwiki link]] and the &lt;code dir=&quot;ltr&quot;&gt;&lt;nowiki&gt;{{&lt;/nowiki&gt;[[mw:Special:MyLanguage/Help:Magic words#interlanguagelink|#interlanguagelink]]&lt;nowiki&gt;}}&lt;/nowiki&gt;&lt;/code&gt; function adds an [[mw:Special:MyLanguage/Help:Links#Interlanguage links|interlanguage link]]. These parser functions are useful on wikis where namespaces conflict with interwiki prefixes. For example, links beginning with &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;MOS:&lt;/code&gt;&lt;/bdi&gt; on English Wikipedia [[phab:T363538|conflict with the &lt;code&gt;mos&lt;/code&gt; language code prefix of Mooré Wikipedia]].
* Starting this week, Wikimedia wikis no longer support connections using old RSA-based HTTPS certificates, specifically rsa-2048. This change is to improve security for all users. Some older, unsupported browser or smartphone devices will be unable to connect; Instead, they will display a connectivity error. See the [[wikitech:HTTPS/Browser_Recommendations|HTTPS Browser Recommendations page]] for more-detailed information. All modern operating systems and browsers are always able to reach Wikimedia projects. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/CTYEHVNSXUD3NFAAMG3BLZVTVQWJXJAH/]
* Starting December 16, Flow/Structured Discussions pages will be automatically archived and set to read-only at the following wikis: &lt;bdi&gt;arwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;cawiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;frwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;mediawikiwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;orwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;wawiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;wawiktionary&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;wikidatawiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;zhwiki&lt;/bdi&gt;. This is done as part of [[mw:Special:MyLanguage/Structured_Discussions/Deprecation|StructuredDiscussions deprecation work]]. If you need any assistance to archive your page in advance, please contact [[m:User:Trizek (WMF)|Trizek (WMF)]]. [https://phabricator.wikimedia.org/T380910]
* This month the Chart extension was deployed to production and is now available on Commons and Testwiki. With the security review complete, pilot wiki deployment is expected to start in the first week of December. You can see a working version [[testwiki:Charts|on Testwiki]] and read [[mw:Special:MyLanguage/Extension:Chart/Project/Updates|the November project update]] for more details.
* View all {{formatnum:23}} community-submitted {{PLURAL:23|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, a bug with the &quot;Download as PDF&quot; system was fixed. [https://phabricator.wikimedia.org/T376438]

'''Updates for technical contributors'''
* In late February, temporary accounts will be rolled out on at least 10 large wikis. This deployment will have a significant effect on the community-maintained code. This is about Toolforge tools, bots, gadgets, and user scripts that use IP address data or that are available for logged-out users. The Trust and Safety Product team wants to identify this code, monitor it, and assist in updating it ahead of the deployment to minimize disruption to workflows. The team asks technical editors and volunteer developers to help identify such tools by adding them to [[mw:Trust and Safety Product/Temporary Accounts/For developers/Impacted tools|this list]]. In addition, review the [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/For developers|updated documentation]] to learn how to adjust the tools. Join the discussions on the [[mw:Talk:Trust and Safety Product/Temporary Accounts|project talk page]] or in the [[discord:channels/221049808784326656/1227616742340034722|dedicated thread]] on the [[w:Wikipedia:Discord|Wikimedia Community Discord server (in English)]] for support and to share feedback. 

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/49|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W49&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:22, 2 December 2024 (UTC)
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== Tech News: 2024-50 ==

&lt;section begin=&quot;technews-2024-W50&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/50|Translations]] are available.

'''Weekly highlight'''
* Technical documentation contributors can find updated resources, and new ways to connect with each other and the Wikimedia Technical Documentation Team, at the [[mw:Special:MyLanguage/Documentation|Documentation hub]] on MediaWiki.org. This page links to: resources for writing and improving documentation, a new &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;#wikimedia-techdocs&lt;/bdi&gt; IRC channel on libera.chat, a listing of past and upcoming documentation events, and ways to request a documentation consultation or review. If you have any feedback or ideas for improvements to the documentation ecosystem, please [[mw:Wikimedia Technical Documentation Team#Contact us|contact the Technical Documentation Team]].

'''Updates for editors'''
[[File:Edit Check on Desktop.png|thumb|Layout change for the Edit Check feature]]
* Later this week, [[mw:Special:MyLanguage/Edit check|Edit Check]] will be relocated to a sidebar on desktop. Edit check is the feature for new editors to help them follow policies and guidelines. This layout change creates space to present people with [[mw:Edit check#1 November 2024|new Checks]] that appear ''while'' they are typing. The [[mw:Special:MyLanguage/Edit check#Reference Check A/B Test|initial results]] show newcomers encountering Edit Check are 2.2 times more likely to publish a new content edit that includes a reference and is not reverted.
* The Chart extension, which enables editors to create data visualizations, was successfully made available on MediaWiki.org and three pilot wikis (Italian, Swedish, and Hebrew Wikipedias). You can see a working examples [[testwiki:Charts|on Testwiki]] and read [[mw:Special:MyLanguage/Extension:Chart/Project/Updates|the November project update]] for more details.
* Translators in wikis where the [[mw:Special:MyLanguage/Content translation/Section translation#Try the tool|mobile experience of Content Translation is available]], can now discover articles in Wikiproject campaigns of their interest from the &quot;[https://test.wikipedia.org/w/index.php?title=Special:ContentTranslation&amp;campaign=specialcx&amp;filter-type=automatic&amp;filter-id=collections&amp;active-list=suggestions&amp;from=es&amp;to=en All collection]&quot; category in the articles suggestion feature. Wikiproject Campaign organizers can use this feature, to help translators to discover articles of interest, by adding the &lt;code dir=ltr&gt;&lt;nowiki&gt;&lt;page-collection&gt; &lt;/page-collection&gt;&lt;/nowiki&gt;&lt;/code&gt; tag to their campaign article list page on Meta-wiki. This will make those articles discoverable in the Content Translation tool. For more detailed information on how to use the tool and tag, please refer to [[mw:Special:MyLanguage/Translation suggestions: Topic-based &amp; Community-defined lists/How to use the features|the step-by-step guide]]. [https://phabricator.wikimedia.org/T378958]
* The [[mw:Special:MyLanguage/Extension:Nuke|Nuke]] feature, which enables administrators to mass delete pages, now has a [[phab:T376379#10310998|multiselect filter for namespace selection]]. This enables users to select multiple specific namespaces, instead of only one or all, when fetching pages for deletion.
* The Nuke feature also now [[phab:T364225#10371365|provides links]] to the userpage of the user whose pages were deleted, and to the pages which were not selected for deletion, after page deletions are queued. This enables easier follow-up admin-actions. Thanks to Chlod and the Moderator Tools team for both of these improvements. [https://phabricator.wikimedia.org/T364225#10371365]
* The Editing Team is working on making it easier to populate citations from archive.org using the [[mw:Special:MyLanguage/Citoid/Enabling Citoid on your wiki|Citoid]] tool, the auto-filled citation generator. They are asking communities to add two parameters preemptively, &lt;code dir=ltr&gt;archiveUrl&lt;/code&gt; and &lt;code dir=ltr&gt;archiveDate&lt;/code&gt;, within the TemplateData for each citation template using Citoid. You can see an [https://en.wikipedia.org/w/index.php?title=Template%3ACite_web%2Fdoc&amp;diff=1261320172&amp;oldid=1260788022 example of a change in a template], and a [https://global-search.toolforge.org/?namespaces=10&amp;q=%5C%22citoid%5C%22%3A%20%5C%7B&amp;regex=1&amp;title= list of all relevant templates]. [https://phabricator.wikimedia.org/T374831]
* One new wiki has been created: a {{int:project-localized-name-group-wikivoyage}} in [[d:Q9240|Indonesian]] ([[voy:id:|&lt;code&gt;voy:id:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T380726]
* Last week, all wikis had problems serving pages to logged-in users and some logged-out users for 30–45 minutes. This was caused by a database problem, and investigation is ongoing. [https://www.wikimediastatus.net/incidents/3g2ckc7bp6l9]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:19}} community-submitted {{PLURAL:19|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, a bug in the [[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add Link]] feature has been fixed. Previously, the list of sections which are excluded from Add Link was partially ignored in certain cases. [https://phabricator.wikimedia.org/T380455][https://phabricator.wikimedia.org/T380329]

'''Updates for technical contributors'''
* [[mw:Special:MyLanguage/Codex|Codex]], the design system for Wikimedia, now has an early-stage [[git:design/codex-php|implementation in PHP]]. It is available for general use in MediaWiki extensions and Toolforge apps through [https://packagist.org/packages/wikimedia/codex Composer], with use in MediaWiki core coming soon. More information is available in [[wmdoc:design-codex-php/main/index.html|the documentation]]. Thanks to Doğu for the inspiration and many contributions to the library. [https://phabricator.wikimedia.org/T379662]
* [https://en.wikipedia.org/api/rest_v1/ Wikimedia REST API] users, such as bot operators and tool maintainers, may be affected by ongoing upgrades. On December 4, the MediaWiki Interfaces team began rerouting page/revision metadata and rendered HTML content endpoints on [[testwiki:|testwiki]] from RESTbase to comparable MediaWiki REST API endpoints. The team encourages active users of these endpoints to verify their tool's behavior on testwiki and raise any concerns on the related [[phab:T374683|Phabricator ticket]] before the end of the year, as they intend to roll out the same change across all Wikimedia projects in early January. These changes are part of the work to replace the outdated [[mw:RESTBase/deprecation|RESTBase]] system.
* The [https://wikimediafoundation.limesurvey.net/986172 2024 Developer Satisfaction Survey] is seeking the opinions of the Wikimedia developer community. Please take the survey if you have any role in developing software for the Wikimedia ecosystem. The survey is open until 3 January 2025, and has an associated [[foundation:Legal:Developer Satisfaction Survey 2024 Privacy Statement|privacy statement]].
* There is no new MediaWiki version this week. [https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

'''Meetings and events'''
* The next meeting in the series of [[c:Commons:WMF support for Commons/Commons community calls|Wikimedia Foundation discussions with the Wikimedia Commons community]] will take place on [[m:Event:Commons community discussion - 12 December 2024 08:00 UTC|December 12 at 8:00 UTC]] and [[m:Event:Commons community discussion - 12_December 2024 16:00 UTC|at 16:00 UTC]]. The topic of this call is new media and new contributors. Contributors from all wikis are welcome to attend.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/50|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W50&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:16, 9 December 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27919424 --&gt;

== Tech News: 2024-51 ==

&lt;section begin=&quot;technews-2024-W51&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/51|Translations]] are available.

'''Weekly highlight'''
* Interested in improving event management on your home wiki? The [[m:Special:MyLanguage/CampaignEvents|CampaignEvents extension]] offers organizers features like event registration management, event/wikiproject promotion, finding potential participants, and more - all directly on-wiki. If you are an organizer or think your community would benefit from this extension, start a discussion to enable it on your wiki today. To learn more about how to enable this extension on your wiki, visit the [[m:CampaignEvents/Deployment status#How to Request the CampaignEvents Extension for your wiki|deployment status page]].

'''Updates for editors'''
* Users of the iOS Wikipedia App in Italy and Mexico on the Italian, Spanish, and English Wikipedias, can see a [[mw:Special:MyLanguage/Wikimedia Apps/Team/iOS/Personalized Wikipedia Year in Review|personalized Year in Review]] with insights based on their reading and editing history.
* Users of the Android Wikipedia App in Sub-Saharan Africa and South Asia can see the new [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android/Rabbit Holes|Rabbit Holes]] feature. This feature shows a suggested search term in the Search bar based on the current article being viewed, and a suggested reading list generated from the user’s last two visited articles.
* The [[m:Special:MyLanguage/Global reminder bot|global reminder bot]] is now active and running on nearly 800 wikis. This service reminds most users holding temporary rights when they are about to expire, so that they can renew should they want to. See [[m:Global reminder bot/Technical details|the technical details page]] for more information.
* The next issue of Tech News will be sent out on 13 January 2025 because of the end of year holidays. Thank you to all of the translators, and people who submitted content or feedback, this year.
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:27}} community-submitted {{PLURAL:27|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, a bug was [[phab:T374988|fixed]] in the Android Wikipedia App which had caused translatable SVG images to show the wrong language when they were tapped.

'''Updates for technical contributors'''
* There is no new MediaWiki version next week. The next deployments will start on 14 January. [https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar/2025] 

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/51|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W51&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:24, 16 December 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27942374 --&gt;</text>
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    </revision>
  </page>
  <page>
    <title>Risk Management</title>
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        <username>Bert Niehaus</username>
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      <comment>/* Main Applications of Risk Management */</comment>
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      <text bytes="12081" sha1="dts688xw0ejeerky3yk08oqjziy34ca" xml:space="preserve">[[File:Basic Risk Response Cycle.png|thumb|Basic Risk and Response Cycle]]
[[File:Risk response cylce.png|thumb|Application: Risk and Response Cycle with intergration of Satellite technology and smartphone (see Learning Task)]]
[[File:Sustainable Development Goal 11SustainableCities.svg|thumb|[[SDG11]]: Sustainable Cities and Communities - Learning Resource supports the SDGs - [http://www.un.org/sustainabledevelopment/wp-content/uploads/2016/10/UN-Guidelines-for-Use-of-SDG-logo-and-17-icons.October-2016.pdf UN-Guidelines]&lt;ref&gt;UN-Guidelines for Use of SDG logo and the 17 SDG icons (2016/10) - http://www.un.org/sustainabledevelopment/wp-content/uploads/2016/10/UN-Guidelines-for-Use-of-SDG-logo-and-17-icons.October-2016.pdf&lt;/ref&gt;]]
[[File:Role playing 110808-A-ZW119-022.jpg|thumb|Role playing - risk management training event]]
'''Risk management''' has two main tasks:
* [[/Content_Matrix/|determine and calculate the risk]] and
* organize the response to the identified risk:
** Improve the preparedness to an event,
** Reduce the probability of an event by risk mitigation activities,
** Reduce the impact of an event
See [[/Content Matrix/|Risk Management Content Matrix RM]]
== Response Time ==
When someone has a [[Wikipedia:heart attack|heart attack]] or a [[Wikipedia:stroke|stroke]] the medical response is urgent in comparison to a broken leg that has in general a long time span in which medical response will not have a serious impact on the health condition of the patient. For the link between risk and response to the available time span is crucial for the assessment of the impact on an hazardous event. Going back to example of a patient with a heart attack, the main question is 
: ''&quot;How long does it take until an ambulance with medical support is available?&quot;''
People living close to the health care facility have a better response time in an case of emergency than people living in rural areas. This leads to that fact that risk depends on spatial allocation of resources and distance of an event to the resource that can be used for risk mitigation (see [[/Spatial risk management/]]).

== Main Applications of Risk Management ==
* [[/Finance|Risk Management in Finance]]&lt;ref&gt;Stulz, R. M. (1996). Rethinking risk management. Journal of applied corporate finance, 9(3), 8-25.&lt;/ref&gt;
* [[/Environment/|Environmental Risk Management]]&lt;ref&gt;Power, M., &amp; McCarty, L. S. (1998). Peer reviewed: a comparative analysis of environmental risk assessment/risk management frameworks. Environmental science &amp; technology, 32(9), 224A-231A.&lt;/ref&gt;
* [[/Health/|Risk Management in the Health Domain]]&lt;ref&gt;Cagliano, Anna Corinna, Sabrina Grimaldi, and Carlo Rafele. &quot;A systemic methodology for risk management in healthcare sector.&quot; Safety Science 49.5 (2011): 695-708.&lt;/ref&gt;
* [[/Global Challenges/|Global Challenges of Risk Management]]: Climate Change and Planetary Boundaries&lt;ref&gt;Rockström, J., Steffen, W., Noone, K., Persson, Å., Chapin III, F. S., Lambin, E., ... &amp; Nykvist, B. (2009). Planetary boundaries: exploring the safe operating space for humanity. Ecology and society, 14(2).&lt;/ref&gt; are linked to economy, health&lt;ref&gt;McMichael, A. J., Woodruff, R. E., &amp; Hales, S. (2006). Climate change and human health: present and future risks. The Lancet, 367(9513), 859-869.&lt;/ref&gt;, and [[Risk Literacy|social, cultural and technical beliefs]]&lt;ref&gt;O'connor, R. E., Bard, R. J., &amp; Fisher, A. (1999). Risk perceptions, general environmental beliefs, and willingness to address climate change. Risk analysis, 19(3), 461-471.&lt;/ref&gt;
* [[Vulnerability assessment|Vulnerability Assessment]] is necessary to improve the preparedness to certain risks.
* [[/Risk path integral/]] - calculate the risk for path as integral - [[Mathematical Modelling]] of [[w:en:Risk|Risk]]

== Learning Task ==

[[File:Earthlights dmsp 1994–1995.jpg|thumb|[[w:light pollution|Artificial lights]] can be detected from satellites. Light emission during night is a proxy variable for the availability of electricity after disasters.]]
* '''([https://en.wikiversity.org/wiki/Risk_Management#/media/File:Basic_Risk_Response_Cycle.png Risk &amp; Response Cycle])''' On this page on right you find an example of an extended risk and response cycle, with a focus on satellite technology and smartphones. Take your personal expertise as starting point and create your own cycle for a domain of your choice. 
* '''([[w:planetary boundaries|Planetary Boundaries]])''' Approach risk management from the angle of [[w:planetary boundaries|planetary boundaries]]&lt;ref&gt;Steffen, W., Richardson, K., Rockström, J., Cornell, S. E., Fetzer, I., Bennett, E. M., ... &amp; Folke, C. (2015). Planetary boundaries: Guiding human development on a changing planet. Science, 347(6223), 1259855.&lt;/ref&gt; and consumption of planetary resources. Create a first risk mitigations concept that might work in your family, school, university, town or region. What are the requirements and constraints you can identify?
* '''([[Machine learning|Machine Learning]])''' Explain how [[Machine learning|Machine Learning (ML)]] can be used to assess risk, allocate resources according to risk and assess the efficiency of the resources to reduce the risk for a specific community. Discuss also ethical aspect in the context of [[Machine learning|Machine Learning]] and Risk Management.
* '''(Sustainable Development Goals)''' Consider people in developing countries that battle to survive the next day, week, month. Describe the challenges for long-term goals in the context of climate change and the planetary boundaries (see [http://www.un.org/sustainabledevelopment/sustainable-development-goals/ Sustainable Development Goals SDG of United Nations]&lt;ref&gt;Griggs, D., Stafford-Smith, M., Gaffney, O., Rockström, J., Öhman, M. C., Shyamsundar, P., ... &amp; Noble, I. (2013). Policy: Sustainable development goals for people and planet. Nature, 495(7441), 305-307.&lt;/ref&gt;). Explain how Risk Management could contribute to sustainable communities and sustainable development in general!
* '''(Systems Thinking)''' If we apply [[/Systems Thinking and Risk/|systems thinking]] in risk management, we will now look on a basic case, where someone living in a [[w:semi-arid |semi-arid]] region will cut down trees for fire wood or for a campfire. Systems analysis will tell us, that cutting down trees will increase erosion of soil and it will cause long-term food problem of agricultural production. 
** Analyze what you would do with the last trees, if you are risk literate and you know about this dependencies, but it gets &lt;math&gt;2^{o}C&lt;/math&gt; during night? 
** What are you implication for [[Risk_Management/Short-term_long-term_drivers_in_Risk_Management|short-term  and long-term risk management]]?
* '''(Risk Management for Electricity as Resource):''' Light emission during night is a proxy variable for the availability of electricity after disasters. Describe the procedure of assessment from the analysis of satellite images (before/after the event), to assessment of the impact of missing electricity for health service delivery, communication or provision of services in general. How would you assess the vulnerability in your hometown in comparison to other areas in the world?
* '''([[COVID-19]])''' Analyze the challenges of risk management in epidemiology and identify different phases:
:* direct implementation of risk mitigations strategies,
:* assessment of the impact of risk mitigation strategies and the improvement of the strategies, 
:* the comparison different risks for the health system, economy, society, ... and explain how the risks are linked, e.g. improvement of protective measures, [[COVID-19/Workflow_Transformation|workflow transformations]] and other elements of [[Risk Literacy]] did not work good enough and a lockdown created economic impacts, ...
* '''[[/Tailored Wikibooks/]]''' Learner have different requirements, constraints and prerequisites. Learning resources that included for one learner might be superfluous for an other learner, because he or she might know the topic very well. Create Wikibook tailored for the learner is an innovative tool to cover this needs. It is applicable in any domain for learning and capacity building. In the context of Risk Management we focus on capacity building and learning and create or adapt a Wikibook tailored to the individual exposure to certain risks and individual skills to respond to the risk.
* '''[[Role Play]]:''' Explore the concept of [[Role Play|role play]] and serious games&lt;ref&gt;Susi, T., Johannesson, M., &amp; Backlund, P. (2007). Serious games: An overview.&lt;/ref&gt; for risk management&lt;ref&gt;Rumore, D., Schenk, T., &amp; Susskind, L. (2016). Role-play simulations for climate change adaptation education and engagement. Nature Climate Change, 6(8), 745-750.&lt;/ref&gt;&lt;ref&gt;Rajbhandari, L., &amp; Snekkenes, E. A. (2013, July). Case study role play for risk analysis research and training. In International Workshop on Security in Information Systems (Vol. 2, pp. 12-23). SCITEPRESS.&lt;/ref&gt;.
* '''([[3D Modelling]])''' How can [[3D Modelling]], [[Virtual Reality]] and [[Augmented Reality]] support risk management&lt;ref&gt;Blanco-Fernández, Y., López-Nores, M., Pazos-Arias, J. J., Gil-Solla, A., Ramos-Cabrer, M., &amp; García-Duque, J. (2014). REENACT: A step forward in immersive learning about Human History by augmented reality, role playing and social networking. Expert Systems with Applications, 41(10), 4811-4828.&lt;/ref&gt;&lt;ref&gt;Wu, H. K., Lee, S. W. Y., Chang, H. Y., &amp; Liang, J. C. (2013). Current status, opportunities and challenges of augmented reality in education. Computers &amp; education, 62, 41-49.&lt;/ref&gt;.

== Resources ==
* [[Risk_Management/Content Matrix|Risk Management - Content Matrix]]
* [[Risk Literacy]] describes the individual skill/expertise to manage the risk
* [[/Spatial risk management/]], risk vary in different geolocation, spatial risk management takes the spatio-temporal variations of risk into account
* [[/Agricultural Water Pollution Management/]]
* [[/Systems Thinking and Risk/]] Economy, environment, public health, social and cultural conditions for risk are connected and risk mitigation activities in Systemic Risk)
* [[/Disaster Management/]] considers risk management for disasters. Disaster Risk Reduction aims to reduce the damage caused by natural hazards like earthquakes, floods, droughts and cyclones, that cause a serious disruption of the functioning of a community or a society&lt;ref&gt;United Nations Office for Disaster Risk Reduction (UNSDR)  (2017) http://www.unisdr.org/we/inform/terminology&lt;/ref&gt;. [[w:Emergency Management|Disasters]] involve widespread human, material, economic or environmental impacts, which exceed the ability of the affected community or society to cope using its own resources. The [[/Systems Thinking and Risk/|systemic risk]] is managed in different phases from immediate response after an event and long-term preparedness measures and capacity building to reduce the [[w:Vulnerability|vulnerability]].
* [[/Tailored Wikibooks/]] to let the learners adapt provided Wikibooks according to their needs and skills.
* [[Vulnerability assessment]] is required to determine the key assets, that must be protected by risk mitigation strategies and may lead to better preparedness.

==See also==
* [[Decision Making]]
* [[Resource Management]]
* [[Risk_Management#Origin_of_Course_Development|Origin of Course Development]]
* [[High risk research]]
* [[Machine learning]]
* [[Project Management/Risk]]
* [[w:Emergency Management|Emergency Management]]
* [[Open Educational Resources]] for Risk management - Reach People provide access to risk mitigation strategies.
* [[w:UNOCHA|UNOCHA United Nations Office for Coordination of Humanitarian Affairs]]
* [[Grand Challenges]]
* [[Climate change]]
* [[Risk Literacy]]
* [[ICT Literacy]]
* [[COVID-19]]
* [[Applied toxicity of chemicals]]
* [[One Health]]

== References ==
&lt;references/&gt;

[[Category:Risk management]]
[[Category:SDG 11 - Sustainable Cities and Communities]]
[[Category:Role-playing games]]

&lt;noinclude&gt;
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    <title>U. S. Government/Federalism vs Separation of Power</title>
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This is a quick guide on &quot;Federalism&quot; and &quot;Separation of Powers&quot;, story behind the two terms, what they are, and the difference. 
==Story of the Constitution==
[[File:Seperation of Powers.PNG|400px|thumbnail|right|Separation of Powers]]
The writers of the Constitution wanted to avoid the government from becoming too strong, so they limited the government's power to only do those things people have given it the power to do. This was the principle of '''Consent of the Governed'''&lt;sup&gt;[[Federalism_vs_Separation_of_Power#Consent_of_the_Governed|[1]]]&lt;/sup&gt;. To give the Constitution legitimacy, the Preamble to the Constitution of the United States of America begins, ''We the People'', which establishes that the power of government comes from the people, this principle became was known as the principle of '''Checks and Balances'''&lt;sup&gt;[[Federalism_vs_Separation_of_Power#Checks_and_Balances|[2]]]&lt;/sup&gt;. 

In addition to the idea that power of government comes from the people, they brought back the idea of electing people to make laws and conduct government on their behalf, which is the fundamental political principle of '''Representative Government'''&lt;sup&gt;[[Federalism_vs_Separation_of_Power#Representative_Government|[3]]]&lt;/sup&gt;. After the states agreed to the Constitution, the Constitution became the Supreme Law of the land in America.

The writers put a lot of thought into what they needed to do. This is when they decided they need three more principles to limit the power of the Central Government.

==Constitutional Principles==
[[File:Federalism.JPG|thumbnail|right|870px|Federalism Chart of the Levels of Government]]
* One of their decisions was to divide the government into levels. The national (U.S.), state (Florida, Idaho, Montana, Virginia, Maine), and local (Chicago, Richmond, New York City) governments are the most popular levels of government. This division of power between ''levels'' would be the principle of '''Federalism'''&lt;sup&gt;[[Federalism_vs_Separation_of_Power#Federalism|[4]]]&lt;/sup&gt;.
* To add to the confusion, they divided the government into three ''branches'' (Legislative, Executive, and Judicial). The principle became known as '''Separation of Powers'''&lt;sup&gt;[[Federalism_vs_Separation_of_Power#Separation_of_Powers|[5]]]&lt;/sup&gt;. This separation would allow each branch of government to do something different. 
* Even though the Declaration of Independence was the thing of the past, the people of the United States were still afraid of any one person or government becoming too strong. They quickly developed the principle of '''Limited Government'''&lt;sup&gt;[[Federalism_vs_Separation_of_Power#Limited_Government|[6]]]&lt;/sup&gt; in order to limit the power of the other branches. This meant that at any time, the Legislative Branch may have the final decision on the Executive and Judicial goals, the Executive Branch could have final decision on the Judicial and Legislative, and Judicial Branch would have final decision on the Executive and Legislative efforts. Even though they were separate branches, the branches could not work without the other two branches. 
=== Vocabulary ===
===== Consent of the Governed =====
People are the source of any and all governmental power
===== Checks and Balances =====
[http://www.congressforkids.net/Constitution_checksandbalances.htm The delegates build a &quot;checks and balances&quot; system, so one branch doesn't have all the power].
===== Representative Government =====
In a representative system of government, people elect public officeholders to make laws and conduct government on the people's behalf.
===== Federalism =====
Divisions of power between levels of the government. 
===== Separation of Powers =====
Divided the government into three BRANCHES (Legislative, Executive, Judicial).
===== Limited Government =====
Not all powerful, only follows what the people order them to do.

[[Category:Atcovi's Work]]
[[Category:Civics]]
[[Category:United States Government]]</text>
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    <title>Animal Phyla</title>
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      <text bytes="13097" sha1="0inty9qfbynapqpu4hc0zqrxoagfph1" xml:space="preserve">[[Image:Animalia diversity.jpg|thumb|300px|right|Some examples of animal diversity.]]
A Phylum (pl. Phyla) is the largest formal major grouping within animal taxonomy.
This list is presented in alphabetical order, and not in any systematic/evolutionary arrangement.
This list is also available in [[/Table/|table form]]. Science is by no means static. There are arguments of all sizes and shapes about the taxonomy of the Animal Phyla. Other sources may combine or split these listed Phyla. However, at this time, the list presented here should stand in good stead for an introduction to the topic of animal diversity.

There are approximately 1.9 million animal species that have been described by science.

This list tries to give the following information on each Phylum:
*Phyllum Name
*A link to a sub-page discussing that Phylum in more detail (if it yet exists)
*Name Meaning (in English)
*An English Common Name, where one is in regular use
*Distinguishing characteristics of animals within the Phylum
*An approximate number of species described within that Phylum. Since zoology fortunately refuses to stand move, this number can change.

Oh! And there's a quiz at the end.
See also [[/Lesson Plan for Animal Phyla/]]. Or head to [[Introduction_to_Taxonomy]] for more on that topic. You can also visit [[Plant Divisions (Phyla)]], the equivalent page for plants.

==Acanthocephalae==
[[Image:Macracanthorhynchus_hirudinaceus_adult_BAM1.jpg|thumb|100px|left|An Acanthocephalan, ''Macracanthorhynchus hirudinaceus'', adult.]]
[[/Acanthocephala/]]

Name Meaning: Horny head

English Common Name: Horny-headed worms

Major distinguishing characteristics: Reversible spiny proboscis

Approximate number of species described: 1,151

==Acoelomorpha==
[[File:Acoelomorpha.jpg|thumb|100px|left|Acoelomorpha]]
[[/Acoelomorpha/]]

Name Meaning: With gut

English Common Name: Acoels

Major distinguishing characteristics: No mouth or alimentary canal

Approximate number of species described: 50,000

== Annelida ==
&lt;div id=&quot;Annelida&quot;&gt;&lt;/div&gt;
[[Image:Reef0200.jpg|thumb|100px|left|An feather duster worm of the family Sabellidae in parchment tube.]]
[[/Annelida/]]

Name Meaning: Little ring

English Common Name: Segmented worms, annelids

Major distinguishing characteristics: Multiple circular segments

Approximate number of species described: 22,000 modern

==Arthropoda==
[[Image:Chelicerae_%26_pedipalps_%288689230685%29.jpg|thumb|100px|left|A scorpion, an example of an Arthropod]]
[[/Arthropoda/]]

Name Meaning: Jointed foot

English Common Name: Arthropods

Major distinguishing characteristics: Chitin exoskeleton

Approximate number of species described: 1,134,000+
==Brachiopoda==
[[Image:Isocrania_costata_Sowerby_1823.jpg|thumb|100px|left|Brachiopod ''Isocrania costata''.]]
[[/Brachiopoda/]]

Name Meaning: Arm foot

English Common Name: Lamp shells, brachiopod

Major distinguishing characteristics: Lophophore and pedicle

Approximate number of species described: between 300 and 500 extant
==Bryozoa==
[[Image:Freshwater_Bryozoan234.JPG|thumb|100px|left|A freshwater Bryozoan, species unknown.]]
[[/Bryozoa/]]

Name Meaning: Moss animals

English Common Name: Moss animals, sea mats, bryozoans

Major distinguishing characteristics: Lophophore, no pedicle, ciliated tentacles

Approximate number of species described: about 5,000 living species
==Chaetognatha==
[[Image:Chaetognatha.PNG|thumb|100px|left|Chaetognathans.]]
[[/Chaetognatha/]]

Name Meaning: Longhair jaw

English Common Name: Arrow worms

Major distinguishing characteristics: Chitinous spines either side of head, fins

Approximate number of species described: about 100 modern species
==Chordata==
[[Image:Clavelina lepadiformis (Müller, 1776) - Banyuls-sur-Mer - 04.83.jpg|thumb|100px|left|''Clavelina lepadiformis'', a non-vertebrate chordates.]]
[[/Chordata/]]

Name Meaning: Cord

English Common Name: Chordates

Major distinguishing characteristics: Hollow dorsal nervous chord

Approximate number of species described: about 100,000+
==Cnidaria==
[[Image:Cynarina_lacrymalis.jpeg|thumb|100px|left|''Cynarina lacrymalis'' an anthozoan member of the Phylum Cnidaria.]]
[[/Cnidaria/]]

Name Meaning: Stinging nettle

English Common Name: Coelenterates, cnidarians, sea anemones, jellies, hydra

Major distinguishing characteristics: Cnidocytes (stinging cells)

Approximate number of species described: about 11,000

==Ctenophora==
[[Image:LightRefractsOf comb-rows of ctenophore Mertensia ovum.jpg|thumb|100px|left|''Mertensia'', a Ctenophore.]]
[[/Ctenophora/]]

Name Meaning: Comb bearer

English Common Name: Comb jellies, Ctenophores

Major distinguishing characteristics: Eight &quot;comb rows&quot; of fused cilia

Approximate number of species described: about 100 modern species
==Cycliophora==
[[Image:Cycliophora - Symbion pandora.png|thumb|100px|left|''Symbion pandora'', a Cycliphoran.]]
[[/Cycliophora/]]

Name Meaning: Wheel carrying

English Common Name: Symbion

Major distinguishing characteristics: Circular mouth surrounded by small cilia

Approximate number of species described: at least 3
==Echinodermata==
[[Image:Crinoideos.jpg|thumb|100px|left|''Crinoideos'', an example of Ecinoderms.]]
[[/Echinodermata/]]

Name Meaning: Spiny skin(Body)

English Common Name: Echinoderms

Major distinguishing characteristics: Five-fold radial symmetry, mesodermal layer, calcified spines

Approximate number of species described: about 7,000 living species and 13,000 extinct ones
==Entoprocta==
[[Image:Barentsa discreta suzukokemusi01.JPG|thumb|100px|left|''Barentsa discreta suzukokemusi'', a species of goblet worm.]]
[[/Entoprocta/]]

Name Meaning: Inside anus

English Common Name: Goblet worm

Major distinguishing characteristics: Anus inside ring of cilia

Approximate number of species described: about 150
==Gastrotricha==
[[Image:Chaetonotus (Chaetonotus) polyspinosus.jpg|thumb|100px|left|''Chaetonotus polyspinosus'' a species of meiofauna.]]
[[/Gastrotricha/]]

Name Meaning: Hair stomach

English Common Name: Meiofauna

Two terminal adhesive tubes

Approximate number of species described: about 690
==Gnathostomulida==
[[Image:Gnathostomula paradoxa Sylt.tif|thumb|100px|left|''Gnathostomula paradoxa'', a jaw worm.]]
[[/Gnathostomulida/]]

Name Meaning: Jaw orifice

English Common Name: Jaw worms

Major distinguishing characteristics: 

Approximate number of species described: about 100
==Hemichordata==
[[Image:Glossograptus whitfieldii 01.jpg|thumb|100px|left|Glossograptus whitfieldii, a fossil hemichordate.]]
[[/Hemichordata/]]

Name Meaning: Half cord

English Common Name: Acorn worms

Major distinguishing characteristics: Stomochord in collar

Approximate number of species described: about 100 living species
==Kinorhyncha==
[[Image:Paracentrophyes quadridentatus.jpg|thumb|100px|left|''Paracentrophyes quadridentatus'', a mud dragon.]]
[[/Kinorhyncha/]]

Name Meaning: Motion snout

English Common Name: Mud dragons

Major distinguishing characteristics: Eleven segments, each with a dorsal plate

Approximate number of species described: about 150
==Loricifera==
[[Image:Pliciloricus enigmatus.jpg|thumb|100px|left|A member of the Phylum Loricifera, ''Pliciloricus enigmatus''.]]
[[/Loricifera/]]

Name Meaning: Corset bearer

English Common Name: Brush heads

Major distinguishing characteristics: Umbrella-like scales at each end

Approximate number of species described: about 122
==Micrognathozoa==
[[Image:Limnonathia, drawing.tif|thumb|100px|left|''Limnonathia'', drawing, a Micrognathozoan.]]
[[/Micrognathozoa/]] (sometimes called Phylum Gnathifera)

Name Meaning: Tiny jaw animals

English Common Name: Micrognathozoans

Major distinguishing characteristics: Accordion like extensible thorax

Approximate number of species described: 1

==Mollusca==
[[Image:Shells01.jpg|thumb|100px|left|A group of Gastropod shells, Phylum Mollusca.]]
[[/Mollusca/]]

Name Meaning: Thin shell

English Common Name: Mollusks or molluscs

Major distinguishing characteristics: Muscular foot and Mantle, round shell

Approximate number of species described: 85,000


==Nematoda==
[[Image:Mermis nigrescens beentree.jpg|thumb|100px|left|''Mermis nigrescens beentree'', a nematode worm.]]
[[/Nematoda/]]

Name Meaning: Thread like

English Common Name: Round worms, Nematodes

Major distinguishing characteristics: Round cross section, keratin, cuticle

Approximate number of species described: 80 000 - 1 million
==Nematomorpha==
[[Image:Gordiidae A MRKVICKA.JPG|thumb|100px|left|A Gordian worm, Phylum Nematomorpha.]]
[[/Nematomorpha/]]

Name Meaning: Thread form

English Common Name: Horsehair worms

Major distinguishing characteristics:

Approximate number of species described: about 320
==Nemertea==
[[Image:Micrura alaskensis.png|thumb|100px|left|Micrura alaskensis, a ribbon worm.]]
[[/Nemertea/]]

Name Meaning: A sea nymph

English Common Name: Ribbon worms

Major distinguishing characteristics:

Approximate number of species described: about 1200
==Onychophora==
[[Image:Euperipatoides kanangrensis crop.jpg|thumb|100px|left|''Euperipatoides kanangrensis'', a velvet worm.]]
[[/Onychophora/]]

Name Meaning: Claw bearer

English Common Name: Velvet worms

Major distinguishing characteristics: Legs tipped by chitinous claws

Approximate number of species described: about 200 modern
==Orthonectida==
&lt;div id=&quot;Orthonectida&quot;&gt;&lt;/div&gt;
[[Image:Orthonetida dict flat and cylinder.png|thumb|100px|left|Orthonetida, an Orthonectidan.]]
[[/Orthonectida/]]

Name Meaning: Straight swim

English Common Name: Orthonectida

Major distinguishing characteristics:

Approximate number of species described: about 20

==Phoronida==
[[Image:Phoronis Maria Grazia Montanucci2.jpg|thumb|100px|left|'' Phoronis hippocrepis'', a species of horseshoe worm.]]
[[/Phoronida/]]

Name Meaning: Zeus' mistress

English Common Name: Horseshoe worms

Major distinguishing characteristics: U-shaped gut

Approximate number of species described: 20
==Placozoa==
[[Image:Trichoplax adhaerens photograph.png|thumb|100px|left|''Trichoplax adhaerens'', the only Placozoan.]]
[[/Placozoa/]]

Name Meaning: Plate animals

English Common Name: none

Major distinguishing characteristics:

Approximate number of species described: 1
==Platyhelminthes==
[[Image:Gigantolina.elongata.in.vivo.png|thumb|100px|left|''Gigantolina elongata'', a flat worm.]]
[[/Platyhelminthes/]]

Name Meaning: Flat worms

English Common Name: Flat worms

Major distinguishing characteristics:

Approximate number of species described: about 25,000
==Porifera==
[[Image:Eponge à déterminer (1).jpg|thumb|100px|left|An unidentified sponge.]]
[[/Porifera/]]

Name Meaning: Pore bearer

English Common Name: Sponges

Major distinguishing characteristics: Perforated interior wall

Approximate number of species described: over 5,000 modern
==Priapulida==
[[Image:Adult priapulid.jpg|thumb|100px|left|''Priapulus caudatus'', a Priapulid worm.]]
[[/Priapulida/]]

Name Meaning: Penis

English Common Name: Priapulid worms

Major distinguishing characteristics: Retractable proboscis surrounded by papillae

Approximate number of species described: 17
==Rhombozoa==
&lt;div id=&quot;Rhombozoa&quot;&gt;&lt;/div&gt;
[[Image:Limnognathia maerski.PNG|thumb|100px|left|''Limnognathia maerski'', a lozenge animal.]]
[[/Rhombozoa/]]

Name Meaning: Lozenge animal

English Common Name: Lozenge animals

Major distinguishing characteristics: Single axial cell surrounded by ciliated cells

Approximate number of species described: 75

==Rotifera==
[[Image:20090730 020239 Rotifer.jpg|thumb|100px|left|An unidentified rotifer.]]
[[/Rotifera/]]

Name Meaning: Wheel bearer

English Common Name: Rotifers

Major distinguishing characteristics: Anterior crown of cilia

Approximate number of species described: about 2000
==Sipuncula==
[[Image:Sipunculus nudus.jpeg|thumb|100px|left|''Sipunculus nudus'', a peanut worm.]]
[[/Sipuncula/]]

Name Meaning: Small tube

English Common Name: Peanut worms

Major distinguishing characteristics: Mouth surrounded by invertible tentacles

Approximate number of species described: 144-320
==Tardigrada==
[[Image:SEM image of Milnesium tardigradum in active state - journal.pone.0045682.g001-2.png|thumb|100px|left|Tardigrade ''Milnesium tardigradum'' in active state.]]
[[/Tardigrada/]]

Name Meaning: Slow step

English Common Name: Water bears

Major distinguishing characteristics: Four segmented body and head

Approximate number of species described: 1,000+
==Xenoturbellida==
[[Image:Xenoturbella bockii longitudinal section.svg|thumb|100px|left|''Xenoturbella bockii'' longitudinal section.]]
[[/Xenoturbellida/]]

Name Meaning: Strange flatworm

English Common Name: none

Major distinguishing characteristics: Ciliated deuterostome

Approximate number of species described: 2
==Quizzes==
* [[/Quiz/|Animal Phyla Quiz]]

==References==
{{reflist}}
*Brusca, Richard C., Wendy Moore, Stephen M. Shuster. Invertebrate Zoology 3rd ed. 2016. Sinauer. 1104 pp.
*[https://en.wikipedia.org/wiki/Phylum Phylum] on Wikipedia
[[Category:Lists]]
[[Category:Taxonomy]]
[[Category:Zoology]]
[[Category:Animals]]
{{science}} {{biology}}</text>
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    <title>User talk:Bnhassin</title>
    <ns>3</ns>
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      <comment>/* Tech News: 2024-51 */ new section</comment>
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      <text bytes="693611" sha1="39hqwvn3ff7602mm0a8onoxzlch1y3c" xml:space="preserve">== First Message Posting ==

Update Talk on Wikiversity [[User:Bnhassin|Bnhassin]] ([[User talk:Bnhassin|discuss]] • [[Special:Contributions/Bnhassin|contribs]]) 21:04, 29 June 2019 (UTC)

== Update Sandbox User ==

==  Posting to sandbox == 

Update Sandbox on Wikiversity [[User:Bnhassin/sandbox]] ([[User talk:Bnhassin|discuss]] • [[Special:Contributions/Bnhassin|contribs]])[[User:Bnhassin|Bnhassin]] ([[User talk:Bnhassin|discuss]] • [[Special:Contributions/Bnhassin|contribs]]) 11:04, 25 October 2020 (UTC)

== [[m:Special:MyLanguage/Tech/News/2020/50|Tech News: 2020-50]] ==

&lt;section begin=&quot;technews-2020-W50&quot;/&gt;&lt;div class=&quot;plainlinks mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2020/50|Translations]] are available.

'''Recent changes'''
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== [[m:Special:MyLanguage/Tech/News/2020/51|Tech News: 2020-51]] ==

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Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2020/51|Translations]] are available.

'''Recent changes'''
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'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.22|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2020-12-15|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2020-12-16|en}}. It will be on all wikis from {{#time:j xg|2020-12-17|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2020/51|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;/div&gt; &lt;section end=&quot;technews-2020-W51&quot;/&gt; 21:34, 14 December 2020 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=20803489 --&gt;

== [[m:Special:MyLanguage/Tech/News/2020/52|Tech News: 2020-52]] ==

&lt;section begin=&quot;technews-2020-W52&quot;/&gt;&lt;div class=&quot;plainlinks mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2020/52|Translations]] are available.

'''Tech News'''
* Because of the [[w:en:Christmas and holiday season|holidays]] the next issue of Tech News will be sent out on 11 January 2021.

'''Recent changes'''
* The &lt;code&gt;&lt;nowiki&gt;{{citation needed}}&lt;/nowiki&gt;&lt;/code&gt; template shows when a statement in a Wikipedia article needs a source. If you click on it when you edit with the visual editor there is a popup that explains this. Now it can also show the reason and when it was added. [https://phabricator.wikimedia.org/T270107]

'''Changes later this week'''
* There is no new MediaWiki version this week or next week.

'''Future changes'''
* You can [[m:WMDE Technical Wishes/Geoinformation/Ideas|propose and discuss]] what technical improvements should be done for geographic information. This could be coordinates, maps or other related things.
* Some wikis use [[mw:Writing systems/LanguageConverter|LanguageConverter]] to switch between writing systems or variants of a language. This can only be done for the entire page. There will be a &lt;code&gt;&lt;nowiki&gt;&lt;langconvert&gt;&lt;/nowiki&gt;&lt;/code&gt; tag that can convert a piece of text on a page. [https://phabricator.wikimedia.org/T263082]
*  Oversighters and stewards can hide entries in [[Special:AbuseLog|Special:AbuseLog]]. They can soon hide multiple entries at once using checkboxes. This works like hiding normal edits. It will happen in early January. [https://phabricator.wikimedia.org/T260904]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2020/52|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;/div&gt; &lt;section end=&quot;technews-2020-W52&quot;/&gt; 20:54, 21 December 2020 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=20833836 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/02|Tech News: 2021-02]] ==

&lt;section begin=&quot;technews-2021-W02&quot;/&gt;&lt;div class=&quot;plainlinks mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/02|Translations]] are available.

'''Recent changes'''
* You can choose to be reminded when you have not added an edit summary. This can be done in your preferences. This could conflict with the [[:w:en:CAPTCHA|CAPTCHA]]. This has now been fixed. [https://phabricator.wikimedia.org/T12729]
* You can link to specific log entries. You can get these links for example by clicking the timestamps in the log. Until now, such links to private log entries showed no entry even if you had permission to view private log entries. The links now show the entry. [https://phabricator.wikimedia.org/T269761]
* Admins can use the [[:mw:Special:MyLanguage/Extension:AbuseFilter|abuse filter tool]] to automatically prevent bad edits. Three changes happened last week:
** The filter editing interface now shows syntax errors while you type. This is similar to JavaScript pages. It also shows a warning for regular expressions that match the empty string. New warnings will be added later. [https://phabricator.wikimedia.org/T187686]
** [[m:Special:MyLanguage/Meta:Oversighters|Oversighters]] can now hide multiple filter log entries at once using checkboxes on [[Special:AbuseLog]]. This is how the usual revision deletion works. [https://phabricator.wikimedia.org/T260904]
** When a filter matches too many actions after it has been changed it is &quot;throttled&quot;. The most powerful actions are disabled. This is to avoid many editors getting blocked when an administrator made a mistake. The administrator will now get a notification about this &quot;throttle&quot;.
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] There is a new tool to [https://skins.wmflabs.org/?#/add build new skins]. You can also [https://skins.wmflabs.org/?#/ see] existing [[mw:Special:MyLanguage/Manual:Skins|skins]]. You can [[mw:User talk:Jdlrobson|give feedback]]. [https://lists.wikimedia.org/pipermail/wikitech-l/2020-December/094130.html]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] Bots using the API no longer watch pages automatically based on account preferences. Setting the &lt;code&gt;watchlist&lt;/code&gt; to &lt;code&gt;watch&lt;/code&gt; will still work. This is to reduce the size of the watchlist data in the database. [https://phabricator.wikimedia.org/T258108]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] [[mw:Special:MyLanguage/Extension:Scribunto|Scribunto's]] [[:mw:Extension:Scribunto/Lua reference manual#File metadata|file metadata]] now includes length. [https://phabricator.wikimedia.org/T209679]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] [[:w:en:CSS|CSS]] and [[:w:en:JavaScript|JavaScript]] code pages now have link anchors to [https://patchdemo.wmflabs.org/wikis/40e4795d4448b55a6d8c46ff414bcf78/w/index.php/MediaWiki:En.js#L-125 line numbers]. You can use wikilinks like [[:w:en:MediaWiki:Common.js#L-50]]. [https://phabricator.wikimedia.org/T29531]
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] There was a [[mw:MediaWiki 1.36/wmf.25|new version]] of MediaWiki last week. You can read [[mw:MediaWiki 1.36/wmf.25/Changelog|a detailed log]] of all 763 changes. Most of them are very small and will not affect you.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.26|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-01-12|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-01-13|en}}. It will be on all wikis from {{#time:j xg|2021-01-14|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/02|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;/div&gt; &lt;section end=&quot;technews-2021-W02&quot;/&gt; 15:42, 11 January 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=20950047 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/03|Tech News: 2021-03]] ==

&lt;section begin=&quot;technews-2021-W03&quot;/&gt;&lt;div class=&quot;plainlinks mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/03|Translations]] are available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.27|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-01-19|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-01-20|en}}. It will be on all wikis from {{#time:j xg|2021-01-21|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]).

'''Future changes'''
* The [[mw:Special:MyLanguage/Growth|Growth team]] plans to add features to [[mw:Special:MyLanguage/Growth/Personalized first day/Newcomer tasks/Experiment analysis, November 2020|get more visitors to edit]] to more Wikipedias. You can help [https://translatewiki.net/w/i.php?title=Special:Translate&amp;group=ext-growthexperiments&amp;language=&amp;filter=&amp;action=translate translating the interface].
* You will be able to read but not to edit Wikimedia Commons for a short time on [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20210126T07 {{#time:j xg|2021-01-26|en}} at 07:00 (UTC)]. [https://phabricator.wikimedia.org/T271791]
* [[m:Special:MyLanguage/MassMessage|MassMessage]] posts could be automatically timestamped in the future. This is because MassMessage senders can now send pages using MassMessage. Pages are more difficult to sign. If there are times when a MassMessage post should not be timestamped you can [[phab:T270435|let the developers know]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/03|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;/div&gt; &lt;section end=&quot;technews-2021-W03&quot;/&gt; 16:10, 18 January 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=20974628 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/04|Tech News: 2021-04]] ==

&lt;section begin=&quot;technews-2021-W04&quot;/&gt;&lt;div class=&quot;plainlinks mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/04|Translations]] are available.

'''Problems'''
* You will be able to read but not to edit Wikimedia Commons for a short time on [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20210126T07 {{#time:j xg|2021-01-26|en}} at 07:00 (UTC)]. You will not be able to read or edit [[:wikitech:Main Page|Wikitech]] for a short time on [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20210128T09 {{#time:j xg|2021-01-28|en}} at 09:00 (UTC)]. [https://phabricator.wikimedia.org/T271791][https://phabricator.wikimedia.org/T272388]

'''Changes later this week'''
* [[m:WMDE Technical Wishes/Bracket Matching|Bracket matching]] will be added to the [[mw:Special:MyLanguage/Extension:CodeMirror|CodeMirror]] syntax highlighter on the first wikis. The first wikis are German and Catalan Wikipedia and maybe other Wikimedia wikis. This will happen on 27 January. [https://phabricator.wikimedia.org/T270238]
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.28|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-01-26|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-01-27|en}}. It will be on all wikis from {{#time:j xg|2021-01-28|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/04|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;/div&gt; &lt;section end=&quot;technews-2021-W04&quot;/&gt; 18:31, 25 January 2021 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2021/05|Tech News: 2021-05]] ==

&lt;section begin=&quot;technews-2021-W05&quot;/&gt;&lt;div class=&quot;plainlinks mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/05|Translations]] are available.

'''Problems'''
* [[:w:en:IPv6|IPv6 addresses]] were written in lowercase letters in diffs. This caused dead links since [[Special:Contributions|Special:Contributions]] only accepted uppercase letters for the IPs. This has been fixed. [https://phabricator.wikimedia.org/T272225]

'''Changes later this week'''
* You can soon use Wikidata to link to pages on the multilingual Wikisource. [https://phabricator.wikimedia.org/T138332]
* Often editors use a &quot;non-breaking space&quot; to make a gap between two items when reading but still show them together. This can be used to avoid a line break. You will now be able to add new ones via the special character tool in the 2010, 2017, and visual editors. The character will be shown in the visual editor as a space with a grey background. [https://phabricator.wikimedia.org/T70429][https://phabricator.wikimedia.org/T96666]
* [[File:Octicons-tools.svg|15px|link=| Advanced item]] Wikis use [[mw:Special:MyLanguage/Extension:AbuseFilter|abuse filters]] to stop bad edits being made. Filter maintainers can now use syntax like &lt;code&gt;1.2.3.4 - 1.2.3.55&lt;/code&gt; as well as the &lt;code&gt;1.2.3.4/27&lt;/code&gt; syntax for IP ranges. [https://phabricator.wikimedia.org/T218074]
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.29|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-02-02|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-02-03|en}}. It will be on all wikis from {{#time:j xg|2021-02-04|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]).

'''Future changes'''
* [[mw:Skin:Minerva Neue|Minerva]] is the skin Wikimedia wikis use for mobile traffic. When a page is protected and you can't edit it you can normally read the source wikicode. This doesn't work on Minerva on mobile devices. This is being fixed. Some text might overlap. This is because your community needs to update [[MediaWiki:Protectedpagetext|MediaWiki:Protectedpagetext]] to work on mobile. You can [[phab:T208827|read more]]. [https://www.mediawiki.org/wiki/Recommendations_for_mobile_friendly_articles_on_Wikimedia_wikis#Inline_styles_should_not_use_properties_that_impact_sizing_and_positioning][https://www.mediawiki.org/wiki/Recommendations_for_mobile_friendly_articles_on_Wikimedia_wikis#Avoid_tables_for_anything_except_data]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] [[:wikitech:Portal:Cloud VPS|Cloud VPS]] and [[:wikitech:Portal:Toolforge|Toolforge]] will change the IP address they use to contact the wikis. The new IP address will be &lt;code&gt;185.15.56.1&lt;/code&gt;. This will happen on February 8. You can [[:wikitech:News/CloudVPS NAT wikis|read more]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/05|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;/div&gt; &lt;section end=&quot;technews-2021-W05&quot;/&gt; 22:38, 1 February 2021 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2021/06|Tech News: 2021-06]] ==

&lt;section begin=&quot;technews-2021-W06&quot;/&gt;&lt;div class=&quot;plainlinks mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/06|Translations]] are available.

'''Recent changes'''
* The [[mw:Special:MyLanguage/Wikimedia Apps|Wikipedia app]] for Android now has watchlists and talk pages in the app. [https://play.google.com/store/apps/details?id=org.wikipedia]

'''Changes later this week'''
* You can see edits to chosen pages on [[Special:Watchlist|Special:Watchlist]]. You can add pages to your watchlist on every wiki you like. The [[:mw:Special:MyLanguage/Extension:GlobalWatchlist|GlobalWatchlist]] extension will come to Meta on 11 February. There you can see entries on watched pages on different wikis on the same page. The new watchlist will be found on [[m:Special:GlobalWatchlist|Special:GlobalWatchlist]] on Meta. You can choose which wikis to watch and other preferences on [[m:Special:GlobalWatchlistSettings|Special:GlobalWatchlistSettings]] on Meta. You can watch up to five wikis. [https://phabricator.wikimedia.org/T260862]
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.30|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-02-09|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-02-10|en}}. It will be on all wikis from {{#time:j xg|2021-02-11|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]).

'''Future changes'''
* When admins [[mw:Special:MyLanguage/Help:Protecting and unprotecting pages|protect]] pages the form will use the [[mw:UX standardization|OOUI look]]. [[Special:Import|Special:Import]] will also get the new look. This will make them easier to use on mobile phones. [https://phabricator.wikimedia.org/T235424][https://phabricator.wikimedia.org/T108792]
* Some services will not work for a short period of time from 07:00 UTC on 17 February. There might be problems with new [[m:Special:MyLanguage/Wikimedia URL Shortener|short links]], new translations, new notifications, adding new items to your [[mw:Reading/Reading Lists|reading lists]] or recording [[:w:en:Email#Tracking of sent mail|email bounces]]. This is because of database maintenance. [https://phabricator.wikimedia.org/T273758]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] [[m:Tech/News/2021/05|Last week]] Tech News reported that the IP address [[:wikitech:Portal:Cloud VPS|Cloud VPS]] and [[:wikitech:Portal:Toolforge|Toolforge]] use to contact the wikis will change on 8 February. This is delayed. It will happen later instead. [https://wikitech.wikimedia.org/wiki/News/CloudVPS_NAT_wikis]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/06|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;/div&gt; &lt;section end=&quot;technews-2021-W06&quot;/&gt; 17:42, 8 February 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21082948 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/07|Tech News: 2021-07]] ==

&lt;section begin=&quot;technews-2021-W07&quot;/&gt;&lt;div class=&quot;plainlinks mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/07|Translations]] are available.

'''Problems'''
* There were problems with recent versions of MediaWiki.  Because the updates caused problems the developers rolled back to an earlier version. Some updates and new functions will come later than planned. [https://lists.wikimedia.org/pipermail/wikitech-l/2021-February/094255.html][https://lists.wikimedia.org/pipermail/wikitech-l/2021-February/094271.html]
* Some services will not work for a short period of time from 07:00 UTC on 17 February. There might be problems with new [[m:Special:MyLanguage/Wikimedia URL Shortener|short links]], new translations, new notifications, adding new items to your [[mw:Reading/Reading Lists|reading lists]] or recording [[:w:en:Email#Tracking of sent mail|email bounces]]. This is because of database maintenance. [https://phabricator.wikimedia.org/T273758]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.31|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-02-16|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-02-17|en}}. It will be on all wikis from {{#time:j xg|2021-02-18|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/07|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;/div&gt; &lt;section end=&quot;technews-2021-W07&quot;/&gt; 17:56, 15 February 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21105437 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/08|Tech News: 2021-08]] ==

&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/08|Translations]] are available.

'''Recent changes'''
* The visual editor will now use [[:c:Commons:Structured data/Media search|MediaSearch]] to find images. You can search for images on Commons in the visual editor when you are looking for illustrations. This is to help editors find better images. [https://phabricator.wikimedia.org/T259896]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] The [[mw:Special:MyLanguage/Extension:SyntaxHighlight|syntax highlighter]] now works with more languages: [[:w:en:Futhark (programming language)|Futhark]], [[:w:en:Graphviz|Graphviz]]/[[:w:en:DOT (graph description language)|DOT]], CDDL and AMDGPU. [https://phabricator.wikimedia.org/T274741]

'''Problems'''
* Editing a [[mw:Special:MyLanguage/Extension:EasyTimeline|timeline]] might have removed all text from it. This was because of a bug and has been fixed. You might need to edit the timeline again for it to show properly. [https://phabricator.wikimedia.org/T274822]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.32|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-02-23|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-02-24|en}}. It will be on all wikis from {{#time:j xg|2021-02-25|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]).

'''Future changes'''
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] There is a [[:m:Wikimedia Rust developers user group|user group]] for developers and users interested in working on Wikimedia wikis with the [[:w:en:Rust (programming language)|Rust programming language]]. You can join or tell others who want to make your wiki better in the future.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/08|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;

----

00:17, 23 February 2021 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2021/09|Tech News: 2021-09]] ==

&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/09|Translations]] are available.

'''Recent changes'''
*  Wikis using the [[mw:Special:MyLanguage/Growth/Feature summary|Growth team tools]] can now show the name of a newcomer's mentor anywhere [[mw:Special:MyLanguage/Help:Growth/Mentorship/Integrating_mentorship|through a magic word]]. This can be used for welcome messages or userboxes.
* A new version of the [[c:Special:MyLanguage/Commons:VideoCutTool|VideoCutTool]] is now available. It enables cropping, trimming, audio disabling, and rotating video content. It is being created as part of the developer outreach programs.

'''Problems'''
* There was a problem with the [[mw:Special:MyLanguage/Manual:Job queue|job queue]]. This meant some functions did not save changes and mass messages were delayed. This did not affect wiki edits. [https://phabricator.wikimedia.org/T275437]
* Some editors may not be logged in to their accounts automatically in the latest versions of Firefox and Safari. [https://phabricator.wikimedia.org/T226797]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.33|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-03-02|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-03-03|en}}. It will be on all wikis from {{#time:j xg|2021-03-04|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/09|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;

----

19:08, 1 March 2021 (UTC)
&lt;!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21161722 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/10|Tech News: 2021-10]] ==

&lt;section begin=&quot;technews-2021-W10&quot;/&gt;&lt;div class=&quot;plainlinks mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/10|Translations]] are available.

'''Recent changes'''
* [[mw:Special:MyLanguage/Content translation/Section translation|Section translation]] now works on Bengali Wikipedia. It helps mobile editors translate sections of articles. It will come to more wikis later. The first focus is active wikis with a smaller number of articles. You can [https://sx.wmflabs.org/index.php/Main_Page test it] and [[mw:Talk:Content translation/Section translation|leave feedback]].
* [[mw:Special:MyLanguage/Help:Extension:FlaggedRevs|Flagged revisions]] now give admins the review right. [https://phabricator.wikimedia.org/T275293]
* When someone links to a Wikipedia article on Twitter this will now show a preview of the article. [https://phabricator.wikimedia.org/T276185]

'''Problems'''
* Many graphs have [[:w:en:JavaScript|JavaScript]] errors. Graph editors can check their graphs in their browser's developer console after editing. [https://phabricator.wikimedia.org/T275833]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.34|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-03-09|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-03-10|en}}. It will be on all wikis from {{#time:j xg|2021-03-11|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]).
* The [[mw:Talk pages project/New discussion|New Discussion]] tool will soon be a new [[mw:Special:MyLanguage/Extension:DiscussionTools|discussion tools]] beta feature for on most Wikipedias. The goal is to make it easier to start new discussions. [https://phabricator.wikimedia.org/T275257]

'''Future changes'''
* There will be a number of changes to make it easier to work with templates. Some will come to the first wikis in March. Other changes will come to the first wikis in June. This is both for those who use templates and those who create or maintain them. You can [[:m:WMDE Technical Wishes/Templates|read more]].
* [[m:WMDE Technical Wishes/ReferencePreviews|Reference Previews]] will become a default feature on some wikis on 17 March. They will share a setting with [[mw:Page Previews|Page Previews]]. If you prefer the Reference Tooltips or Navigation-Popups gadget you can keep using them. If so Reference Previews won't be shown. [https://phabricator.wikimedia.org/T271206][https://meta.wikimedia.org/wiki/Talk:WMDE_Technical_Wishes/ReferencePreviews]
* New JavaScript-based functions will not work in [[:w:en:Internet Explorer 11|Internet Explorer 11]]. This is because Internet Explorer is an old browser that doesn't work with how JavaScript is written today. Everything that works in Internet Explorer 11 today will continue working in Internet Explorer for now. You can [[mw:Compatibility/IE11|read more]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/10|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;/div&gt; &lt;section end=&quot;technews-2021-W10&quot;/&gt; 17:51, 8 March 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21175593 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/11|Tech News: 2021-11]] ==

&lt;section begin=&quot;technews-2021-W11&quot;/&gt;&lt;div class=&quot;plainlinks mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/11|Translations]] are available.

'''Recent changes'''
* Wikis that are part of the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|desktop improvements]] project can now use a new [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Features/Search|search function]]. The desktop improvements and the new search will come to more wikis later. You can also [[mw:Reading/Web/Desktop Improvements#Deployment plan and timeline|test it early]].
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] Editors who put up banners or change site-wide [[:w:en:JavaScript|JavaScript]] code should use the [https://grafana.wikimedia.org/d/000000566/overview?viewPanel=16&amp;orgId=1 client error graph] to see that their changes has not caused problems. You can [https://diff.wikimedia.org/2021/03/08/sailing-steady%e2%80%8a-%e2%80%8ahow-you-can-help-keep-wikimedia-sites-error-free read more]. [https://phabricator.wikimedia.org/T276296]

'''Problems'''
* Due to [[phab:T276968|database issues]] the [https://meta.wikimedia.beta.wmflabs.org Wikimedia Beta Cluster] was read-only for over a day.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.34|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-03-16|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-03-17|en}}. It will be on all wikis from {{#time:j xg|2021-03-18|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]).

'''Future changes'''
* You can add a [[:w:en:Newline|newline]] or [[:w:en:Carriage return|carriage return]] character to a custom signature if you use a template. There is a proposal to not allow them in the future. This is because they can cause formatting problems. [https://www.mediawiki.org/wiki/New_requirements_for_user_signatures#Additional_proposal_(2021)][https://phabricator.wikimedia.org/T272322]
* You will be able to read but not edit [[phab:T276899|12 wikis]] for a short period of time on [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20210323T06 {{#time:j xg|2021-03-23|en}} at 06:00 (UTC)]. This could take 30 minutes but will probably be much faster.
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] You can use [https://quarry.wmflabs.org/ Quarry] for [[:w:en:SQL|SQL]] queries to the [[wikitech:Wiki replicas|Wiki Replicas]]. Cross-database &lt;code&gt;JOINS&lt;/code&gt; will no longer work from 23 March. There will be a new field to specify the database to connect to. If you think this affects you and you need help you can [[phab:T268498|post on Phabricator]] or on [[wikitech:Talk:News/Wiki Replicas 2020 Redesign|Wikitech]]. [https://wikitech.wikimedia.org/wiki/PAWS PAWS] and other ways to do [[:w:en:SQL|SQL]] queries to the Wiki Replicas will be affected later. [https://wikitech.wikimedia.org/wiki/News/Wiki_Replicas_2020_Redesign]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/11|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;/div&gt; &lt;section end=&quot;technews-2021-W11&quot;/&gt; 23:22, 15 March 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21226057 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/12|Tech News: 2021-12]] ==

&lt;section begin=&quot;technews-2021-W12&quot;/&gt;&lt;div class=&quot;plainlinks mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/12|Translations]] are available.

'''Recent changes'''
* There is a [[mw:Wikipedia for KaiOS|Wikipedia app]] for [[:w:en:KaiOS|KaiOS]] phones. They don't have a touch screen so readers navigate with the phone keys. There is now a [https://wikimedia.github.io/wikipedia-kaios/sim.html simulator] so you can see what it looks like.
* The [[mw:Special:MyLanguage/Talk pages project/Replying|reply tool]] and [[mw:Special:MyLanguage/Talk pages project/New discussion|new discussion tool]] are now available as the &quot;{{int:discussiontools-preference-label}}&quot; [[Special:Preferences#mw-prefsection-betafeatures|beta feature]] in almost all wikis except German Wikipedia.

'''Problems'''
* You will be able to read but not edit [[phab:T276899|twelve wikis]] for a short period of time on [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20210323T06 {{#time:j xg|2021-03-23|{{PAGELANGUAGE}}}} at 06:00 (UTC)]. This can also affect password changes, logging in to new wikis, global renames and changing or confirming emails. This could take 30 minutes but will probably be much faster.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.36|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-03-23|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-03-24|en}}. It will be on all wikis from {{#time:j xg|2021-03-25|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]).
* [[:w:en:Syntax highlighting|Syntax highlighting]] colours will change to be easier to read. This will soon come to the [[phab:T276346|first wikis]]. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Improved_Color_Scheme_of_Syntax_Highlighting]

'''Future changes'''
* [[mw:Special:MyLanguage/Extension:FlaggedRevs|Flagged revisions]] will no longer have multiple tags like &quot;tone&quot; or &quot;depth&quot;. It will also only have one tier. This was changed because very few wikis used these features and they make the tool difficult to maintain. [https://phabricator.wikimedia.org/T185664][https://phabricator.wikimedia.org/T277883]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] Gadgets and user scripts can access variables about the current page in JavaScript. In 2015 this was moved from &lt;code dir=ltr&gt;wg*&lt;/code&gt; to &lt;code dir=ltr&gt;mw.config&lt;/code&gt;. &lt;code dir=ltr&gt;wg*&lt;/code&gt; will soon no longer work. [https://phabricator.wikimedia.org/T72470]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]] • [[m:Special:MyLanguage/Tech/News#contribute|Contribute]] • [[m:Special:MyLanguage/Tech/News/2021/12|Translate]] • [[m:Tech|Get help]] • [[m:Talk:Tech/News|Give feedback]] • [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;/div&gt; &lt;section end=&quot;technews-2021-W12&quot;/&gt; 16:53, 22 March 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21244806 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/13|Tech News: 2021-13]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/13|Translations]] are available.

'''Recent changes'''
* Some very old [[:w:en:Web browser|web browsers]] [[:mw:Special:MyLanguage/Compatibility|don’t work]] well with the Wikimedia wikis. Some old code for browsers that used to be supported is being removed. This could cause issues in those browsers. [https://phabricator.wikimedia.org/T277803]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] [[:m:IRC/Channels#Raw_feeds|IRC recent changes feeds]] have been moved to a new server. Make sure all tools automatically reconnect to &lt;code&gt;irc.wikimedia.org&lt;/code&gt; and not to the name of any specific server. Users should also consider switching to the more modern [[:wikitech:Event Platform/EventStreams|EventStreams]]. [https://phabricator.wikimedia.org/T224579]

'''Problems'''
* When you move a page that many editors have on their watchlist the history can be split. It might also not be possible to move it again for a while. This is because of a [[:w:en:Job queue|job queue]] problem. [https://phabricator.wikimedia.org/T278350]
* Some translatable pages on Meta could not be edited. This was because of a bug in the translation tool. The new MediaWiki version was delayed because of problems like this. [https://phabricator.wikimedia.org/T278429][https://phabricator.wikimedia.org/T274940]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.37|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-03-30|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-03-31|en}}. It will be on all wikis from {{#time:j xg|2021-04-01|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/13|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

17:30, 29 March 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21267131 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/14|Tech News: 2021-14]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/14|Translations]] are available.

'''Recent changes'''
* Editors can collapse part of an article so you have to click on it to see it. When you click a link to a section inside collapsed content it will now expand to show the section. The browser will scroll down to the section. Previously such links didn't work unless you manually expanded the content first. [https://phabricator.wikimedia.org/T276741]

'''Changes later this week'''
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] The [[mw:Special:MyLanguage/Citoid|citoid]] [[:w:en:API|API]] will use for example &lt;code&gt;2010-12-XX&lt;/code&gt; instead of &lt;code&gt;2010-12&lt;/code&gt; for dates with a month but no days. This is because &lt;code&gt;2010-12&lt;/code&gt; could be confused with &lt;code&gt;2010-2012&lt;/code&gt; instead of &lt;code&gt;December 2010&lt;/code&gt;. This is called level 1 instead of level 0 in the [https://www.loc.gov/standards/datetime/ Extended Date/Time Format]. [https://phabricator.wikimedia.org/T132308]
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.36/wmf.38|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-04-06|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-04-07|en}}. It will be on all wikis from {{#time:j xg|2021-04-08|en}} ([[mw:MediaWiki 1.36/Roadmap|calendar]]).

'''Future changes'''
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] [[:wikitech:PAWS|PAWS]] can now connect to the new [[:wikitech:Wiki Replicas|Wiki Replicas]]. Cross-database &lt;code&gt;JOINS&lt;/code&gt; will no longer work from 28 April. There is [[:wikitech:News/Wiki Replicas 2020 Redesign#How should I connect to databases in PAWS?|a new way to connect]] to the databases. Until 28 April both ways to connect to the databases will work. If you think this affects you and you need help you can post [[phab:T268498|on Phabricator]] or on [[wikitech:Talk:News/Wiki Replicas 2020 Redesign|Wikitech]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/14|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

19:41, 5 April 2021 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21287348 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/16|Tech News: 2021-16]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/16|Translations]] are available.

'''Recent changes'''
* Email to the Wikimedia wikis are handled by groups of Wikimedia editors. These volunteer response teams now use [https://github.com/znuny/Znuny Znuny] instead of [[m:Special:MyLanguage/OTRS|OTRS]]. The functions and interface remain the same. The volunteer administrators will give more details about the next steps soon. [https://phabricator.wikimedia.org/T279303][https://phabricator.wikimedia.org/T275294]
* If you use [[Mw:Special:MyLanguage/Extension:CodeMirror|syntax highlighting]], you can see line numbers in the 2010 and 2017 wikitext editors when editing templates. This is to make it easier to see line breaks or talk about specific lines. Line numbers will soon come to all namespaces. [https://phabricator.wikimedia.org/T267911][https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Line_Numbering][https://meta.wikimedia.org/wiki/Talk:WMDE_Technical_Wishes/Line_Numbering]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] Because of a technical change there could be problems with gadgets and scripts that have an edit summary area that looks [https://phab.wmfusercontent.org/file/data/llvdqqnb5zpsfzylbqcg/PHID-FILE-25vs4qowibmtysl7cbml/Screen_Shot_2021-04-06_at_2.34.04_PM.png similar to this one]. If they look strange they should use &lt;code&gt;mw.loader.using('mediawiki.action.edit.styles')&lt;/code&gt; to go back to how they looked before. [https://phabricator.wikimedia.org/T278898]
* The [[mw:MediaWiki 1.37/wmf.1|latest version]] of MediaWiki came to the Wikimedia wikis last week. There was no Tech News issue last week.

'''Changes later this week'''
*  There is no new MediaWiki version this week.

'''Future changes'''
* The user group &lt;code&gt;oversight&lt;/code&gt; will be renamed &lt;code&gt;suppress&lt;/code&gt;. This is for [[phab:T109327|technical reasons]]. This is the technical name. It doesn't affect what you call the editors with this user right on your wiki. This is planned to happen in two weeks. You can comment [[phab:T112147|in Phabricator]] if you have objections.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/16|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

16:48, 19 April 2021 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2021/17|Tech News: 2021-17]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/17|Translations]] are available.

'''Recent changes'''
* Templates have parameters that can have specific values. It is possible to suggest values for editors with [[mw:Special:MyLanguage/Extension:TemplateData|TemplateData]]. You can soon see them as a drop-down list in the visual editor. This is to help template users find the right values faster. [https://phabricator.wikimedia.org/T273857][https://meta.wikimedia.org/wiki/Special:MyLanguage/WMDE_Technical_Wishes/Suggested_values_for_template_parameters][https://meta.wikimedia.org/wiki/Talk:WMDE_Technical_Wishes/Suggested_values_for_template_parameters]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.3|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-04-27|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-04-28|en}}. It will be on all wikis from {{#time:j xg|2021-04-29|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/17|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

21:24, 26 April 2021 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2021/18|Tech News: 2021-18]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/18|Translations]] are available.

'''Recent changes'''
* [[w:en:Wikipedia:Twinkle|Twinkle]] is a gadget on English Wikipedia. It can help with maintenance and patrolling. It can [[m:Grants:Project/Rapid/SD0001/Twinkle localisation/Report|now be used on other wikis]]. You can get Twinkle on your wiki using the [https://github.com/wikimedia-gadgets/twinkle-starter twinkle-starter] GitHub repository.

'''Problems'''
* The [[mw:Special:MyLanguage/Content translation|content translation tool]] did not work for many articles for a little while. This was because of a bug. [https://phabricator.wikimedia.org/T281346]
* Some things will not work for about a minute on 5 May. This will happen [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20210505T0600 around 06:00 UTC]. This will affect the content translation tool and notifications among other things. This is because of an upgrade to avoid crashes. [https://phabricator.wikimedia.org/T281212]

'''Changes later this week'''
* [[mw:Special:MyLanguage/Help:Reference Previews|Reference Previews]] will become a default feature on a number of wikis on 5 May. This is later than planned because of some changes. You can use it without using [[mw:Special:MyLanguage/Page Previews|Page Previews]] if you want to. The earlier plan was to have the preference to use both or none. [https://phabricator.wikimedia.org/T271206][https://meta.wikimedia.org/wiki/Talk:WMDE_Technical_Wishes/ReferencePreviews]
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.4|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-05-04|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-05-05|en}}. It will be on all wikis from {{#time:j xg|2021-05-06|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]).

'''Future changes'''
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] The [[:w:en:CSS|CSS]] classes &lt;code dir=ltr&gt;.error&lt;/code&gt;, &lt;code dir=ltr&gt;.warning&lt;/code&gt; and &lt;code dir=ltr&gt;.success&lt;/code&gt; do not work for mobile readers if they have not been specifically defined on your wiki. From June they will not work for desktop readers. This can affect gadgets and templates. The classes can be defined in [[MediaWiki:Common.css]] or template styles instead. [https://phabricator.wikimedia.org/T280766]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/18|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

15:43, 3 May 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21418010 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/19|Tech News: 2021-19]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/19|Translations]] are available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.5|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-05-11|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-05-12|en}}. It will be on all wikis from {{#time:j xg|2021-05-13|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]).

'''Future changes'''
* You can see what participants plan to work on at the online [[mw:Wikimedia Hackathon 2021|Wikimedia hackathon]] 22–23 May.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/19|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

15:10, 10 May 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21428676 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/20|Tech News: 2021-20]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/20|Translations]] are available.

'''Recent changes'''
* There is a new toolbar in [[mw:Talk pages project/Replying|the Reply tool]]. It works in the wikitext source mode. You can enable it in [[Special:Preferences#mw-htmlform-discussion|your preferences]]. [https://phabricator.wikimedia.org/T276608] [https://www.mediawiki.org/wiki/Talk_pages_project/Replying#13_May_2021] [https://www.mediawiki.org/wiki/Talk_pages_project/New_discussion#13_May_2021]
* Wikimedia [https://lists.wikimedia.org/mailman/listinfo mailing lists] are being moved to [[:w:en:GNU Mailman|Mailman 3]]. This is a newer version. For the [[:w:en:Character encoding|character encoding]] to work it will change from &lt;code&gt;[[:w:en:UTF-8|UTF-8]]&lt;/code&gt; to &lt;code&gt;utf8mb3&lt;/code&gt;. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/IEYQ2HS3LZF2P3DAYMNZYQDGHWPVMTPY/][https://phabricator.wikimedia.org/T282621]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] An [[m:Special:MyLanguage/Tech/News/2021/14|earlier issue]] of Tech News said that the  [[mw:Special:MyLanguage/Citoid|citoid]] [[:w:en:API|API]] would handle dates with a month but no days in a new way. This has been reverted for now. There needs to be more discussion of how it affects different wikis first. [https://phabricator.wikimedia.org/T132308]

'''Changes later this week'''
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] &lt;code&gt;MediaWiki:Pageimages-blacklist&lt;/code&gt; will be renamed &lt;code&gt;MediaWiki:Pageimages-denylist&lt;/code&gt;. The list can be copied to the new name. It will happen on 19 May for some wikis and 20 May for some wikis. Most wikis don't use it. It lists images that should never be used as thumbnails for articles. [https://phabricator.wikimedia.org/T282626]
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.6|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-05-18|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-05-19|en}}. It will be on all wikis from {{#time:j xg|2021-05-20|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/20|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

13:49, 17 May 2021 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2021/21|Tech News: 2021-21]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/21|Translations]] are available.

'''Recent changes'''
* The Wikimedia movement has been using [[:m:Special:MyLanguage/IRC|IRC]] on a network called [[:w:en:Freenode|Freenode]]. There have been changes around who is in control of the network. The [[m:Special:MyLanguage/IRC/Group_Contacts|Wikimedia IRC Group Contacts]] have [[m:Special:Diff/21476411|decided]] to move to the new [[:w:en:Libera Chat|Libera Chat]] network instead. This is not a formal decision for the movement to move all channels but most Wikimedia IRC channels will probably leave Freenode. There is a [[:m:IRC/Migrating_to_Libera_Chat|migration guide]] and ongoing Wikimedia [[m:Wikimedia Forum#Freenode (IRC)|discussions about this]].

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.7|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-05-25|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-05-26|en}}. It will be on all wikis from {{#time:j xg|2021-05-27|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/21|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

17:07, 24 May 2021 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2021/22|Tech News: 2021-22]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/22|Translations]] are available.

'''Problems'''
* There was an issue on the Vector skin with the text size of categories and notices under the page title. It was fixed last Monday. [https://phabricator.wikimedia.org/T283206]

'''Changes later this week'''
* There is no new MediaWiki version this week.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/22|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

17:05, 31 May 2021 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2021/23|Tech News: 2021-23]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/23|Translations]] are available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.9|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-06-08|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-06-09|en}}. It will be on all wikis from {{#time:j xg|2021-06-10|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]).

'''Future changes'''
* The Wikimedia movement uses [[:mw:Special:MyLanguage/Phabricator|Phabricator]] for technical tasks. This is where we collect technical suggestions, bugs and what developers are working on. The company behind Phabricator will stop working on it. This will not change anything for the Wikimedia movement now. It could lead to changes in the future. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/message/YAXOD46INJLAODYYIJUVQWOZFIV54VUI/][https://admin.phacility.com/phame/post/view/11/phacility_is_winding_down_operations/][https://phabricator.wikimedia.org/T283980]
* Searching on Wikipedia will find more results in some languages. This is mainly true for when those who search do not use the correct [[:w:en:Diacritic|diacritics]] because they are not seen as necessary in that language. For example searching for &lt;code&gt;Bedusz&lt;/code&gt; doesn't find &lt;code&gt;Będusz&lt;/code&gt; on German Wikipedia. The character &lt;code&gt;ę&lt;/code&gt; isn't used in German so many would write &lt;code&gt;e&lt;/code&gt; instead. This will work better in the future in some languages. [https://phabricator.wikimedia.org/T219550]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] The [[:w:en:Cross-site request forgery|CSRF token parameters]] in the [[:mw:Special:MyLanguage/API:Main page|action API]] were changed in 2014. The old parameters from before 2014 will stop working soon. This can affect bots, gadgets and user scripts that still use the old parameters. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/IMP43BNCI32C524O5YCUWMQYP4WVBQ2B/][https://phabricator.wikimedia.org/T280806]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/23|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

20:02, 7 June 2021 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2021/24|Tech News: 2021-24]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/24|Translations]] are available.

'''Recent changes'''
* Logged-in users on the mobile web can choose to use the [[:mw:Special:MyLanguage/Reading/Web/Advanced mobile contributions|advanced mobile mode]]. They now see categories in a similar way as users on desktop do. This means that some gadgets that have just been for desktop users could work for users of the mobile site too. If your wiki has such gadgets you could decide to turn them on for the mobile site too. Some gadgets probably need to be fixed to look good on mobile. [https://phabricator.wikimedia.org/T284763]
* Language links on Wikidata now works for [[:oldwikisource:Main Page|multilingual Wikisource]]. [https://phabricator.wikimedia.org/T275958]

'''Changes later this week'''
* There is no new MediaWiki version this week.

'''Future changes'''
* In the future we [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation|can't show the IP]] of unregistered editors to everyone. This is because privacy regulations and norms have changed. There is now a rough draft of how [[m:IP Editing: Privacy Enhancement and Abuse Mitigation#Updates|showing the IP to those who need to see it]] could work.
* German Wikipedia, English Wikivoyage and 29 smaller wikis will be read-only for a few minutes on 22 June. This is planned between 5:00 and 5:30 UTC. [https://phabricator.wikimedia.org/T284530]
* All wikis will be read-only for a few minutes in the week of 28 June. More information will be published in Tech News later. It will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T281515][https://phabricator.wikimedia.org/T281209]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/24|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

20:26, 14 June 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21587625 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/25|Tech News: 2021-25]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/25|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] The &lt;code&gt;otrs-member&lt;/code&gt; group name is now &lt;code&gt;vrt-permissions&lt;/code&gt;. This could affect abuse filters. [https://phabricator.wikimedia.org/T280615]

'''Problems'''
* You will be able to read but not edit German Wikipedia, English Wikivoyage and 29 smaller wikis for a few minutes on 22 June. This is planned between [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20210623T0500 5:00 and 5:30 UTC]. [https://phabricator.wikimedia.org/T284530]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.11|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-06-22|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-06-23|en}}. It will be on all wikis from {{#time:j xg|2021-06-24|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/25|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

15:49, 21 June 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21593987 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/26|Tech News: 2021-26]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/26|Translations]] are available.

'''Recent changes'''
* Wikis with the [[mw:Special:MyLanguage/Growth|Growth features]] now can [[mw:Special:MyLanguage/Growth/Community configuration|configure Growth features directly on their wiki]]. This uses the new special page &lt;code&gt;Special:EditGrowthConfig&lt;/code&gt;. [https://phabricator.wikimedia.org/T285423]
* Wikisources have a new [[m:Special:MyLanguage/Community Tech/OCR Improvements|OCR tool]]. If you don't want to see the &quot;extract text&quot; button on Wikisource you can add &lt;code&gt;.ext-wikisource-ExtractTextWidget { display: none; }&lt;/code&gt; to your [[Special:MyPage/common.css|common.css page]]. [https://phabricator.wikimedia.org/T285311]

'''Problems'''
*You will be able to read but not edit the Wikimedia wikis for a few minutes on 29 June. This is planned at [https://www.timeanddate.com/worldclock/fixedtime.html?iso=20210629T1400 14:00 UTC]. [https://phabricator.wikimedia.org/T281515][https://phabricator.wikimedia.org/T281209]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.12|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-06-29|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-06-30|en}}. It will be on all wikis from {{#time:j xg|2021-07-01|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]).

'''Future changes'''
* &lt;code&gt;Threshold for stub link formatting&lt;/code&gt;, &lt;code&gt;thumbnail size&lt;/code&gt; and &lt;code&gt;auto-number headings&lt;/code&gt; can be set in preferences. They are expensive to maintain and few editors use them. The developers are planning to remove them. Removing them will make pages load faster. You can [[mw:Special:MyLanguage/User:SKim (WMF)/Performance Dependent User Preferences|read more and give feedback]].
* A toolbar will be added to the [[mw:Talk pages project/Replying|Reply tool]]'s wikitext source mode. This will make it easier to link to pages and to ping other users. [https://phabricator.wikimedia.org/T276609][https://www.mediawiki.org/wiki/Talk_pages_project/Replying#Status_updates]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/26|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

16:32, 28 June 2021 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2021/27|Tech News: 2021-27]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/27|Translations]] are available.

'''Tech News'''
* The next issue of Tech News will be sent out on 19 July.

'''Recent changes'''
* [[:wikidata:Q4063270|AutoWikiBrowser]] is a tool to make repetitive tasks easier. It now uses [[:w:en:JSON|JSON]]. &lt;code&gt;Wikipedia:AutoWikiBrowser/CheckPage&lt;/code&gt; has moved to &lt;code&gt;Wikipedia:AutoWikiBrowser/CheckPageJSON&lt;/code&gt; and &lt;code&gt;Wikipedia:AutoWikiBrowser/Config&lt;/code&gt;. &lt;code&gt;Wikipedia:AutoWikiBrowser/CheckPage/Version&lt;/code&gt; has moved to &lt;code&gt;Wikipedia:AutoWikiBrowser/CheckPage/VersionJSON&lt;/code&gt;. The tool will eventually be configured on the wiki so that you don't have to wait until the new version to add templates or regular expression fixes. [https://phabricator.wikimedia.org/T241196]

'''Problems'''
* [[m:Special:MyLanguage/InternetArchiveBot|InternetArchiveBot]] helps saving online sources on some wikis. It adds them to [[:w:en:Wayback Machine|Wayback Machine]] and links to them there. This is so they don't disappear if the page that was linked to is removed. It currently has a problem with linking to the wrong date when it moves pages from &lt;code&gt;archive.is&lt;/code&gt; to &lt;code&gt;web.archive.org&lt;/code&gt;. [https://phabricator.wikimedia.org/T283432]

'''Changes later this week'''
* The tool to [[m:WMDE Technical Wishes/Finding and inserting templates|find, add and remove templates]] will be updated. This is to make it easier to find and use the right templates. It will come to the first wikis on 7 July. It will come to more wikis later this year. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Removing_a_template_from_a_page_using_the_VisualEditor][https://phabricator.wikimedia.org/T284553]
* There is no new MediaWiki version this week.

'''Future changes'''
* Some Wikimedia wikis use [[m:Special:MyLanguage/Flagged Revisions|Flagged Revisions]] or pending changes. It hides edits from new and unregistered accounts for readers until they have been patrolled. The auto review action in Flagged Revisions will no longer be logged. All old logs of auto-review will be removed. This is because it creates a lot of logs that are not very useful. [https://phabricator.wikimedia.org/T285608]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/27|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

17:33, 5 July 2021 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21694636 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/29|Tech News: 2021-29]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/29|Translations]] are available.

'''Recent changes'''
* The tool to [[m:WMDE Technical Wishes/Finding and inserting templates|find, add and remove templates]] was updated. This is to make it easier to find and use the right templates. It was supposed to come to the first wikis on 7 July. It was delayed to 12 July instead. It will come to more wikis later this year. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Removing_a_template_from_a_page_using_the_VisualEditor][https://phabricator.wikimedia.org/T284553]
* [[Special:UnconnectedPages|Special:UnconnectedPages]] lists pages that are not connected to Wikidata. This helps you find pages that can be connected to Wikidata items. Some pages should not be connected to Wikidata. You can use the magic word &lt;code&gt;&lt;nowiki&gt;__EXPECTED_UNCONNECTED_PAGE__&lt;/nowiki&gt;&lt;/code&gt; on pages that should not be listed on the special page. [https://phabricator.wikimedia.org/T97577]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.15|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-07-20|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-07-21|en}}. It will be on all wikis from {{#time:j xg|2021-07-22|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]).

'''Future changes'''
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] How media is structured in the [[:w:en:Parsing|parser's]] HTML output will soon change. This can affect bots, gadgets, user scripts and extensions. You can [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/L2UQJRHTFK5YG3IOZEC7JSLH2ZQNZRVU/ read more]. You can test it on [[:testwiki:Main Page|Testwiki]] or [[:test2wiki:Main Page|Testwiki 2]].
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] The parameters for how you obtain [[mw:API:Tokens|tokens]] in the MediaWiki API were changed in 2014. The old way will no longer work from 1 September. Scripts, bots and tools that use the parameters from before the 2014 change need to be updated. You can [[phab:T280806#7215377|read more]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/29|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

15:31, 19 July 2021 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2021/30|Tech News: 2021-30]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/30|Translations]] are available.

'''Recent changes'''
* A [[mw:MediaWiki 1.37/wmf.14|new version]] of MediaWiki came to the Wikimedia wikis the week before last week. This was not in Tech News because there was no newsletter that week. 

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.16|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-07-27|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-07-28|en}}. It will be on all wikis from {{#time:j xg|2021-07-29|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]).

'''Future changes'''
* If you use the [[mw:Special:MyLanguage/Skin:MonoBook|Monobook skin]] you can choose to switch off [[:w:en:Responsive web design|responsive design]] on mobile. This will now work for more skins. If &lt;code&gt;{{int:monobook-responsive-label}}&lt;/code&gt; is unticked you need to also untick the new [[Special:Preferences#mw-prefsection-rendering|preference]] &lt;code&gt;{{int:prefs-skin-responsive}}&lt;/code&gt;. Otherwise it will stop working. Interface admins can automate this process on your wiki. You can [[phab:T285991|read more]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/30|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

21:11, 26 July 2021 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2021/31|Tech News: 2021-31]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/31|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] If your wiki uses markup like &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;&lt;div class=&quot;mw-content-ltr&quot;&gt;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; or &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;&lt;div class=&quot;mw-content-rtl&quot;&gt;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; without the required &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;dir&lt;/code&gt;&lt;/bdi&gt; attribute, then these will no longer work in 2 weeks. There is a short-term fix that can be added to your local wiki's Common.css page, which is explained at [[phab:T287701|T287701]]. From now on, all usages should include the full attributes, for example: &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;&lt;div class=&quot;mw-content-ltr&quot; dir=&quot;ltr&quot; lang=&quot;en&quot;&gt;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; or &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;&lt;div class=&quot;mw-content-rtl&quot; dir=&quot;rtl&quot; lang=&quot;he&quot;&gt;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. This also applies to some other HTML tags, such as &lt;code&gt;span&lt;/code&gt; or &lt;code&gt;code&lt;/code&gt;. You can find existing examples on your wiki that need to be updated, using the instructions at [[phab:T287701|T287701]].
* Reminder: Wikimedia has [[m:Special:MyLanguage/IRC/Migrating to Libera Chat|migrated to the Libera Chat IRC network]], from the old Freenode network. Local documentation should be updated.

'''Problems'''
* Last week, all wikis had slow access or no access for 30 minutes. There was a problem with generating dynamic lists of articles on the Russian Wikinews, due to the bulk import of 200,000+ new articles over 3 days, which led to database problems. The problematic feature has been disabled on that wiki and developers are discussing if it can be fixed properly. [https://phabricator.wikimedia.org/T287380][https://wikitech.wikimedia.org/wiki/Incident_documentation/2021-07-26_ruwikinews_DynamicPageList]

'''Changes later this week'''
* When adding links to a page using [[mw:VisualEditor|VisualEditor]] or the [[mw:Special:MyLanguage/2017 wikitext editor|2017 wikitext editor]], [[mw:Special:MyLanguage/Extension:Disambiguator|disambiguation pages]] will now only appear at the bottom of search results. This is because users do not often want to link to disambiguation pages. [https://phabricator.wikimedia.org/T285510]
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.17|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-08-03|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-08-04|en}}. It will be on all wikis from {{#time:j xg|2021-08-05|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]).

'''Future changes'''
* The [[mw:Wikimedia Apps/Team/Android|team of the Wikipedia app for Android]] is working on communication in the app. The developers are working on how to talk to other editors and get notifications. You can [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android/Communication|read more]]. They are looking for users who want to [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android/Communication/UsertestingJuly2021|test the plans]]. Any editor who has an Android phone and is willing to download the app can do this.
* The [[Special:Preferences#mw-prefsection-betafeatures|Beta Feature]] for {{int:discussiontools-preference-label}} will be updated in the coming weeks.  You will be able to [[mw:Talk pages project/Notifications|subscribe to individual sections]] on a talk page at more wikis.  You can test this now by adding &lt;code&gt;?dtenable=1&lt;/code&gt; to the end of the talk page's URL ([https://meta.wikimedia.org/wiki/Meta_talk:Sandbox?dtenable=1 example]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/31|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

20:47, 2 August 2021 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2021/32|Tech News: 2021-32]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/32|Translations]] are available.

'''Problems'''
* You can read but not edit 17 wikis for a few minutes on 10 August. This is planned at [https://zonestamp.toolforge.org/1628571650 05:00 UTC]. This is because of work on the database. [https://phabricator.wikimedia.org/T287449]

'''Changes later this week'''
* The [[wmania:Special:MyLanguage/2021:Hackathon|Wikimania Hackathon]] will take place remotely on 13 August, starting at 5:00 UTC, for 24 hours. You can participate in many ways. You can still propose projects and sessions.
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.18|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-08-10|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-08-11|en}}. It will be on all wikis from {{#time:j xg|2021-08-12|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]).
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] The old CSS &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;&lt;div class=&quot;visualClear&quot;&gt;&lt;/div&gt;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; will not be supported after 12 August. Instead, templates and pages should use &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;&lt;div style=&quot;clear:both;&quot;&gt;&lt;/div&gt;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. Please help to replace any existing uses on your wiki. There are global-search links available at [[phab:T287962|T287962]].

'''Future changes'''
* [[m:Special:MyLanguage/The Wikipedia Library|The Wikipedia Library]] is a place for Wikipedia editors to get access to sources. There is an [[mw:Special:MyLanguage/Extension:TheWikipediaLibrary|extension]] which has a new function to tell users when they can take part in it. It will use notifications. It will start pinging the first users in September. It will ping more users later. [https://phabricator.wikimedia.org/T288070]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] [[w:en:Vue.js|Vue.js]] will be the [[w:en:JavaScript|JavaScript]] framework for MediaWiki in the future. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/SOZREBYR36PUNFZXMIUBVAIOQI4N7PDU/]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/32|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

16:21, 9 August 2021 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2021/33|Tech News: 2021-33]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/33|Translations]] are available.

'''Recent changes'''
* You can add language links in the sidebar in the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|new Vector skin]] again. You do this by connecting the page to a Wikidata item. The new Vector skin has moved the language links but the new language selector cannot add language links yet. [https://phabricator.wikimedia.org/T287206]

'''Problems'''
* There was a problem on wikis which use the Translate extension. Translations were not updated or were replaced with the English text. The problems have been fixed. [https://phabricator.wikimedia.org/T288700][https://phabricator.wikimedia.org/T288683][https://phabricator.wikimedia.org/T288719]

'''Changes later this week'''
* A [[mw:Help:Tags|revision tag]] will soon be added to edits that add links to [[mw:Special:MyLanguage/Extension:Disambiguator|disambiguation pages]]. This is because these links are usually added by accident. The tag will allow editors to easily find the broken links and fix them. If your wiki does not like this feature, it can be [[mw:Help:Tags#Deleting a tag added by the software|hidden]]. [https://phabricator.wikimedia.org/T287549]
*Would you like to help improve the information about tools? Would you like to attend or help organize a small virtual meetup for your community to discuss the list of tools? Please get in touch on the [[m:Toolhub/The Quality Signal Sessions|Toolhub Quality Signal Sessions]] talk page. We are also looking for feedback [[m:Talk:Toolhub/The Quality Signal Sessions#Discussion topic for &quot;Quality Signal Sessions: The Tool Maintainers edition&quot;|from tool maintainers]] on some specific questions.
* In the past, edits to any page in your user talk space ignored your [[mw:Special:MyLanguage/Help:Notifications#mute|mute list]], e.g. sub-pages. Starting this week, this is only true for edits to your talk page. [https://phabricator.wikimedia.org/T288112]
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.19|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-08-17|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-08-18|en}}. It will be on all wikis from {{#time:j xg|2021-08-19|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/33|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

19:27, 16 August 2021 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=21889213 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/34|Tech News: 2021-34]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/34|Translations]] are available.

'''Recent changes'''
* The [[mw:Special:MyLanguage/Extension:Score|Score]] extension (&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;&lt;score&gt;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; notation) has been re-enabled on public wikis and upgraded to a newer version. Some musical score functionality may no longer work because the extension is only enabled in &quot;safe mode&quot;. The security issue has been fixed and an [[mw:Special:MyLanguage/Extension:Score/2021 security advisory|advisory published]].

'''Problems'''
* You will be able to read but not edit [[phab:T289130|some wikis]] for a few minutes on {{#time:j xg|2021-08-25|en}}. This will happen around [https://zonestamp.toolforge.org/1629871217 06:00 UTC]. This is for database maintenance. During this time, operations on the CentralAuth will also not be possible.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.20|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-08-24|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-08-25|en}}. It will be on all wikis from {{#time:j xg|2021-08-26|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/34|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

21:58, 23 August 2021 (UTC)
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== Read-only reminder ==

&lt;section begin=&quot;MassMessage&quot;/&gt;
A maintenance operation will be performed on [https://zonestamp.toolforge.org/1629871231 {{#time: l F d H:i e|2021-08-25T06:00|en}}]. It should only last for a few minutes.

This will affect your wiki as well as 11 other wikis. During this time, publishing edits will not be possible. 

Also during this time, operations on the CentralAuth will not be possible (GlobalRenames, changing/confirming e-mail addresses, logging into new wikis, password changes).

For more details about the operation and on all impacted services, please check [[phab:T289130|on Phabricator]].

A banner will be displayed 30 minutes before the operation.

Please help your community to be aware of this maintenance operation. {{Int:Feedback-thanks-title}}&lt;section end=&quot;MassMessage&quot;/&gt;

20:35, 24 August 2021 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2021/35|Tech News: 2021-35]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/35|Translations]] are available.

'''Recent changes'''
* Some musical score syntax no longer works and may needed to be updated, you can check [[:Category:{{MediaWiki:score-error-category}}]] on your wiki for a list of pages with errors.

'''Problems'''
* Musical scores were unable to render lyrics in some languages because of missing fonts. This has been fixed now. If your language would prefer a different font, please file a request in Phabricator. [https://phabricator.wikimedia.org/T289554]

'''Changes later this week'''
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] The parameters for how you obtain [[mw:API:Tokens|tokens]] in the MediaWiki API were changed in 2014. The old way will no longer work from 1 September. Scripts, bots and tools that use the parameters from before the 2014 change need to be updated. You can [[phab:T280806#7215377|read more]] about this.
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.21|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-08-31|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-09-01|en}}. It will be on all wikis from {{#time:j xg|2021-09-02|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]).

'''Future changes'''
* You will be able to read but not edit [[phab:T289660|Commons]] for a few minutes on {{#time:j xg|2021-09-06|en}}. This will happen around [https://zonestamp.toolforge.org/1630818058 05:00 UTC]. This is for database maintenance.
* All wikis will be read-only for a few minutes in the week of 13 September. More information will be published in Tech News later. It will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T287539]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/35|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

16:01, 30 August 2021 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2021/36|Tech News: 2021-36]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/36|Translations]] are available.

'''Recent changes'''
* The wikis that have [[mw:Special:MyLanguage/Growth/Feature_summary|Growth features]] deployed have been part of A/B testing since deployment, in which some newcomers did not receive the new features. Now, all of the newcomers on 21 of the smallest of those wikis will be receiving the features. [https://phabricator.wikimedia.org/T289786]

'''Changes later this week'''
* There is no new MediaWiki version this week.

'''Future changes'''
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] In 2017, the provided jQuery library was upgraded from version 1 to 3, with a compatibility layer. The migration will soon finish, to make the site load faster for everyone. If you maintain a gadget or user script, check if you have any JQMIGRATE errors and fix them, or they will break. [https://phabricator.wikimedia.org/T280944][https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/6Z2BVLOBBEC2QP4VV4KOOVQVE52P3HOP/]
* Last year, the Portuguese Wikipedia community embarked on an experiment to make log-in compulsory for editing.  The [[m:IP Editing: Privacy Enhancement and Abuse Mitigation/Impact report for Login Required Experiment on Portuguese Wikipedia|impact report of this trial]] is ready. Moving forward, the Anti-Harassment Tools team is looking for projects that are willing to experiment with restricting IP editing on their wiki for a short-term experiment. [[m:IP Editing: Privacy Enhancement and Abuse Mitigation/Login Required Experiment|Learn more]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/36|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

15:20, 6 September 2021 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2021/37|Tech News: 2021-37]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/37|Translations]] are available.

'''Recent changes'''
* 45 new Wikipedias now have access to the [[mw:Special:MyLanguage/Growth/Feature summary|Growth features]]. [https://phabricator.wikimedia.org/T289680]
* [[mw:Special:MyLanguage/Growth/Deployment table|A majority of Wikipedias]] now have access to the Growth features. The Growth team [[mw:Special:MyLanguage/Growth/FAQ|has published an FAQ page]] about the features. This translatable FAQ covers the description of the features, how to use them, how to change the configuration, and more.

'''Problems'''
* [[m:Special:MyLanguage/Tech/Server switch|All wikis will be read-only]] for a few minutes on 14 September. This is planned at [https://zonestamp.toolforge.org/1631628002 14:00 UTC]. [https://phabricator.wikimedia.org/T287539]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.37/wmf.23|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-09-14|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-09-15|en}}. It will be on all wikis from {{#time:j xg|2021-09-16|en}} ([[mw:MediaWiki 1.37/Roadmap|calendar]]).
* Starting this week, Wikipedia in Italian will receive weekly software updates on Wednesdays. It used to receive the updates on Thursdays. Due to this change, bugs will be noticed and fixed sooner. [https://phabricator.wikimedia.org/T286664]
* You can add language links in the sidebar in [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|the new Vector skin]] again. You do this by connecting the page to a Wikidata item. The new Vector skin has moved the language links but the new language selector cannot add language links yet. [https://phabricator.wikimedia.org/T287206]
* The [[mw:Special:MyLanguage/Extension:SyntaxHighlight|syntax highlight]] tool marks up code with different colours. It now can highlight 23 new code languages. Additionally, &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;golang&lt;/code&gt;&lt;/bdi&gt; can now be used as an alias for the [[d:Q37227|Go programming language]], and a special &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;output&lt;/code&gt;&lt;/bdi&gt; mode has been added to show a program's output. [https://phabricator.wikimedia.org/T280117][https://gerrit.wikimedia.org/r/c/mediawiki/extensions/SyntaxHighlight_GeSHi/+/715277/]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/37|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

15:35, 13 September 2021 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2021/38|Tech News: 2021-38]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/38|Translations]] are available.

'''Recent changes'''
* Growth features are now deployed to almost all Wikipedias. [[phab:T290582|For the majority of small Wikipedias]], the features are only available for experienced users, to [[mw:Special:MyLanguage/Growth/FAQ#enable|test the features]] and [[mw:Special:MyLanguage/Growth/FAQ#config|configure them]]. Features will be available for newcomers starting on 20 September 2021.
* MediaWiki had a feature that would highlight local links to short articles in a different style. Each user could pick the size at which &quot;stubs&quot; would be highlighted. This feature was very bad for performance, and following a consultation, has been removed. [https://phabricator.wikimedia.org/T284917]
* A technical change was made to the MonoBook skin to allow for easier maintenance and upkeep. This has resulted in some minor changes to HTML that make MonoBook's HTML consistent with other skins. Efforts have been made to minimize the impact on editors, but please ping [[m:User:Jon (WMF)|Jon (WMF)]] on wiki or in [[phab:T290888|phabricator]] if any problems are reported.

'''Problems'''
* There was a problem with search last week. Many search requests did not work for 2 hours because of an accidental restart of the search servers. [https://wikitech.wikimedia.org/wiki/Incident_documentation/2021-09-13_cirrussearch_restart]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.1|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-09-21|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-09-22|en}}. It will be on all wikis from {{#time:j xg|2021-09-23|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] The [[s:Special:ApiHelp/query+proofreadinfo|meta=proofreadpage API]] has changed. The &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;piprop&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; parameter has been renamed to &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;prpiprop&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. API users should update their code to avoid unrecognized parameter warnings. Pywikibot users should upgrade to 6.6.0. [https://phabricator.wikimedia.org/T290585]

'''Future changes'''
* The [[mw:Special:MyLanguage/Help:DiscussionTools#Replying|Reply tool]] will be deployed to the remaining wikis in the coming weeks.  It is currently part of &quot;{{int:discussiontools-preference-label}}&quot; in [[Special:Preferences#mw-prefsection-betafeatures|Beta features]] at most wikis. You will be able to turn it off in [[Special:Preferences#mw-prefsection-editing-discussion|Editing Preferences]]. [https://phabricator.wikimedia.org/T262331]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] The [[mw:MediaWiki_1.37/Deprecation_of_legacy_API_token_parameters|previously announced]] change to how you obtain tokens from the API has been delayed to September 21 because of an incompatibility with Pywikibot. Bot operators using Pywikibot can follow [[phab:T291202|T291202]] for progress on a fix, and should plan to upgrade to 6.6.1 when it is released.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/38|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

18:32, 20 September 2021 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2021/39|Tech News: 2021-39]] ==

&lt;section begin=&quot;technews-2021-W39&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/39|Translations]] are available.

'''Recent changes'''
* [[w:en:IOS|iOS 15]] has a new function called [https://support.apple.com/en-us/HT212614 Private Relay] (Apple website). This can hide the user's IP when they use [[w:en:Safari (software)|Safari]] browser. This is like using a [[w:en:Virtual private network|VPN]] in that we see another IP address instead. It is opt-in and only for those who pay extra for [[w:en:ICloud|iCloud]]. It will come to Safari users on [[:w:en:OSX|OSX]] later. There is a [[phab:T289795|technical discussion]] about what this means for the Wikimedia wikis.

'''Problems'''
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] Some gadgets and user-scripts add items to the [[m:Customization:Explaining_skins#Portlets|portlets]] (article tools) part of the skin. A recent change to the HTML may have made those links a different font-size. This can be fixed by adding the CSS class &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;.vector-menu-dropdown-noicon&lt;/code&gt;&lt;/bdi&gt;. [https://phabricator.wikimedia.org/T291438]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.2|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-09-28|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-09-29|en}}. It will be on all wikis from {{#time:j xg|2021-09-30|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).
* The [[mw:Special:MyLanguage/Onboarding_new_Wikipedians#New_experience|GettingStarted extension]] was built in 2013, and provides an onboarding process for new account holders in a few versions of Wikipedia. However, the recently developed [[mw:Special:MyLanguage/Growth/Feature_summary|Growth features]] provide a better onboarding experience. Since the vast majority of Wikipedias now have access to the Growth features, GettingStarted will be deactivated starting on 4 October. [https://phabricator.wikimedia.org/T235752]
* A small number of users will not be able to connect to the Wikimedia wikis after 30 September. This is because an old [[:w:en:root certificate|root certificate]] will no longer work. They will also have problems with many other websites. Users who have updated their software in the last five years are unlikely to have problems. Users in Europe, Africa and Asia are less likely to have immediate problems even if their software is too old. You can [[m:Special:MyLanguage/HTTPS/2021 Let's Encrypt root expiry|read more]].
* You can [[mw:Special:MyLanguage/Help:Notifications|receive notifications]] when someone leaves a comment on user talk page or mentions you in a talk page comment. Clicking the notification link will now bring you to the comment and highlight it. Previously, doing so brought you to the top of the section that contained the comment. You can find [[phab:T282029|more information in T282029.]]

'''Future changes'''
* The [[mw:Special:MyLanguage/Help:DiscussionTools#Replying|Reply tool]] will be deployed to the remaining wikis in the coming weeks. It is currently part of &quot;{{int:discussiontools-preference-label}}&quot; in [[Special:Preferences#mw-prefsection-betafeatures|Beta features]] at most wikis. You will be able to turn it off in [[Special:Preferences#mw-prefsection-editing-discussion|Editing Preferences]]. [[phab:T288485|See the list of wikis.]] [https://phabricator.wikimedia.org/T262331]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/39|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2021-W39&quot;/&gt;

22:23, 27 September 2021 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=22077885 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/40|Tech News: 2021-40]] ==

&lt;section begin=&quot;tech-newsletter-content&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/40|Translations]] are available.

'''Recent changes'''
* A more efficient way of sending changes from Wikidata to Wikimedia wikis that show them has been enabled for the following 10 wikis: mediawiki.org, the Italian, Catalan, Hebrew and Vietnamese Wikipedias, French Wikisource, and English Wikivoygage, Wikibooks, Wiktionary and Wikinews. If you notice anything strange about how changes from Wikidata appear in recent changes or your watchlist on those wikis you can [[phab:T48643|let the developers know]].

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.3|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-10-05|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-10-06|en}}. It will be on all wikis from {{#time:j xg|2021-10-07|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] Some gadgets and bots that use the API to read the AbuseFilter log might break. The &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;hidden&lt;/code&gt;&lt;/bdi&gt; property will no longer say an entry is &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;implicit&lt;/code&gt;&lt;/bdi&gt; for unsuppressed log entries about suppressed edits. If your bot needs to know this, do a separate revision query. Additionally, the property will have the value &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;false&lt;/code&gt;&lt;/bdi&gt; for visible entries; previously, it wasn't included in the response. [https://phabricator.wikimedia.org/T291718]
* A more efficient way of sending changes from Wikidata to Wikimedia wikis that show them will be enabled for ''all production wikis''. If you notice anything strange about how changes from Wikidata appear in recent changes or your watchlist you can [[phab:T48643|let the developers know]].

'''Future changes'''
* You can soon get cross-wiki notifications in the [[mw:Wikimedia Apps/Team/iOS|iOS Wikipedia app]]. You can also get notifications as push notifications. More notification updates will follow in later versions. [https://www.mediawiki.org/wiki/Wikimedia_Apps/Team/iOS/Notifications#September_2021_update]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] The JavaScript variables &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;wgExtraSignatureNamespaces&lt;/code&gt;&lt;/bdi&gt;, &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;wgLegalTitleChars&lt;/code&gt;&lt;/bdi&gt;, and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;wgIllegalFileChars&lt;/code&gt;&lt;/bdi&gt; will soon be removed from &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;[[mw:Special:MyLanguage/Manual:Interface/JavaScript#mw.config|mw.config]]&lt;/code&gt;&lt;/bdi&gt;. These are not part of the &quot;stable&quot; variables available for use in wiki JavaScript. [https://phabricator.wikimedia.org/T292011] 
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] The JavaScript variables &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;wgCookiePrefix&lt;/code&gt;&lt;/bdi&gt;, &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;wgCookieDomain&lt;/code&gt;&lt;/bdi&gt;, &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;wgCookiePath&lt;/code&gt;&lt;/bdi&gt;, and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;wgCookieExpiration&lt;/code&gt;&lt;/bdi&gt; will soon be removed from mw.config. Scripts should instead use &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;mw.cookie&lt;/code&gt;&lt;/bdi&gt; from the &quot;&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;[[mw:ResourceLoader/Core_modules#mediawiki.cookie|mediawiki.cookie]]&lt;/bdi&gt;&quot; module. [https://phabricator.wikimedia.org/T291760]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/40|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;tech-newsletter-content&quot;/&gt;

16:32, 4 October 2021 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2021/41|Tech News: 2021-41]] ==

&lt;section begin=&quot;technews-2021-W41&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/41|Translations]] are available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.4|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-10-12|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-10-13|en}}. It will be on all wikis from {{#time:j xg|2021-10-14|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).

* The [[mw:Manual:Table_of_contents#Auto-numbering|&quot;auto-number headings&quot; preference]] is being removed. You can read [[phab:T284921]] for the reasons and discussion. This change was [[m:Tech/News/2021/26|previously]] announced. [[mw:Snippets/Auto-number_headings|A JavaScript snippet]] is available which can be used to create a Gadget on wikis that still want to support auto-numbering.

'''Meetings'''
* You can join a meeting about the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Desktop Improvements]]. A demonstration version of the [[mw:Reading/Web/Desktop Improvements/Features/Sticky Header|newest feature]] will be shown. The event will take place on Tuesday, 12 October at 16:00 UTC. [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/Talk to Web/12-10-2021|See how to join]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/41|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2021-W41&quot;/&gt;

15:30, 11 October 2021 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2021/42|Tech News: 2021-42]] ==

&lt;section begin=&quot;technews-2021-W42&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/42|Translations]] are available.

'''Recent changes'''
*[[m:Toolhub|Toolhub]] is a catalogue to make it easier to find software tools that can be used for working on the Wikimedia projects. You can [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/LF4SSR4QRCKV6NPRFGUAQWUFQISVIPTS/ read more].

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.5|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-10-19|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-10-20|en}}. It will be on all wikis from {{#time:j xg|2021-10-21|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).

'''Future changes'''
* The developers of the [[mw:Wikimedia Apps/Team/Android|Wikipedia Android app]] are working on [[mw:Wikimedia Apps/Team/Android/Communication|communication in the app]]. You can now answer questions in [[mw:Wikimedia Apps/Team/Android/Communication/UsertestingOctober2021|survey]] to help the development.
* 3–5% of editors may be blocked in the next few months. This is because of a new service in Safari, which is similar to a [[w:en:Proxy server|proxy]] or a [[w:en:VPN|VPN]]. It is called iCloud Private Relay. There is a [[m:Special:MyLanguage/Apple iCloud Private Relay|discussion about this]] on Meta. The goal is to learn what iCloud Private Relay could mean for the communities.
* [[m:Special:MyLanguage/Wikimedia Enterprise|Wikimedia Enterprise]] is a new [[w:en:API|API]] for those who use a lot of information from the Wikimedia projects on other sites. It is a way to get big commercial users to pay for the data. There will soon be a copy of the Wikimedia Enterprise dataset. You can [https://lists.wikimedia.org/hyperkitty/list/wikitech-ambassadors@lists.wikimedia.org/message/B2AX6PWH5MBKB4L63NFZY3ADBQG7MSBA/ read more]. You can also ask the team questions [https://wikimedia.zoom.us/j/88994018553 on Zoom] on [https://www.timeanddate.com/worldclock/fixedtime.html?hour=15&amp;min=00&amp;sec=0&amp;day=22&amp;month=10&amp;year=2021 22 October 15:00 UTC].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/42|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2021-W42&quot;/&gt;

20:53, 18 October 2021 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=22176877 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/43|Tech News: 2021-43]] ==

&lt;section begin=&quot;technews-2021-W43&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/43|Translations]] are available.

'''Recent changes'''
* The [[m:Special:MyLanguage/Coolest_Tool_Award|Coolest Tool Award 2021]] is looking for nominations. You can recommend tools until 27 October.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.6|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-10-26|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-10-27|en}}. It will be on all wikis from {{#time:j xg|2021-10-28|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).

'''Future changes'''
*[[m:Special:MyLanguage/Help:Diff|Diff pages]] will have an improved copy and pasting experience. [[m:Special:MyLanguage/Community Wishlist Survey 2021/Copy paste diffs|The changes]] will allow the text in the diff for before and after to be treated as separate columns and will remove any unwanted syntax. [https://phabricator.wikimedia.org/T192526]
* The version of the [[w:en:Liberation fonts|Liberation fonts]] used in SVG files will be upgraded. Only new thumbnails will be affected. Liberation Sans Narrow will not change. [https://phabricator.wikimedia.org/T253600]

'''Meetings'''
* You can join a meeting about the [[m:Special:MyLanguage/Community Wishlist Survey|Community Wishlist Survey]]. News about the [[m:Special:MyLanguage/Community Wishlist Survey 2021/Warn when linking to disambiguation pages|disambiguation]] and the [[m:Special:MyLanguage/Community Wishlist Survey 2021/Real Time Preview for Wikitext|real-time preview]] wishes will be shown. The event will take place on Wednesday, 27 October at 14:30 UTC. [[m:Special:MyLanguage/Community Wishlist Survey/Updates/Talk to Us|See how to join]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/43|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2021-W43&quot;/&gt;

20:08, 25 October 2021 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=22232718 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/44|Tech News: 2021-44]] ==

&lt;section begin=&quot;technews-2021-W44&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/44|Translations]] are available.

'''Recent changes'''
* There is a limit on the amount of emails a user can send each day. This limit is now global instead of per-wiki. This change is to prevent abuse. [https://phabricator.wikimedia.org/T293866]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.7|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-11-02|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-11-03|en}}. It will be on all wikis from {{#time:j xg|2021-11-04|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/44|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2021-W44&quot;/&gt;

20:28, 1 November 2021 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=22269406 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/45|Tech News: 2021-45]] ==

&lt;section begin=&quot;technews-2021-W45&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/45|Translations]] are available.

'''Recent changes'''
* Mobile IP editors are now able to receive warning notices indicating they have a talk page message on the mobile website (similar to the orange banners available on desktop). These notices will be displayed on every page outside of the main namespace and every time the user attempts to edit. The notice on desktop now has a slightly different colour. [https://phabricator.wikimedia.org/T284642][https://phabricator.wikimedia.org/T278105]

'''Changes later this week'''
* [[phab:T294321|Wikidata will be read-only]] for a few minutes on 11 November. This will happen around [https://zonestamp.toolforge.org/1636610400 06:00 UTC]. This is for database maintenance. [https://phabricator.wikimedia.org/T294321]
* There is no new MediaWiki version this week.

'''Future changes'''
* In the future, unregistered editors will be given an identity that is not their [[:w:en:IP address|IP address]]. This is for legal reasons. A new user right will let editors who need to know the IPs of unregistered accounts to fight vandalism, spam, and harassment, see the IP. You can read the [[m:IP Editing: Privacy Enhancement and Abuse Mitigation#IP Masking Implementation Approaches (FAQ)|suggestions for how that identity could work]] and [[m:Talk:IP Editing: Privacy Enhancement and Abuse Mitigation|discuss on the talk page]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/45|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2021-W45&quot;/&gt;

20:36, 8 November 2021 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=22311003 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/46|Tech News: 2021-46]] ==

&lt;section begin=&quot;technews-2021-W46&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/46|Translations]] are available.

'''Recent changes'''
* Most [[c:Special:MyLanguage/Commons:Maximum_file_size#MAXTHUMB|large file uploads]] errors that had messages like &quot;&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;stashfailed&lt;/code&gt;&lt;/bdi&gt;&quot; or &quot;&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;DBQueryError&lt;/code&gt;&lt;/bdi&gt;&quot; have now been fixed. An [[wikitech:Incident documentation/2021-11-04 large file upload timeouts|incident report]] is available.

'''Problems'''
* Sometimes, edits made on iOS using the visual editor save groups of numbers as telephone number links, because of a feature in the operating system. This problem is under investigation. [https://phabricator.wikimedia.org/T116525]
* There was a problem with search last week. Many search requests did not work for 2 hours because of a configuration error. [https://wikitech.wikimedia.org/wiki/Incident_documentation/2021-11-10_cirrussearch_commonsfile_outage]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.9|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-11-16|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-11-17|en}}. It will be on all wikis from {{#time:j xg|2021-11-18|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/46|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2021-W46&quot;/&gt;

22:06, 15 November 2021 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=22338097 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/47|Tech News: 2021-47]] ==

&lt;section begin=&quot;technews-2021-W47&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/47|Translations]] are available.

'''Changes later this week'''
* There is no new MediaWiki version this week.
*The template dialog in VisualEditor and in the [[Special:Preferences#mw-prefsection-betafeatures|new wikitext mode]] Beta feature will be [[m:WMDE Technical Wishes/VisualEditor template dialog improvements|heavily improved]] on [[phab:T286992|a few wikis]]. Your [[m:Talk:WMDE Technical Wishes/VisualEditor template dialog improvements|feedback is welcome]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/47|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2021-W47&quot;/&gt;

20:02, 22 November 2021 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=22366010 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/48|Tech News: 2021-48]] ==

&lt;section begin=&quot;technews-2021-W48&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/48|Translations]] are available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.11|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-11-30|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-12-01|en}}. It will be on all wikis from {{#time:j xg|2021-12-02|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/48|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2021-W48&quot;/&gt;

21:15, 29 November 2021 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=22375666 --&gt;

== [[m:Special:MyLanguage/Tech/News/2021/49|Tech News: 2021-49]] ==

&lt;section begin=&quot;technews-2021-W49&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/49|Translations]] are available.

'''Problems'''
* MediaWiki 1.38-wmf.11 was scheduled to be deployed on some wikis last week. The deployment was delayed because of unexpected problems.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.12|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-12-07|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-12-08|en}}. It will be on all wikis from {{#time:j xg|2021-12-09|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).
* At all Wikipedias, a Mentor Dashboard is now available at &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;Special:MentorDashboard&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. It allows registered mentors, who take care of newcomers' first steps, to monitor their assigned newcomers' activity. It is part of the [[mw:Special:MyLanguage/Growth/Feature summary|Growth features]]. You can learn more about [[mw:Special:MyLanguage/Growth/Communities/How_to_configure_the_mentors%27_list|activating the mentor list]] on your wiki and about [[mw:Special:MyLanguage/Growth/Mentor dashboard|the mentor dashboard project]].
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] The predecessor to the current [[mw:API|MediaWiki Action API]] (which was created in 2008), &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;action=ajax&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;, will be removed this week. Any scripts or bots using it will need to switch to the corresponding API module. [https://phabricator.wikimedia.org/T42786]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] An old ResourceLoader module, &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;jquery.jStorage&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;, which was deprecated in 2016, will be removed this week. Any scripts or bots using it will need to switch to &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;mediawiki.storage&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; instead. [https://phabricator.wikimedia.org/T143034]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/49|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2021-W49&quot;/&gt;

21:59, 6 December 2021 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2021/50|Tech News: 2021-50]] ==

&lt;section begin=&quot;technews-2021-W50&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/50|Translations]] are available.

'''Recent changes'''
* There are now default [[m:Special:MyLanguage/Help:Namespace#Other_namespace_aliases|short aliases]] for the &quot;Project:&quot; namespace on most wikis. E.g. On Wikibooks wikis, &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;[[WB:]]&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; will go to the local language default for the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;[[Project:]]&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; namespace. This change is intended to help the smaller communities have easy access to this feature. Additional local aliases can still be requested via [[m:Special:MyLanguage/Requesting wiki configuration changes|the usual process]]. [https://phabricator.wikimedia.org/T293839]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.13|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2021-12-14|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2021-12-15|en}}. It will be on all wikis from {{#time:j xg|2021-12-16|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/50|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2021-W50&quot;/&gt;

22:27, 13 December 2021 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2021/51|Tech News: 2021-51]] ==

&lt;section begin=&quot;technews-2021-W51&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2021/51|Translations]] are available.

'''Tech News'''
* Because of the [[w:en:Christmas and holiday season|holidays]] the next issue of Tech News will be sent out on 10 January 2022.

'''Recent changes'''
* Queries made by the DynamicPageList extension (&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;&lt;DynamicPageList&gt;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;) are now only allowed to run for 10 seconds and error if they take longer. This is in response to multiple outages where long-running queries caused an outage on all wikis. [https://phabricator.wikimedia.org/T287380#7575719]

'''Changes later this week'''
* There is no new MediaWiki version this week or next week.

'''Future changes'''
* The developers of the Wikipedia iOS app are looking for testers who edit in multiple languages. You can [[mw:Wikimedia Apps/Team/iOS/202112 testing|read more and let them know if you are interested]].
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] The Wikimedia [[wikitech:Portal:Cloud VPS|Cloud VPS]] hosts technical projects for the Wikimedia movement. Developers need to [[wikitech:News/Cloud VPS 2021 Purge|claim projects]] they use. This is because old and unused projects are removed once a year. Unclaimed projects can be shut down from February. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/2B7KYL5VLQNHGQQHMYLW7KTUKXKAYY3T/]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2021/51|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2021-W51&quot;/&gt;

22:05, 20 December 2021 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2022/02|Tech News: 2022-02]] ==

&lt;section begin=&quot;technews-2022-W02&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/02|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] A &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;oauth_consumer&lt;/code&gt;&lt;/bdi&gt; variable has been added to the [[mw:Special:MyLanguage/AbuseFilter|AbuseFilter]] to enable identifying changes made by specific tools. [https://phabricator.wikimedia.org/T298281]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] Gadgets are [[mw:Special:MyLanguage/ResourceLoader/Migration_guide_(users)#Package_Gadgets|now able to directly include JSON pages]]. This means some gadgets can now be configured by administrators without needing the interface administrator permission, such as with the Geonotice gadget. [https://phabricator.wikimedia.org/T198758]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] Gadgets [[mw:Extension:Gadgets#Options|can now specify page actions]] on which they are available. For example, &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;|actions=edit,history&lt;/code&gt;&lt;/bdi&gt; will load a gadget only while editing and on history pages. [https://phabricator.wikimedia.org/T63007]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] Gadgets can now be loaded on demand with the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;withgadget&lt;/code&gt;&lt;/bdi&gt; URL parameter. This can be used to replace [[mw:Special:MyLanguage/Snippets/Load JS and CSS by URL|an earlier snippet]] that typically looks like &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;withJS&lt;/code&gt;&lt;/bdi&gt; or &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;withCSS&lt;/code&gt;&lt;/bdi&gt;. [https://phabricator.wikimedia.org/T29766]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] At wikis where [[mw:Special:MyLanguage/Growth/Communities/How to configure the mentors' list|the Mentorship system is configured]], you can now use the Action API to get a list of a [[mw:Special:MyLanguage/Growth/Mentor_dashboard|mentor's]] mentees. [https://phabricator.wikimedia.org/T291966]
* The heading on the main page can now be configured using &lt;span class=&quot;mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[MediaWiki:Mainpage-title-loggedin]]&lt;/span&gt; for logged-in users and &lt;span class=&quot;mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[MediaWiki:Mainpage-title]]&lt;/span&gt; for logged-out users. Any CSS that was previously used to hide the heading should be removed. [https://meta.wikimedia.org/wiki/Special:MyLanguage/Small_wiki_toolkits/Starter_kit/Main_page_customization#hide-heading] [https://phabricator.wikimedia.org/T298715]
* Four special pages (and their API counterparts) now have a maximum database query execution time of 30 seconds. These special pages are: RecentChanges, Watchlist, Contributions, and Log. This change will help with site performance and stability. You can read [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/IPJNO75HYAQWIGTHI5LJHTDVLVOC4LJP/ more details about this change] including some possible solutions if this affects your workflows. [https://phabricator.wikimedia.org/T297708]
* The [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Features/Sticky Header|sticky header]] has been deployed for 50% of logged-in users on [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Frequently asked questions#pilot-wikis|more than 10 wikis]]. This is part of the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Desktop Improvements]]. See [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Participate|how to take part in the project]]. 

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.17|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-01-11|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-01-12|en}}. It will be on all wikis from {{#time:j xg|2022-01-13|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).

'''Events'''
* [[m:Special:MyLanguage/Community Wishlist Survey 2022|Community Wishlist Survey 2022]] begins. All contributors to the Wikimedia projects can propose for tools and platform improvements. The proposal phase takes place from {{#time:j xg|2022-01-10|en}} 18:00 UTC to {{#time:j xg|2022-01-23|en}} 18:00 UTC. [[m:Special:MyLanguage/Community_Wishlist_Survey/FAQ|Learn more]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/02|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W02&quot;/&gt;

01:23, 11 January 2022 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2022/03|Tech News: 2022-03]] ==

&lt;section begin=&quot;technews-2022-W03&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/03|Translations]] are available.

'''Recent changes'''
* When using [[mw:Special:MyLanguage/Extension:WikiEditor|WikiEditor]] (also known as the 2010 wikitext editor), people will now see a warning if they link to disambiguation pages.  If you click &quot;{{int:Disambiguator-review-link}}&quot; in the warning, it will ask you to correct the link to a more specific term. You can [[m:Community Wishlist Survey 2021/Warn when linking to disambiguation pages#Jan 12, 2021: Turning on the changes for all Wikis|read more information]] about this completed 2021 Community Wishlist item.
* You can [[mw:Special:MyLanguage/Help:DiscussionTools#subscribe|automatically subscribe to all of the talk page discussions]] that you start or comment in using [[mw:Special:MyLanguage/Talk pages project/Feature summary|DiscussionTools]].  You will receive [[mw:Special:MyLanguage/Notifications|notifications]] when another editor replies.  This is available at most wikis.  Go to your [[Special:Preferences#mw-prefsection-editing-discussion|Preferences]] and turn on &quot;{{int:discussiontools-preference-autotopicsub}}&quot;. [https://phabricator.wikimedia.org/T263819]
* When asked to create a new page or talk page section, input fields can be [[mw:Special:MyLanguage/Manual:Creating_pages_with_preloaded_text|&quot;preloaded&quot; with some text]]. This feature is now limited to wikitext pages. This is so users can't be tricked into making malicious edits. There is a discussion about [[phab:T297725|if this feature should be re-enabled]] for some content types.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.18|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-01-18|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-01-19|en}}. It will be on all wikis from {{#time:j xg|2022-01-20|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).

'''Events'''
* [[m:Special:MyLanguage/Community Wishlist Survey 2022|Community Wishlist Survey 2022]] continues. All contributors to the Wikimedia projects can propose for tools and platform improvements. The proposal phase takes place from {{#time:j xg|2022-01-10|en}} 18:00 UTC to {{#time:j xg|2022-01-23|en}} 18:00 UTC. [[m:Special:MyLanguage/Community_Wishlist_Survey/FAQ|Learn more]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/03|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W03&quot;/&gt;

19:55, 17 January 2022 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2022/04|Tech News: 2022-04]] ==

&lt;section begin=&quot;technews-2022-W04&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/04|Translations]] are available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.19|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-01-25|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-01-26|en}}. It will be on all wikis from {{#time:j xg|2022-01-27|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).
* The following languages can now be used with [[mw:Special:MyLanguage/Extension:SyntaxHighlight|syntax highlighting]]: BDD, Elpi, LilyPond, Maxima, Rita, Savi, Sed, Sophia, Spice, .SRCINFO.
* You can now access your watchlist from outside of the user menu in the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|new Vector skin]]. The watchlist link appears next to the notification icons if you are at the top of the page. [https://phabricator.wikimedia.org/T289619]

'''Events'''
* You can see the results of the [[m:Special:MyLanguage/Coolest Tool Award|Coolest Tool Award 2021]] and learn more about 14 tools which were selected this year.
* You can [[m:Special:MyLanguage/Community_Wishlist_Survey/Help_us|translate, promote]], or comment on [[m:Special:MyLanguage/Community Wishlist Survey 2022/Proposals|the proposals]] in the Community Wishlist Survey. Voting will begin on {{#time:j xg|2022-01-28|en}}.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/04|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W04&quot;/&gt;

21:38, 24 January 2022 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2022/05|Tech News: 2022-05]] ==

&lt;section begin=&quot;technews-2022-W05&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/05|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] If a gadget should support the new &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;?withgadget&lt;/code&gt;&lt;/bdi&gt; URL parameter that was [[m:Special:MyLanguage/Tech/News/2022/02|announced]] 3 weeks ago, then it must now also specify &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;supportsUrlLoad&lt;/code&gt;&lt;/bdi&gt; in the gadget definition ([[mw:Special:MyLanguage/Extension:Gadgets#supportsUrlLoad|documentation]]). [https://phabricator.wikimedia.org/T29766]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.20|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-02-01|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-02-02|en}}. It will be on all wikis from {{#time:j xg|2022-02-03|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).

'''Future changes'''
* A change that was [[m:Special:MyLanguage/Tech/News/2021/16|announced]] last year was delayed. It is now ready to move ahead:
** The user group &lt;code&gt;oversight&lt;/code&gt; will be renamed &lt;code&gt;suppress&lt;/code&gt;. This is for [[phab:T109327|technical reasons]]. This is the technical name. It doesn't affect what you call the editors with this user right on your wiki. This is planned to happen in three weeks. You can comment [[phab:T112147|in Phabricator]] if you have objections. As usual, these labels can be translated on translatewiki ([[phab:T112147|direct links are available]]) or by administrators on your wiki.

'''Events'''
* You can vote on proposals in the [[m:Special:MyLanguage/Community Wishlist Survey 2022|Community Wishlist Survey]] between 28 January and 11 February. The survey decides what the [[m:Special:MyLanguage/Community Tech|Community Tech team]] will work on.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/05|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W05&quot;/&gt;

17:42, 31 January 2022 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2022/06|Tech News: 2022-06]] ==

&lt;section begin=&quot;technews-2022-W06&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/06|Translations]] are available.

'''Recent changes'''
* English Wikipedia recently set up a gadget for dark mode. You can enable it there, or request help from an [[m:Special:MyLanguage/Interface administrators|interface administrator]] to set it up on your wiki ([[w:en:Wikipedia:Dark mode (gadget)|instructions and screenshot]]).
* Category counts are sometimes wrong. They will now be completely recounted at the beginning of every month. [https://phabricator.wikimedia.org/T299823]

'''Problems'''
* A code-change last week to fix a bug with [[mw:Special:MyLanguage/Manual:Live preview|Live Preview]] may have caused problems with some local gadgets and user-scripts. Any code with skin-specific behaviour for &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;vector&lt;/code&gt;&lt;/bdi&gt; should be updated to also check for &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;vector-2022&lt;/code&gt;&lt;/bdi&gt;. [[phab:T300987|A code-snippet, global search, and example are available]].

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.21|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-02-08|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-02-09|en}}. It will be on all wikis from {{#time:j xg|2022-02-10|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/06|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W06&quot;/&gt;

21:15, 7 February 2022 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2022/07|Tech News: 2022-07]] ==

&lt;section begin=&quot;technews-2022-W07&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/07|Translations]] are available.

'''Recent changes'''
* [[mw:Special:MyLanguage/Manual:Purge|Purging]] a category page with fewer than 5,000 members will now recount it completely. This will allow editors to fix incorrect counts when it is wrong. [https://phabricator.wikimedia.org/T85696]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.22|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-02-15|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-02-16|en}}. It will be on all wikis from {{#time:j xg|2022-02-17|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).
* [[File:Octicons-tools.svg|15px|link=|Advanced item]] In the [[mw:Special:MyLanguage/Extension:AbuseFilter|AbuseFilter]] extension, the &lt;code dir=ltr&gt;rmspecials()&lt;/code&gt; function has been updated so that it does not remove the &quot;space&quot; character. Wikis are advised to wrap all the uses of &lt;code dir=ltr&gt;rmspecials()&lt;/code&gt; with &lt;code dir=ltr&gt;rmwhitespace()&lt;/code&gt; wherever necessary to keep filters' behavior unchanged. You can use the search function on [[Special:AbuseFilter]] to locate its usage. [https://phabricator.wikimedia.org/T263024]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/07|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W07&quot;/&gt;

19:18, 14 February 2022 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2022/08|Tech News: 2022-08]] ==

&lt;section begin=&quot;technews-2022-W08&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/08|Translations]] are available.

'''Recent changes'''
* [[Special:Nuke|Special:Nuke]] will now provide the standard deletion reasons (editable at &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[MediaWiki:Deletereason-dropdown]]&lt;/bdi&gt;) to use when mass-deleting pages. This was [[m:Community Wishlist Survey 2022/Admins and patrollers/Mass-delete to offer drop-down of standard reasons, or templated reasons.|a request in the 2022 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T25020]
* At Wikipedias, all new accounts now get the [[mw:Special:MyLanguage/Growth/Feature_summary|Growth features]] by default when creating an account. Communities are encouraged to [[mw:Special:MyLanguage/Help:Growth/Tools/Account_creation|update their help resources]]. Previously, only 80% of new accounts would get the Growth features. A few Wikipedias remain unaffected by this change. [https://phabricator.wikimedia.org/T301820]
* You can now prevent specific images that are used in a page from appearing in other locations, such as within PagePreviews or Search results. This is done with the markup &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;class=notpageimage&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. For example, &lt;code&gt;&lt;nowiki&gt;[[File:Example.png|class=notpageimage]]&lt;/nowiki&gt;&lt;/code&gt;. [https://phabricator.wikimedia.org/T301588]
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] There has been a change to the HTML of Special:Contributions, Special:MergeHistory, and History pages, to support the grouping of changes by date in [[mw:Special:MyLanguage/Skin:Minerva_Neue|the mobile skin]]. While unlikely, this may affect gadgets and user scripts. A [[phab:T298638|list of all the HTML changes]] is on Phabricator.

'''Events'''
* [[m:Special:MyLanguage/Community Wishlist Survey 2022/Results|Community Wishlist Survey results]] have been published. The [[m:Special:MyLanguage/Community Wishlist Survey/Updates/2022 results#leaderboard|ranking of prioritized proposals]] is also available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.23|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-02-22|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-02-23|en}}. It will be on all wikis from {{#time:j xg|2022-02-24|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).

'''Future changes'''
* The software to play videos and audio files on pages will change soon on all wikis. The old player will be removed. Some audio players will become wider after this change. [[mw:Special:MyLanguage/Extension:TimedMediaHandler/VideoJS_Player|The new player]] has been a beta feature for over four years. [https://phabricator.wikimedia.org/T100106][https://phabricator.wikimedia.org/T248418]
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Toolforge's underlying operating system is being updated. If you maintain any tools there, there are two options for migrating your tools into the new system.  There are [[wikitech:News/Toolforge Stretch deprecation|details, deadlines, and instructions]] on Wikitech. [https://lists.wikimedia.org/hyperkitty/list/cloud-announce@lists.wikimedia.org/thread/EPJFISC52T7OOEFH5YYMZNL57O4VGSPR/]
* Administrators will soon have [[m:Special:MyLanguage/Community Wishlist Survey 2021/(Un)delete associated talk page|the option to delete/undelete]] the associated &quot;talk&quot; page when they are deleting a given page. An API endpoint with this option will also be available. This was [[m:Community Wishlist Survey 2021/Admins and patrollers/(Un)delete associated talk page|a request from the 2021 Wishlist Survey]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/08|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W08&quot;/&gt;

19:12, 21 February 2022 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2022/09|Tech News: 2022-09]] ==

&lt;section begin=&quot;technews-2022-W09&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/09|Translations]] are available.

'''Recent changes'''
* When searching for edits by [[mw:Special:MyLanguage/Help:Tags|change tags]], e.g. in page history or user contributions, there is now a dropdown list of possible tags. This was [[m:Community Wishlist Survey 2022/Miscellaneous/Improve plain-text change tag selector|a request in the 2022 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T27909]
* Mentors using the [[mw:Special:MyLanguage/Growth/Mentor_dashboard|Growth Mentor dashboard]] will now see newcomers assigned to them who have made at least one edit, up to 200 edits. Previously, all newcomers assigned to the mentor were visible on the dashboard, even ones without any edit or ones who made hundred of edits. Mentors can still change these values using the filters on their dashboard. Also, the last choice of filters will now be saved. [https://phabricator.wikimedia.org/T301268][https://phabricator.wikimedia.org/T294460]
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] The user group &lt;code&gt;oversight&lt;/code&gt; was renamed &lt;code&gt;suppress&lt;/code&gt;. This is for [[phab:T109327|technical reasons]]. You may need to update any local references to the old name, e.g. gadgets, links to Special:Listusers, or uses of [[mw:Special:MyLanguage/Help:Magic_words|NUMBERINGROUP]].

'''Problems'''
* The recent change to the HTML of [[mw:Special:MyLanguage/Help:Tracking changes|tracking changes]] pages caused some problems for screenreaders. This is being fixed. [https://phabricator.wikimedia.org/T298638]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.24|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-03-01|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-03-02|en}}. It will be on all wikis from {{#time:j xg|2022-03-03|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).

'''Future changes'''
* Working with templates will become easier. [[m:WMDE_Technical_Wishes/Templates|Several improvements]] are planned for March 9 on most wikis and on March 16 on English Wikipedia. The improvements include: Bracket matching, syntax highlighting colors, finding and inserting templates, and related visual editor features.
* If you are a template developer or an interface administrator, and you are intentionally overriding or using the default CSS styles of user feedback boxes (the classes: &lt;code dir=ltr&gt;successbox, messagebox, errorbox, warningbox&lt;/code&gt;), please note that these classes and associated CSS will soon be removed from MediaWiki core. This is to prevent problems when the same class-names are also used on a wiki. Please let us know by commenting at [[phab:T300314]] if you think you might be affected.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/09|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W09&quot;/&gt;

22:59, 28 February 2022 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2022/10|Tech News: 2022-10]] ==

&lt;section begin=&quot;technews-2022-W10&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/10|Translations]] are available.

'''Problems'''
* There was a problem with some interface labels last week. It will be fixed this week. This change was part of ongoing work to simplify the support for skins which do not have active maintainers. [https://phabricator.wikimedia.org/T301203]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.25|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-03-08|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-03-09|en}}. It will be on all wikis from {{#time:j xg|2022-03-10|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/10|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W10&quot;/&gt;

21:16, 7 March 2022 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2022/11|Tech News: 2022-11]] ==

&lt;section begin=&quot;technews-2022-W11&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/11|Translations]] are available.

'''Recent changes'''
* In the Wikipedia Android app [[mw:Special:MyLanguage/Wikimedia_Apps/Team/Android/Communication#Updates|it is now possible]] to change the toolbar at the bottom so the tools you use more often are easier to click on. The app now also has a focused reading mode. [https://phabricator.wikimedia.org/T296753][https://phabricator.wikimedia.org/T254771]

'''Problems'''
* There was a problem with the collection of some page-view data from June 2021 to January 2022 on all wikis. This means the statistics are incomplete. To help calculate which projects and regions were most affected, relevant datasets are being retained for 30 extra days. You can [[m:Talk:Data_retention_guidelines#Added_exception_for_page_views_investigation|read more on Meta-wiki]].
* There was a problem with the databases on March 10. All wikis were unreachable for logged-in users for 12 minutes. Logged-out users could read pages but could not edit or access uncached content then. [https://wikitech.wikimedia.org/wiki/Incident_documentation/2022-03-10_MediaWiki_availability]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.38/wmf.26|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-03-15|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-03-16|en}}. It will be on all wikis from {{#time:j xg|2022-03-17|en}} ([[mw:MediaWiki 1.38/Roadmap|calendar]]).
* When [[mw:Special:MyLanguage/Help:System_message#Finding_messages_and_documentation|using &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;uselang=qqx&lt;/code&gt;&lt;/bdi&gt; to find localisation messages]], it will now show all possible message keys for navigation tabs such as &quot;{{int:vector-view-history}}&quot;. [https://phabricator.wikimedia.org/T300069]
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Access to [[{{#special:RevisionDelete}}]] has been expanded to include users who have &lt;code dir=ltr&gt;deletelogentry&lt;/code&gt; and &lt;code dir=ltr&gt;deletedhistory&lt;/code&gt; rights through their group memberships. Before, only those with the &lt;code dir=ltr&gt;deleterevision&lt;/code&gt; right could access this special page. [https://phabricator.wikimedia.org/T301928]
* On the [[{{#special:Undelete}}]] pages for diffs and revisions, there will be a link back to the main Undelete page with the list of revisions. [https://phabricator.wikimedia.org/T284114]

'''Future changes'''
*  The Wikimedia Foundation has announced the IP Masking implementation strategy and next steps. The [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation#feb25|announcement can be read here]].
* The [[mw:Special:MyLanguage/Wikimedia Apps/Android FAQ|Wikipedia Android app]] developers are working on [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android/Communication|new functions]] for user talk pages and article talk pages. [https://phabricator.wikimedia.org/T297617]

'''Events'''
* The [[mw:Wikimedia Hackathon 2022|Wikimedia Hackathon 2022]] will take place as a hybrid event on 20-22 May 2022. The Hackathon will be held online and there are grants available to support local in-person meetups around the world. Grants can be requested until 20 March.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/11|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W11&quot;/&gt;

22:07, 14 March 2022 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2022/12|Tech News: 2022-12]] ==

&lt;section begin=&quot;technews-2022-W12&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/12|Translations]] are available.

'''New code release schedule for this week'''
* There will be four MediaWiki releases this week, instead of just one. This is an experiment which should lead to fewer problems and to faster feature updates. The releases will be on all wikis, at different times, on Monday, Tuesday, and Wednesday. You can [[mw:Special:MyLanguage/Wikimedia Release Engineering Team/Trainsperiment week|read more about this project]].

'''Recent changes'''
* You can now set how many search results to show by default in [[Special:Preferences#mw-prefsection-searchoptions|your Preferences]]. This was the 12th most popular wish in the [[m:Special:MyLanguage/Community Wishlist Survey 2022/Results|Community Wishlist Survey 2022]]. [https://phabricator.wikimedia.org/T215716]
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] The Jupyter notebooks tool [[wikitech:PAWS|PAWS]] has been updated to a new interface. [https://phabricator.wikimedia.org/T295043]

'''Future changes'''
* Interactive maps via [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] will soon work on wikis using the [[mw:Special:MyLanguage/Extension:FlaggedRevs|FlaggedRevisions]] extension. [https://wikimedia.sslsurvey.de/Kartographer-Workflows-EN/ Please tell us] which improvements you want to see in Kartographer. You can take this survey in simple English. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Geoinformation]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/12|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W12&quot;/&gt;

16:01, 21 March 2022 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2022/13|Tech News: 2022-13]] ==

&lt;section begin=&quot;technews-2022-W13&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/13|Translations]] are available.

'''Recent changes'''
* There is a simple new Wikimedia Commons upload tool available for macOS users, [[c:Commons:Sunflower|Sunflower]].

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.5|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-03-29|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-03-30|en}}. It will be on all wikis from {{#time:j xg|2022-03-31|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* Some wikis will be in read-only for a few minutes because of regular database maintenance. It will be performed on {{#time:j xg|2022-03-29|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s3.dblist targeted wikis]) and on {{#time:j xg|2022-03-31|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s5.dblist targeted wikis]). [https://phabricator.wikimedia.org/T301850][https://phabricator.wikimedia.org/T303798]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/13|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W13&quot;/&gt;

19:54, 28 March 2022 (UTC)
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== [[m:Special:MyLanguage/Tech/News/2022/14|Tech News: 2022-14]] ==

&lt;section begin=&quot;technews-2022-W14&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/14|Translations]] are available.

'''Problems'''
* For a few days last week, edits that were suggested to newcomers were not tagged in the [[{{#special:recentchanges}}]] feed. This bug has been fixed. [https://phabricator.wikimedia.org/T304747]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.6|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-04-05|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-04-06|en}}. It will be on all wikis from {{#time:j xg|2022-04-07|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-04-07|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s4.dblist targeted wikis]).

'''Future changes'''
* Starting next week, Tech News' title will be translatable. When the newsletter is distributed, its title may not be &lt;code dir=ltr&gt;Tech News: 2022-14&lt;/code&gt; anymore. It may affect some filters that have been set up by some communities. [https://phabricator.wikimedia.org/T302920]
* Over the next few months, the &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]&quot; Growth feature [[phab:T304110|will become available to more Wikipedias]]. Each week, a few wikis will get the feature. You can test this tool at [[mw:Special:MyLanguage/Growth#deploymentstable|a few wikis where &quot;Link recommendation&quot; is already available]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/14|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W14&quot;/&gt;

21:01, 4 April 2022 (UTC)
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== Tech News: 2022-15 ==

&lt;section begin=&quot;technews-2022-W15&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/15|Translations]] are available.

'''Recent changes'''
* There is a new public status page at &lt;span class=&quot;mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[https://www.wikimediastatus.net/ www.wikimediastatus.net]&lt;/span&gt;. This site shows five automated high-level metrics where you can see the overall health and performance of our wikis' technical environment. It also contains manually-written updates for widespread incidents, which are written as quickly as the engineers are able to do so while also fixing the actual problem. The site is separated from our production infrastructure and hosted by an external service, so that it can be accessed even if the wikis are briefly unavailable. You can [https://diff.wikimedia.org/2022/03/31/announcing-www-wikimediastatus-net/ read more about this project].
* On Wiktionary wikis, the software to play videos and audio files on pages has now changed. The old player has been removed. Some audio players will become wider after this change. [[mw:Special:MyLanguage/Extension:TimedMediaHandler/VideoJS_Player|The new player]] has been a beta feature for over four years. [https://phabricator.wikimedia.org/T100106][https://phabricator.wikimedia.org/T248418]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.7|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-04-12|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-04-13|en}}. It will be on all wikis from {{#time:j xg|2022-04-14|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/15|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W15&quot;/&gt;

19:44, 11 April 2022 (UTC)
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== Tech News: 2022-16 ==

&lt;section begin=&quot;technews-2022-W16&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/16|Translations]] are available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.8|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-04-19|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-04-20|en}}. It will be on all wikis from {{#time:j xg|2022-04-21|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-04-19|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s7.dblist targeted wikis]) and on {{#time:j xg|2022-04-21|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s8.dblist targeted wikis]).
* Administrators will now have [[m:Community Wishlist Survey 2021/(Un)delete associated talk page|the option to delete/undelete the associated &quot;Talk&quot; page]] when they are deleting a given page. An API endpoint with this option is also available. This concludes the [[m:Community Wishlist Survey 2021/Admins and patrollers/(Un)delete associated talk page|11th wish of the 2021 Community Wishlist Survey]].
* On [[mw:Special:MyLanguage/Reading/Web/Desktop_Improvements#test-wikis|selected wikis]], 50% of logged-in users will see the new [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Features/Table of contents|table of contents]]. When scrolling up and down the page, the table of contents will stay in the same place on the screen. This is part of the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Desktop Improvements]] project. [https://phabricator.wikimedia.org/T304169]
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Message boxes produced by MediaWiki code will no longer have these CSS classes: &lt;code dir=ltr&gt;successbox&lt;/code&gt;, &lt;code dir=ltr&gt;errorbox&lt;/code&gt;, &lt;code dir=ltr&gt;warningbox&lt;/code&gt;. The styles for those classes and &lt;code dir=ltr&gt;messagebox&lt;/code&gt; will be removed from MediaWiki core. This only affects wikis that use these classes in wikitext, or change their appearance within site-wide CSS. Please review any local usage and definitions for these classes you may have. This was previously announced in the [[m:Special:MyLanguage/Tech/News/2022/09|28 February issue of Tech News]].

'''Future changes'''
* [[mw:Special:MyLanguage/Extension:Kartographer|Kartographer]] will become compatible with [[mw:Special:MyLanguage/Extension:FlaggedRevs|FlaggedRevisions page stabilization]]. Kartographer maps will also work on pages with [[mw:Special:MyLanguage/Help:Pending changes|pending changes]]. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Geoinformation#Project_descriptions] The Kartographer documentation has been thoroughly updated. [https://www.mediawiki.org/wiki/Special:MyLanguage/Help:Extension:Kartographer/Getting_started] [https://www.mediawiki.org/wiki/Special:MyLanguage/Help:VisualEditor/Maps] [https://www.mediawiki.org/wiki/Special:MyLanguage/Help:Extension:Kartographer]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/16|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W16&quot;/&gt;

23:11, 18 April 2022 (UTC)
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== Tech News: 2022-17 ==

&lt;section begin=&quot;technews-2022-W17&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/17|Translations]] are available.

'''Recent changes'''
*  On [https://noc.wikimedia.org/conf/dblists/group1.dblist many wikis] (group 1), the software to play videos and audio files on pages has now changed. The old player has been removed. Some audio players will become wider after this change. [[mw:Special:MyLanguage/Extension:TimedMediaHandler/VideoJS_Player|The new player]] has been a beta feature for over four years. [https://phabricator.wikimedia.org/T100106][https://phabricator.wikimedia.org/T248418]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.9|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-04-26|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-04-27|en}}. It will be on all wikis from {{#time:j xg|2022-04-28|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-04-26|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s2.dblist targeted wikis]).
* Some very old browsers and operating systems are no longer supported. Some things on the wikis might look weird or not work in very old browsers like Internet Explorer 9 or 10, Android 4, or Firefox 38 or older. [https://phabricator.wikimedia.org/T306486]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/17|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W17&quot;/&gt;

22:56, 25 April 2022 (UTC)
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== Tech News: 2022-18 ==

&lt;section begin=&quot;technews-2022-W18&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/18|Translations]] are available.

'''Recent changes'''
* On [https://noc.wikimedia.org/conf/dblists/group2.dblist all remaining wikis] (group 2), the software to play videos and audio files on pages has now changed. The old player has been removed. Some audio players will become wider after this change. [[mw:Special:MyLanguage/Extension:TimedMediaHandler/VideoJS_Player|The new player]] has been a beta feature for over four years. [https://phabricator.wikimedia.org/T100106][https://phabricator.wikimedia.org/T248418]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.10|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-05-03|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-05-04|en}}. It will be on all wikis from {{#time:j xg|2022-05-05|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).

'''Future changes'''
* The developers are working on talk pages in the [[mw:Wikimedia Apps/Team/iOS|Wikipedia app for iOS]]. You can [https://wikimedia.qualtrics.com/jfe/form/SV_9GBcHczQGLbQWTY give feedback]. You can take the survey in English, German, Hebrew or Chinese.
* [[m:WMDE_Technical_Wishes/VisualEditor_template_dialog_improvements#Status_and_next_steps|Most wikis]] will receive an [[m:WMDE_Technical_Wishes/VisualEditor_template_dialog_improvements|improved template dialog]] in VisualEditor and New Wikitext mode. [https://phabricator.wikimedia.org/T296759] [https://phabricator.wikimedia.org/T306967]
* If you use syntax highlighting while editing wikitext, you can soon activate a [[m:WMDE_Technical_Wishes/Improved_Color_Scheme_of_Syntax_Highlighting#Color-blind_mode|colorblind-friendly color scheme]]. [https://phabricator.wikimedia.org/T306867]
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Several CSS IDs related to MediaWiki interface messages will be removed. Technical editors should please [[phab:T304363|review the list of IDs and links to their existing uses]]. These include &lt;code dir=ltr&gt;#mw-anon-edit-warning&lt;/code&gt;, &lt;code dir=ltr&gt;#mw-undelete-revision&lt;/code&gt; and 3 others.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/18|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W18&quot;/&gt;

19:33, 2 May 2022 (UTC)
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== Tech News: 2022-19 ==

&lt;section begin=&quot;technews-2022-W19&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/19|Translations]] are available.

'''Recent changes'''
* You can now see categories in the [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android|Wikipedia app for Android]]. [https://phabricator.wikimedia.org/T73966]

'''Problems'''
* Last week, there was a problem with Wikidata's search autocomplete. This has now been fixed. [https://phabricator.wikimedia.org/T307586]
* Last week, all wikis had slow access or no access for 20 minutes, for logged-in users and non-cached pages. This was caused by a problem with a database change. [https://phabricator.wikimedia.org/T307647]

'''Changes later this week'''
* There is no new MediaWiki version this week. [https://phabricator.wikimedia.org/T305217#7894966]
*  [[m:WMDE Technical Wishes/Geoinformation#Current issues|Incompatibility issues]] with [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] and the [[mw:Special:MyLanguage/Help:Extension:FlaggedRevs|FlaggedRevs extension]] will be fixed: Deployment is planned for May 10 on all wikis. Kartographer will then be enabled on the [[phab:T307348|five wikis which have not yet enabled the extension]] on May 24.
* The [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Vector (2022)]] skin will be set as the default on several more wikis, including Arabic and Catalan Wikipedias. Logged-in users will be able to switch back to the old Vector (2010). See the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/2022-04 for the largest wikis|latest update]] about Vector (2022).

'''Future meetings'''
* The next [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/Talk to Web|open meeting with the Web team]] about Vector (2022) will take place on 17 May. The following meetings are currently planned for: 7 June, 21 June, 5 July, 19 July.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/19|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W19&quot;/&gt;

15:22, 9 May 2022 (UTC)
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== Tech News: 2022-20 ==

&lt;section begin=&quot;technews-2022-W20&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/20|Translations]] are available.

'''Changes later this week'''
* Some wikis can soon use the [[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|add a link]] feature. This will start on Wednesday. The wikis are {{int:project-localized-name-cawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hiwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ptwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-simplewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-svwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ukwiki/en}}. This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T304542]
* The [[mw:Special:MyLanguage/Wikimedia Hackathon 2022|Wikimedia Hackathon 2022]] will take place online on May 20–22. It will be in English. There are also local [[mw:Special:MyLanguage/Wikimedia Hackathon 2022/Meetups|hackathon meetups]] in Germany, Ghana, Greece, India, Nigeria and the United States. Technically interested Wikimedians can work on software projects and learn new skills. You can also host a session or post a project you want to work on.
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.12|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-05-17|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-05-18|en}}. It will be on all wikis from {{#time:j xg|2022-05-19|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).

'''Future changes'''
* You can soon edit translatable pages in the visual editor. Translatable pages exist on for examples Meta and Commons. [https://diff.wikimedia.org/2022/05/12/mediawiki-1-38-brings-support-for-editing-translatable-pages-with-the-visual-editor/]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/20|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W20&quot;/&gt;

18:58, 16 May 2022 (UTC)
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== Tech News: 2022-21 ==

&lt;section begin=&quot;technews-2022-W21&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/21|Translations]] are available.

'''Recent changes'''
* Administrators using the mobile web interface can now access Special:Block directly from user pages. [https://phabricator.wikimedia.org/T307341]
* The &lt;span class=&quot;mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[https://www.wiktionary.org/ www.wiktionary.org]&lt;/span&gt; portal page now uses an automated update system. Other [[m:Project_portals|project portals]] will be updated over the next few months. [https://phabricator.wikimedia.org/T304629]

'''Problems'''
* The Growth team maintains a mentorship program for newcomers. Previously, newcomers weren't able to opt out from the program. Starting May 19, 2022, newcomers are able to fully opt out from Growth mentorship, in case they do not wish to have any mentor at all. [https://phabricator.wikimedia.org/T287915]
* Some editors cannot access the content translation tool if they load it by clicking from the contributions menu. This problem is being worked on. It should still work properly if accessed directly via Special:ContentTranslation. [https://phabricator.wikimedia.org/T308802]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.13|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-05-24|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-05-25|en}}. It will be on all wikis from {{#time:j xg|2022-05-26|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).

'''Future changes'''
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Gadget and user scripts developers are invited to give feedback on a [[mw:User:Jdlrobson/Extension:Gadget/Policy|proposed technical policy]] aiming to improve support from MediaWiki developers. [https://phabricator.wikimedia.org/T308686]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/21|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W21&quot;/&gt;

00:21, 24 May 2022 (UTC)
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== Tech News: 2022-22 ==

&lt;section begin=&quot;technews-2022-W22&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/22|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|15px|link=|Advanced item]]  In the [[mw:Special:MyLanguage/Extension:AbuseFilter|AbuseFilter]] extension, an &lt;code dir=ltr&gt;ip_in_ranges()&lt;/code&gt; function has been introduced to check if an IP is in any of the ranges. Wikis are advised to combine multiple &lt;code dir=ltr&gt;ip_in_range()&lt;/code&gt; expressions joined by &lt;code&gt;|&lt;/code&gt; into a single expression for better performance. You can use the search function on [[Special:AbuseFilter|Special:AbuseFilter]] to locate its usage. [https://phabricator.wikimedia.org/T305017]
* The [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation/IP Info feature|IP Info feature]] which helps abuse fighters access information about IPs, [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation/IP Info feature#May 24, 2022|has been deployed]] to all wikis as a beta feature. This comes after weeks of beta testing on test.wikipedia.org.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.14|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-05-31|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-06-01|en}}. It will be on all wikis from {{#time:j xg|2022-06-02|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-05-31|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s5.dblist targeted wikis]).
* The [[mw:Special:MyLanguage/Help:DiscussionTools#New topic tool|New Topic Tool]] will be deployed for all editors at most wikis soon. You will be able to opt out from within the tool and in [[Special:Preferences#mw-prefsection-editing-discussion|Preferences]]. [https://www.mediawiki.org/wiki/Special:MyLanguage/Talk_pages_project/New_discussion][https://phabricator.wikimedia.org/T287804]
* [[File:Octicons-tools.svg|15px|link=|Advanced item]]  The [[:mw:Special:ApiHelp/query+usercontribs|list=usercontribs API]] will support fetching contributions from an [[mw:Special:MyLanguage/Help:Range blocks#Non-technical explanation|IP range]] soon. API users can set the &lt;code&gt;uciprange&lt;/code&gt; parameter to get contributions from any IP range within [[:mw:Manual:$wgRangeContributionsCIDRLimit|the limit]]. [https://phabricator.wikimedia.org/T177150]
* A new parser function will be introduced: &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;{{=}}&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. It will replace existing templates named &quot;=&quot;. It will insert an [[w:en:Equals sign|equal sign]]. This can be used to escape the equal sign in the parameter values of templates. [https://phabricator.wikimedia.org/T91154]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/22|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W22&quot;/&gt;

20:28, 30 May 2022 (UTC)
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== Tech News: 2022-23 ==

&lt;section begin=&quot;technews-2022-W23&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/23|Translations]] are available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.15|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-06-07|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-06-08|en}}. It will be on all wikis from {{#time:j xg|2022-06-09|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] A new &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;str_replace_regexp()&lt;/code&gt;&lt;/bdi&gt; function can be used in [[Special:AbuseFilter|abuse filters]] to replace parts of text using a [[w:en:Regular expression|regular expression]]. [https://phabricator.wikimedia.org/T285468]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/23|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W23&quot;/&gt;

02:46, 7 June 2022 (UTC)
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== Tech News: 2022-24 ==

&lt;section begin=&quot;technews-2022-W24&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/24|Translations]] are available.

'''Recent changes'''
* All wikis can now use [[mw:Special:MyLanguage/Extension:Kartographer|Kartographer]] maps. Kartographer maps now also work on pages with [[mw:Special:MyLanguage/Help:Pending changes|pending changes]]. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Geoinformation#Project_descriptions][https://phabricator.wikimedia.org/T307348]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.16|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-06-14|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-06-15|en}}. It will be on all wikis from {{#time:j xg|2022-06-16|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-06-14|en}} at 06:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s6.dblist targeted wikis]). [https://phabricator.wikimedia.org/T300471]
* Starting on Wednesday, a new set of Wikipedias will get &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]&quot; ({{int:project-localized-name-abwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-acewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-adywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-afwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-akwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-alswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-amwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-anwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-angwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-arcwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-arzwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-astwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-atjwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-avwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-aywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-azwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-azbwiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T304548]
* The [[mw:Special:MyLanguage/Help:DiscussionTools#New topic tool|New Topic Tool]] will be deployed for all editors at Commons, Wikidata, and some other wikis soon. You will be able to opt out from within the tool and in [[Special:Preferences#mw-prefsection-editing-discussion|Preferences]]. [https://www.mediawiki.org/wiki/Special:MyLanguage/Talk_pages_project/New_discussion][https://phabricator.wikimedia.org/T287804]

'''Future meetings'''
* The next [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/Talk to Web|open meeting with the Web team]] about Vector (2022) will take place today (13 June). The following meetings will take place on: 28 June, 12 July, 26 July.

'''Future changes'''
* By the end of July, the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Vector 2022]] skin should be ready to become the default across all wikis. Discussions on how to adjust it to the communities' needs will begin in the next weeks. It will always be possible to revert to the previous version on an individual basis. [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/2022-04 for the largest wikis|Learn more]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/24|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W24&quot;/&gt;

16:58, 13 June 2022 (UTC)
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== Tech News: 2022-25 ==

&lt;section begin=&quot;technews-2022-W25&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/25|Translations]] are available.

'''Recent changes'''
* The [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android|Wikipedia App for Android]] now has an option for editing the whole page at once, located in the overflow menu (three-dots menu [[File:Ic more vert 36px.svg|15px|link=|alt=]]). [https://phabricator.wikimedia.org/T103622]
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Some recent database changes may affect queries using the [[m:Research:Quarry|Quarry tool]]. Queries for &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;site_stats&lt;/code&gt;&lt;/bdi&gt; at English Wikipedia, Commons, and Wikidata will need to be updated. [[phab:T306589|Read more]].
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] A new &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;user_global_editcount&lt;/code&gt;&lt;/bdi&gt; variable can be used in [[Special:AbuseFilter|abuse filters]] to avoid affecting globally active users. [https://phabricator.wikimedia.org/T130439]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.17|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-06-21|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-06-22|en}}. It will be on all wikis from {{#time:j xg|2022-06-23|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* Users of non-responsive skins (e.g. MonoBook or Vector) on mobile devices may notice a slight change in the default zoom level. This is intended to optimize zooming and ensure all interface elements are present on the page (for example the table of contents on Vector 2022). In the unlikely event this causes any problems with how you use the site, we'd love to understand better, please ping &lt;span class=&quot;mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[m:User:Jon (WMF)|Jon (WMF)]]&lt;/span&gt; to any on-wiki conversations. [https://phabricator.wikimedia.org/T306910]

'''Future changes'''
* The Beta Feature for [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] will be updated throughout July.  Discussions will look different.  You can see [[mw:Special:MyLanguage/Talk pages project/Usability/Prototype|some of the proposed changes]].
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Parsoid's HTML output will soon stop annotating file links with different &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;typeof&lt;/code&gt;&lt;/bdi&gt; attribute values, and instead use &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;mw:File&lt;/code&gt;&lt;/bdi&gt; for all types. Tool authors should adjust any code that expects: &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;mw:Image&lt;/code&gt;&lt;/bdi&gt;, &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;mw:Audio&lt;/code&gt;&lt;/bdi&gt;, or &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;mw:Video&lt;/code&gt;&lt;/bdi&gt;. [https://phabricator.wikimedia.org/T273505]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/25|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W25&quot;/&gt;

20:18, 20 June 2022 (UTC)
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== Tech News: 2022-26 ==

&lt;section begin=&quot;technews-2022-W26&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/26|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] [[m:Special:MyLanguage/Wikimedia Enterprise|Wikimedia Enterprise]] API service now has self-service accounts with free on-demand requests and monthly snapshots ([https://enterprise.wikimedia.com/docs/ API documentation]). Community access [[m:Special:MyLanguage/Wikimedia Enterprise/FAQ#community-access|via database dumps &amp; Wikimedia Cloud Services]] continues.
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] [[d:Special:MyLanguage/Wikidata:Wiktionary#lua|All Wikimedia wikis can now use Wikidata Lexemes in Lua]] after creating local modules and templates. Discussions are welcome [[d:Wikidata_talk:Lexicographical_data#You_can_now_reuse_Wikidata_Lexemes_on_all_wikis|on the project talk page]].

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.18|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-06-28|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-06-29|en}}. It will be on all wikis from {{#time:j xg|2022-06-30|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-06-28|en}} at 06:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s7.dblist targeted wikis]). [https://phabricator.wikimedia.org/T311033]
*  Some global and cross-wiki services will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-06-30|en}} at 06:00 UTC. This will impact ContentTranslation, Echo, StructuredDiscussions, Growth experiments and a few more services. [https://phabricator.wikimedia.org/T300472]
* Users will be able to sort columns within sortable tables in the mobile skin. [https://phabricator.wikimedia.org/T233340]

'''Future meetings'''
* The next [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/Talk to Web|open meeting with the Web team]] about Vector (2022) will take place tomorrow (28 June). The following meetings will take place on 12 July and 26 July.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/26|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W26&quot;/&gt;

20:02, 27 June 2022 (UTC)
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== Tech News: 2022-27 ==

&lt;section begin=&quot;technews-2022-W27&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/27|Translations]] are available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.19|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-07-05|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-07-06|en}}. It will be on all wikis from {{#time:j xg|2022-07-07|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-07-05|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s6.dblist targeted wikis]) and on {{#time:j xg|2022-07-07|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s4.dblist targeted wikis]).
* The Beta Feature for [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] will be updated throughout July.  Discussions will look different.  You can see [[mw:Special:MyLanguage/Talk pages project/Usability/Prototype|some of the proposed changes]].
* [[File:Octicons-tools.svg|15px|link=|alt=| Advanced item]] This change only affects pages in the main namespace in Wikisource. The Javascript config variable &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;proofreadpage_source_href&lt;/code&gt;&lt;/bdi&gt; will be removed from &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;[[mw:Special:MyLanguage/Manual:Interface/JavaScript#mw.config|mw.config]]&lt;/code&gt;&lt;/bdi&gt; and be replaced with the variable &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;prpSourceIndexPage&lt;/code&gt;&lt;/bdi&gt;.  [https://phabricator.wikimedia.org/T309490]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/27|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W27&quot;/&gt;

19:32, 4 July 2022 (UTC)
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== Tech News: 2022-28 ==

&lt;section begin=&quot;technews-2022-W28&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/28|Translations]] are available.

'''Recent changes'''
* In the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Vector 2022 skin]], the page title is now displayed above the tabs such as Discussion, Read, Edit, View history, or More. [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates#Page title/tabs switch|Learn more]]. [https://phabricator.wikimedia.org/T303549]
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] It is now possible to easily view most of the configuration settings that apply to just one wiki, and to compare settings between two wikis if those settings are different. For example: [https://noc.wikimedia.org/wiki.php?wiki=jawiktionary Japanese Wiktionary settings], or [https://noc.wikimedia.org/wiki.php?wiki=eswiki&amp;compare=eowiki settings that are different between the Spanish and Esperanto Wikipedias]. Local communities may want to [[m:Special:MyLanguage/Requesting_wiki_configuration_changes|discuss and propose changes]] to their local settings. Details about each of the named settings can be found by [[mw:Special:Search|searching MediaWiki.org]]. [https://phabricator.wikimedia.org/T308932]
*The Anti-Harassment Tools team [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation/IP Info feature#May|recently deployed]] the IP Info Feature as a [[Special:Preferences#mw-prefsection-betafeatures|Beta Feature at all wikis]]. This feature allows abuse fighters to access information about IP addresses. Please check our update on [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation/IP Info feature#April|how to find and use the tool]]. Please share your feedback using a link you will be given within the tool itself.

'''Changes later this week'''
* There is no new MediaWiki version this week.
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-07-12|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s3.dblist targeted wikis]).

'''Future changes'''
* The Beta Feature for [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] will be updated throughout July.  Discussions will look different.  You can see [[mw:Special:MyLanguage/Talk pages project/Usability/Prototype|some of the proposed changes]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/28|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W28&quot;/&gt;

19:24, 11 July 2022 (UTC)
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== Tech News: 2022-29 ==

&lt;section begin=&quot;technews-2022-W29&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/29|Translations]] are available.

'''Problems'''
* The feature on mobile web for [[mw:Special:MyLanguage/Extension:NearbyPages|Nearby Pages]] was missing last week. It will be fixed this week. [https://phabricator.wikimedia.org/T312864]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.21|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-07-19|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-07-20|en}}. It will be on all wikis from {{#time:j xg|2022-07-21|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).

'''Future changes'''
* The [[mw:Technical_decision_making/Forum|Technical Decision Forum]] is seeking [[mw:Technical_decision_making/Community_representation|community representatives]]. You can apply on wiki or by emailing &lt;span class=&quot;mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;TDFSupport@wikimedia.org&lt;/span&gt; before 12 August.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/29|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W29&quot;/&gt;

22:59, 18 July 2022 (UTC)
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== Tech News: 2022-30 ==

&lt;section begin=&quot;technews-2022-W30&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/30|Translations]] are available.

'''Recent changes'''
* The &lt;span class=&quot;mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[https://www.wikibooks.org/ www.wikibooks.org]&lt;/span&gt; and &lt;span class=&quot;mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[https://www.wikiquote.org/ www.wikiquote.org]&lt;/span&gt; portal pages now use an automated update system. Other [[m:Project_portals|project portals]] will be updated over the next few months. [https://phabricator.wikimedia.org/T273179]

'''Problems'''
*  Last week, some wikis were in read-only mode for a few minutes because of an emergency switch of their main database ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s7.dblist targeted wikis]). [https://phabricator.wikimedia.org/T313383]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.22|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-07-26|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-07-27|en}}. It will be on all wikis from {{#time:j xg|2022-07-28|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* The external link icon will change slightly in the skins Vector legacy and Vector 2022. The new icon uses simpler shapes to be more recognizable on low-fidelity screens. [https://phabricator.wikimedia.org/T261391]
* Administrators will now see buttons on user pages for &quot;{{int:changeblockip}}&quot; and &quot;{{int:unblockip}}&quot; instead of just &quot;{{int:blockip}}&quot; if the user is already blocked. [https://phabricator.wikimedia.org/T308570]

'''Future meetings'''
* The next [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/Talk to Web|open meeting with the Web team]] about Vector (2022) will take place tomorrow (26 July).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/30|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W30&quot;/&gt;

19:27, 25 July 2022 (UTC)
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== Tech News: 2022-31 ==

&lt;section begin=&quot;technews-2022-W31&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/31|Translations]] are available.

'''Recent changes'''
* Improved [[m:Special:MyLanguage/Help:Displaying_a_formula#Phantom|LaTeX capabilities for math rendering]] are now available in the wikis thanks to supporting &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;Phantom&lt;/code&gt;&lt;/bdi&gt; tags. This completes part of [[m:Community_Wishlist_Survey_2022/Editing/Missing_LaTeX_capabilities_for_math_rendering|the #59 wish]] of the 2022 Community Wishlist Survey.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.23|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-08-02|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-08-03|en}}. It will be on all wikis from {{#time:j xg|2022-08-04|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* The [[mw:Special:MyLanguage/Help:Extension:WikiEditor/Realtime_Preview|Realtime Preview]] will be available as a Beta Feature on wikis in [https://noc.wikimedia.org/conf/highlight.php?file=dblists%2Fgroup0.dblist Group 0]. This feature was built in order to fulfill [[m:Special:MyLanguage/Community_Wishlist_Survey_2021/Real_Time_Preview_for_Wikitext|one of the Community Wishlist Survey proposals]].

'''Future changes'''
* The Beta Feature for [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] will be updated throughout August.  Discussions will look different.  You can see [[mw:Special:MyLanguage/Talk pages project/Usability/Prototype|some of the proposed changes]].

'''Future meetings'''
* This week, three meetings about [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Vector (2022)]] with live interpretation will take place. On Tuesday, interpretation in Russian will be provided. On Thursday, meetings for Arabic and Spanish speakers will take place. [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/Talk to Web|See how to join]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/31|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W31&quot;/&gt;

21:21, 1 August 2022 (UTC)
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== Tech News: 2022-32 ==

&lt;section begin=&quot;technews-2022-W32&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/32|Translations]] are available.

'''Recent changes'''
* [[:m:Special:MyLanguage/Meta:GUS2Wiki/Script|GUS2Wiki]] copies the information from [[{{#special:GadgetUsage}}]] to an on-wiki page so you can review its history. If your project isn't already listed on the [[d:Q113143828|Wikidata entry for Project:GUS2Wiki]] you can either run GUS2Wiki yourself or [[:m:Special:MyLanguage/Meta:GUS2Wiki/Script#Opting|make a request to receive updates]]. [https://phabricator.wikimedia.org/T121049]

'''Changes later this week'''
* There is no new MediaWiki version this week.
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-08-09|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s5.dblist targeted wikis]) and on {{#time:j xg|2022-08-11|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s2.dblist targeted wikis]).

'''Future meetings'''
* The [[wmania:Special:MyLanguage/Hackathon|Wikimania Hackathon]] will take place online from August 12–14. Don't miss [[wmania:Special:MyLanguage/Hackathon/Schedule|the pre-hacking showcase]] to learn about projects and find collaborators. Anyone can [[phab:/project/board/6030/|propose a project]] or [[wmania:Special:MyLanguage/Hackathon/Schedule|host a session]]. [[wmania:Special:MyLanguage/Hackathon/Newcomers|Newcomers are welcome]]!

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/32|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W32&quot;/&gt;

19:50, 8 August 2022 (UTC)
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== Tech News: 2022-33 ==

&lt;section begin=&quot;technews-2022-W33&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/33|Translations]] are available.

'''Recent changes'''
* The Persian (Farsi) Wikipedia community decided to block IP editing from October 2021 to April 2022. The Wikimedia Foundation's Product Analytics team tracked the impact of this change. [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation/IP Editing Restriction Study/Farsi Wikipedia|An impact report]] is now available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.25|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-08-16|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-08-17|en}}. It will be on all wikis from {{#time:j xg|2022-08-18|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-08-16|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s1.dblist targeted wikis]) and on {{#time:j xg|2022-08-18|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s8.dblist targeted wikis]).
* The [[mw:Special:MyLanguage/Help:Extension:WikiEditor/Realtime_Preview|Realtime Preview]] will be available as a Beta Feature on wikis in [https://noc.wikimedia.org/conf/highlight.php?file=dblists%2Fgroup1.dblist Group 1]. This feature was built in order to fulfill [[m:Special:MyLanguage/Community_Wishlist_Survey_2021/Real_Time_Preview_for_Wikitext|one of the Community Wishlist Survey proposals]].

'''Future changes'''
* The Beta Feature for [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] will be updated throughout August.  Discussions will look different.  You can see [[mw:Special:MyLanguage/Talk pages project/Usability/Prototype|some of the proposed changes]]. [https://www.mediawiki.org/wiki/Talk_pages_project/Usability#4_August_2022][https://www.mediawiki.org/wiki/Talk_pages_project/Usability#Phase_1:_Topic_containers][https://phabricator.wikimedia.org/T312672]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/33|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W33&quot;/&gt;

21:08, 15 August 2022 (UTC)
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== Tech News: 2022-34 ==

&lt;section begin=&quot;technews-2022-W34&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/34|Translations]] are available.

'''Recent changes'''
* Two problems with [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] maps have been fixed. Maps are no longer shown as empty when a geoline was created via VisualEditor. Geolines consisting of points with QIDs (e.g., subway lines) are no longer shown with pushpins. [https://phabricator.wikimedia.org/T292613][https://phabricator.wikimedia.org/T308560]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.26|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-08-23|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-08-24|en}}. It will be on all wikis from {{#time:j xg|2022-08-25|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-08-25|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s4.dblist targeted wikis]).
* The colours of links and visited links will change. This is to make the difference between links and other text more clear. [https://phabricator.wikimedia.org/T213778]

'''Future changes'''
* The new [{{int:discussiontools-topicsubscription-button-subscribe}}] button [[mw:Talk pages project/Notifications#12 August 2022|helps newcomers get answers]].  The Editing team is enabling this tool everywhere.  You can turn it off in [[Special:Preferences#mw-prefsection-editing-discussion|your preferences]]. [https://phabricator.wikimedia.org/T284489]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/34|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W34&quot;/&gt;

00:12, 23 August 2022 (UTC)
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== Tech News: 2022-35 ==

&lt;section begin=&quot;technews-2022-W35&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/35|Translations]] are available.

'''Recent changes'''
* The [[mw:Special:MyLanguage/Help:Extension:WikiEditor/Realtime_Preview|Realtime Preview]] is available as a Beta Feature on wikis in [https://noc.wikimedia.org/conf/highlight.php?file=dblists%2Fgroup2.dblist Group 2]. This feature was built in order to fulfill [[m:Special:MyLanguage/Community_Wishlist_Survey_2021/Real_Time_Preview_for_Wikitext|one of the Community Wishlist Survey proposals]]. Please note that when this Beta feature is enabled, it may cause conflicts with some wiki-specific Gadgets.

'''Problems'''
* In recent months, there have been inaccurate numbers shown for various [[{{#special:statistics}}]] at Commons, Wikidata, and English Wikipedia. This has now been fixed. [https://phabricator.wikimedia.org/T315693]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.27|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-08-30|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-08-31|en}}. It will be on all wikis from {{#time:j xg|2022-09-01|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-08-30|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s6.dblist targeted wikis]) and on {{#time:j xg|2022-09-01|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s7.dblist targeted wikis]).

'''Future changes'''
* The Wikimedia Foundation wants to improve how Wikimedia communities report harmful incidents by building the [[m:Special:MyLanguage/Private Incident Reporting System|Private Incident Reporting System (PIRS)]] to make it easy and safe for users to make reports. You can leave comments on the talk page, by answering the [[m:Special:MyLanguage/Private Incident Reporting System#Phase 1|questions provided]].  If you have ever faced a harmful situation that you wanted to report/reported, join a PIRS interview to share your experience. To sign up [[m:Special:EmailUser/MAna_(WMF)|please email]] &lt;span class=&quot;mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[m:User:MAna (WMF)|Madalina Ana]]&lt;/span&gt;.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/35|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W35&quot;/&gt;

23:05, 29 August 2022 (UTC)
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== Tech News: 2022-36 ==

&lt;section begin=&quot;technews-2022-W36&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/36|Translations]] are available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.39/wmf.28|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-09-06|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-09-07|en}}. It will be on all wikis from {{#time:j xg|2022-09-08|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-09-06|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s1.dblist targeted wikis]) and on {{#time:j xg|2022-09-08|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s3.dblist targeted wikis]).
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] On Special pages that only have one tab, the tab-bar's row will be hidden in the Vector-2022 skin to save space. The row will still show if Gadgets use it. Gadgets that currently append directly to the CSS id of &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;#p-namespaces&lt;/code&gt;&lt;/bdi&gt; should be updated to use the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;[[mw:ResourceLoader/Core_modules#addPortletLink|mw.util.addPortletLink]]&lt;/code&gt;&lt;/bdi&gt; function instead. Gadgets that style this id should consider also targeting &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;#p-associated-pages&lt;/code&gt;&lt;/bdi&gt;, the new id for this row. [[phab:T316908|Examples are available]]. [https://phabricator.wikimedia.org/T316908][https://phabricator.wikimedia.org/T313409]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/36|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W36&quot;/&gt;

23:22, 5 September 2022 (UTC)
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== Tech News: 2022-37 ==

&lt;section begin=&quot;technews-2022-W37&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/37|Translations]] are available.

'''Recent changes'''
* The search servers have been upgraded to a new major version. If you notice any issues with searching, please report them on [[phab:project/view/1849/|Phabricator]]. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/message/XPCTYYTN67FVFKN6XOHULJVGUO44J662]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.1|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-09-13|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-09-14|en}}. It will be on all wikis from {{#time:j xg|2022-09-15|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* [[mw:Special:MyLanguage/Extension:SyntaxHighlight|Syntax highlighting]] is now tracked as an [[mw:Special:MyLanguage/Manual:$wgExpensiveParserFunctionLimit|expensive parser function]]. Only 500 expensive function calls can be used on a single page. Pages that exceed the limit are added to a [[:Category:{{MediaWiki:expensive-parserfunction-category}}|tracking category]]. [https://phabricator.wikimedia.org/T316858]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/37|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W37&quot;/&gt;

01:50, 13 September 2022 (UTC)
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== Tech News: 2022-38 ==

&lt;section begin=&quot;technews-2022-W38&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/38|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Two database fields in the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;templatelinks&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; table are now being dropped: &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;tl_namespace&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;tl_title&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. Any queries that rely on these fields need to be changed to use the new normalization field called &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;tl_target_id&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. See &lt;span class=&quot;mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[phab:T299417|T299417]]&lt;/span&gt; for more information. This is part of [[w:Database normalization|normalization]] of links tables. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/message/U2U6TXIBABU3KDCVUOITIGI5OJ4COBSW/][https://www.mediawiki.org/wiki/User:ASarabadani_(WMF)/Database_for_devs_toolkit/Concepts/Normalization]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.2|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-09-20|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-09-21|en}}. It will be on all wikis from {{#time:j xg|2022-09-22|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* In [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] maps, you can use icons on markers for common points of interest. On Tuesday, the [[mw:Special:MyLanguage/Help:Extension:Kartographer/Icons|previous icon set]] will be updated to [https://de.wikipedia.beta.wmflabs.org/wiki/Hilfe:Extension:Kartographer/Icons version maki 7.2]. That means, around 100 new icons will be available.  Additionally, all existing icons were updated for clarity and to make them work better in international contexts. [https://phabricator.wikimedia.org/T302861][https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Geoinformation#Update_maki_icons]

'''Future changes'''
* In a [[m:Content_Partnerships_Hub/Software/Volunteer_developers_discussion_at_Wikimania_2022|group discussion at Wikimania]], more than 30 people talked about how to make content partnership software in the Wikimedia movement more sustainable. What kind of support is acceptable for volunteer developers? Read the summary and [[m:Talk:Content Partnerships Hub/Software/Volunteer developers discussion at Wikimania 2022|leave your feedback]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/38|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W38&quot;/&gt;

&lt;span class=&quot;mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/span&gt; 22:16, 19 September 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23826293 --&gt;

== Tech News: 2022-39 ==

&lt;section begin=&quot;technews-2022-W39&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/39|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Parsoid clients should be updated to allow for space-separated multi-values in the &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;rel&lt;/code&gt;&lt;/bdi&gt; attribute of links. Further details are in &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[phab:T315209|T315209]]&lt;/bdi&gt;.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.3|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-09-27|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-09-28|en}}. It will be on all wikis from {{#time:j xg|2022-09-29|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* [[mw:Special:MyLanguage/VisualEditor/Diffs|Visual diffs]] will become available to all users, except at the Wiktionaries and Wikipedias. [https://phabricator.wikimedia.org/T314588]
* [[mw:Special:MyLanguage/Help:DiscussionTools#Mobile|Talk pages on the mobile site]] will change at the Arabic, Bangla, Chinese, French, Haitian Creole, Hebrew, Korean, and Vietnamese Wikipedias.  They should be easier to use and provide more information. [https://phabricator.wikimedia.org/T318302] [https://www.mediawiki.org/wiki/Talk_pages_project/Mobile]
* In the [[mw:Lua/Scripting|{{ns:828}}]] namespace, pages ending with &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;.json&lt;/code&gt;&lt;/bdi&gt; will be treated as JSON, just like they already are in the {{ns:2}} and {{ns:8}} namespaces. [https://phabricator.wikimedia.org/T144475]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/39|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W39&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:30, 27 September 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23860085 --&gt;

== Tech News: 2022-40 ==

&lt;section begin=&quot;technews-2022-W40&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/40|Translations]] are available.

'''Recent changes'''
* [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] maps can now show geopoints from Wikidata, via QID or SPARQL query. Previously, this was only possible for geoshapes and geolines. [https://phabricator.wikimedia.org/T307695] [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Geoinformation/Geopoints_via_QID]
* The [[m:Special:MyLanguage/Coolest_Tool_Award|Coolest Tool Award 2022]] is looking for nominations. You can recommend tools until 12 October.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.4|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-10-04|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-10-05|en}}. It will be on all wikis from {{#time:j xg|2022-10-06|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* [[mw:Special:MyLanguage/Help:DiscussionTools#Mobile|Talk pages on the mobile site]] will change at the Arabic, Bangla, Chinese, French, Haitian Creole, Hebrew, Korean, and Vietnamese Wikipedias.  They should be easier to use and provide more information. (Last week's release was delayed) [https://phabricator.wikimedia.org/T318302] [https://www.mediawiki.org/wiki/Talk_pages_project/Mobile]
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] The &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;scribunto-console&lt;/code&gt;&lt;/bdi&gt; API module will require a [[mw:Special:MyLanguage/API:Tokens|CSRF token]]. This module is documented as internal and use of it is not supported. [[phab:T212071|[5]]] 
* The Vector 2022 skin will become the default across the smallest Wikimedia projects. [[mw:Special:MyLanguage/Reading/Web/Desktop_Improvements#Deployment_plan_and_timeline|Learn more]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/40|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W40&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:23, 4 October 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23885489 --&gt;

== Tech News: 2022-41 ==

&lt;section begin=&quot;technews-2022-W41&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/41|Translations]] are available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.5|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-10-11|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-10-12|en}}. It will be on all wikis from {{#time:j xg|2022-10-13|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* On some wikis, [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] maps in full size view will be able to display nearby articles. After a feedback period, more wikis will follow. [https://phabricator.wikimedia.org/T316782][https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Geoinformation/Nearby_articles]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/41|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W41&quot;/&gt;

14:08, 10 October 2022 (UTC)
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== Tech News: 2022-42 ==

&lt;section begin=&quot;technews-2022-W42&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/42|Translations]] are available.

'''Recent changes'''
* The recently implemented feature of [[phab:T306883|article thumbnails in Special:Search]] will be limited to Wikipedia projects only. Further details are in [[phab:T320510|T320510]]. [https://www.mediawiki.org/wiki/Special:MyLanguage/Structured_Data_Across_Wikimedia/Search_Improvements]
* A bug that caused problems in loading article thumbnails in Special:Search has been fixed. Further details are in [[phab:T320406|T320406]].

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.6|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-10-18|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-10-19|en}}. It will be on all wikis from {{#time:j xg|2022-10-20|en}} ([[mw:MediaWiki 1.39/Roadmap|calendar]]).
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Lua module authors can use &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;[[mw:Special:MyLanguage/Extension:Scribunto/Lua_reference_manual#mw.loadJsonData|mw.loadJsonData()]]&lt;/code&gt;&lt;/bdi&gt; to load data from JSON pages. [https://phabricator.wikimedia.org/T217500]
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Lua module authors can enable &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;[[mw:Special:MyLanguage/Extension:Scribunto/Lua_reference_manual#Strict_library|require( &quot;strict&quot; )]]&lt;/code&gt;&lt;/bdi&gt; to add errors for some possible code problems. This replaces &quot;[[wikidata:Q16748603|Module:No globals]]&quot; on most wikis. [https://phabricator.wikimedia.org/T209310]

'''Future changes'''
* The [[Special:Preferences#mw-prefsection-betafeatures|Beta Feature]] for [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] will be updated at most wikis. The &quot;{{int:discussiontools-replylink}}&quot; button will look different after this change. [https://phabricator.wikimedia.org/T320683]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/42|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W42&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:46, 17 October 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23943992 --&gt;

== Tech News: 2022-43 ==

&lt;section begin=&quot;technews-2022-W43&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/43|Translations]] are available.

'''Recent changes'''
* There have been some minor visual fixes in Special:Search, regarding audio player alignment and image placeholder height. Further details are in [[phab:T319230|T319230]].
* On Wikipedias, a new [[Special:Preferences#mw-prefsection-searchoptions|preference]] has been added to hide article thumbnails in Special:Search. Full details are in [[phab:T320337|T320337]].

'''Problems'''
* Last week, three wikis ({{int:project-localized-name-frwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-jawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ruwiki/en}}) had read-only access for 25 minutes. This was caused by a hardware problem. [https://phabricator.wikimedia.org/T320990]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.7|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-10-25|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-10-26|en}}. It will be on all wikis from {{#time:j xg|2022-10-27|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-10-25|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s5.dblist targeted wikis]) and on {{#time:j xg|2022-10-27|en}} at 7:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s4.dblist targeted wikis]).
* Starting on Wednesday, a new set of Wikipedias will get &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]&quot; ({{int:project-localized-name-aswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-banwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-barwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bat smgwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bclwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-be x oldwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bgwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bhwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-biwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bjnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bmwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bpywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-brwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bugwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bxrwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-idwiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T304549] 
* Starting on Wednesday October 26, 2022, the list of mentors will be upgraded [[d:Q14339834 | at wikis where Growth mentorship is available]]. The mentorship system will continue to work as it does now. The signup process [[mw:Special:MyLanguage/Growth/Communities/How to configure the mentors' list#add|will be replaced]], and a new management option will be provided. Also, this change simplifies [[mw:Special:MyLanguage/Growth/Communities/How to configure the mentors' list#create|the creation of mentorship systems at Wikipedias]]. [https://phabricator.wikimedia.org/T314858][https://phabricator.wikimedia.org/T310905][https://www.mediawiki.org/wiki/Special:MyLanguage/Growth/Structured_mentor_list]
* Pages with titles that start with a lower-case letter according to Unicode 11 will be renamed or deleted. There is a list of affected pages at &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[m:Unicode 11 case map migration]]&lt;/bdi&gt;. More information can be found at [[phab:T292552|T292552]].
* The Vector 2022 skin will become the default across the smallest Wikipedias. [[mw:Special:MyLanguage/Reading/Web/Desktop_Improvements#smallest-1|Learn more]].

'''Future changes'''
* The [[mw:Special:MyLanguage/Talk pages project/Replying|Reply tool]] and [[mw:Special:MyLanguage/Talk pages project/New discussion|New Topic tool]] will soon get a [[mw:Special:MyLanguage/VisualEditor/Special characters|special characters menu]]. [https://phabricator.wikimedia.org/T249072]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/43|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W43&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:22, 24 October 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23975411 --&gt;

== Tech News: 2022-44 ==

&lt;section begin=&quot;technews-2022-W44&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/44|Translations]] are available.

'''Recent changes'''
* When using keyboard navigation on a [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] map, the focus will become more visible. [https://phabricator.wikimedia.org/T315997]
* In {{#special:RecentChanges}}, you can now hide the log entries for new user creations with the filter for &quot;{{int:rcfilters-filter-newuserlogactions-label}}&quot;. [https://phabricator.wikimedia.org/T321155]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.8|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-11-01|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-11-02|en}}. It will be on all wikis from {{#time:j xg|2022-11-03|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* The [[mw:Special:MyLanguage/Help:Extension:Kartographer|maps dialog]] in VisualEditor now has some help texts. [https://phabricator.wikimedia.org/T318818]
* It is now possible to select the language of a [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] map in VisualEditor via a dropdown menu. [https://phabricator.wikimedia.org/T318817]
* It is now possible to add a caption to a [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] map in VisualEditor. [https://phabricator.wikimedia.org/T318815]
* It is now possible to hide the frame of a [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] map in VisualEditor. [https://phabricator.wikimedia.org/T318813]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/44|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W44&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:15, 31 October 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=23977539 --&gt;

== Tech News: 2022-45 ==

&lt;section begin=&quot;technews-2022-W45&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/45|Translations]] are available.

'''Recent changes'''
* An updated version of the [[m:Special:MyLanguage/EventCenter/Registration|Event Registration]] tool is now available for testing at [[testwiki:|testwiki]] and [[test2wiki:| test2wiki]]. The tool provides features for event organizers and participants. Your feedback is welcome at our [[m:Talk:Campaigns/Foundation Product Team/Registration|project talkpage]]. More information about [[m:Campaigns/Foundation Product Team/Registration|the project]] is available. [https://phabricator.wikimedia.org/T318592]

'''Problems'''
* Twice last week, for about 45 minutes, some files and thumbnails failed to load and uploads failed, mostly for logged-in users. The cause is being investigated and an incident report will be available soon.

'''Changes later this week'''
* There is no new MediaWiki version this week.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/45|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W45&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:32, 8 November 2022 (UTC)
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== Tech News: 2022-46 ==

&lt;section begin=&quot;technews-2022-W46&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/46|Translations]] are available.

'''Recent changes'''
* At Wikidata, an interwiki link can now point to a redirect page if certain conditions are met. This new feature is called [[wikidata:Special:MyLanguage/Wikidata:Sitelinks_to_redirects|sitelinks to redirects]]. It is needed when one wiki uses one page to cover multiple concepts but another wiki uses more pages to cover the same concepts. Your [[wikidata:Special:MyLanguage/Wikidata talk:Sitelinks to redirects|feedback on the talkpage]] of the new proposed guideline is welcome. [https://phabricator.wikimedia.org/T278962]
* The &lt;span class=&quot;mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[https://www.wikinews.org/ www.wikinews.org]&lt;/span&gt;, &lt;span class=&quot;mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[https://www.wikiversity.org/ www.wikiversity.org]&lt;/span&gt;, and &lt;span class=&quot;mw-content-ltr&quot; lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[https://www.wikivoyage.org/ www.wikivoyage.org]&lt;/span&gt; portal pages now use an automated update system. [https://phabricator.wikimedia.org/T273179]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.10|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-11-15|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-11-16|en}}. It will be on all wikis from {{#time:j xg|2022-11-17|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* There will be a new link to directly &quot;Edit template data&quot; on Template pages. [https://phabricator.wikimedia.org/T316759]

'''Future changes'''
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Wikis where mobile [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] are enabled ([[mw:Special:MyLanguage/Talk pages project/Deployment Status|these ones]]) will soon use full CSS styling to display any templates that are placed at the top of talk pages. To adapt these “talk page boxes” for narrow mobile devices you can use media queries, such as in [https://en.wikipedia.org/w/index.php?title=Module:Message_box/tmbox.css&amp;oldid=1097618699#L-69 this example]. [https://phabricator.wikimedia.org/T312309]
* Starting in January 2023, [[m:Special:MyLanguage/Community Tech|Community Tech]] will be [[m:Special:MyLanguage/Community Wishlist Survey/Updates/2023 Changes Update|running the Community Wishlist Survey (CWS) every two years]]. This means that in 2024, there will be no new proposals or voting.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/46|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W46&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:54, 14 November 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24071290 --&gt;

== Tech News: 2022-47 ==

&lt;section begin=&quot;technews-2022-W47&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/47|Translations]] are available.

'''Recent changes'''
* The display of non-free media in the search bar and for article thumbnails in Special:Search has been deactivated. Further details are in [[phab:T320661|T320661]].

'''Changes later this week'''
* There is no new MediaWiki version this week.
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-11-22|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s2.dblist targeted wikis]) and on {{#time:j xg|2022-11-24|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s7.dblist targeted wikis]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/47|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W47&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:22, 21 November 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24071290 --&gt;

== Tech News: 2022-48 ==

&lt;section begin=&quot;technews-2022-W48&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/48|Translations]] are available.

'''Recent changes'''
* A new preference, “Enable limited width mode”, has been added to the [[Special:Preferences#mw-prefsection-rendering|Vector 2022 skin]]. The preference is also available as a toggle on every page if your monitor is 1600 pixels or wider. It allows for increasing the width of the page for logged-out and logged-in users. [https://phabricator.wikimedia.org/T319449]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.12|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-11-29|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-11-30|en}}. It will be on all wikis from {{#time:j xg|2022-12-01|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2022-11-29|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s3.dblist targeted wikis]) and on {{#time:j xg|2022-12-01|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s1.dblist targeted wikis]).
* Mathematical formulas shown in SVG image format will no longer have PNG fall-backs for browsers that don't support them. This is part of work to modernise the generation system. Showing only PNG versions was the default option until in February 2018. [https://lists.wikimedia.org/hyperkitty/list/wikimedia-l@lists.wikimedia.org/message/3BGOKWJIZGL4TC4HJ22ICRU2SEPWGCR4/][https://phabricator.wikimedia.org/T311620][https://phabricator.wikimedia.org/T186327]
* On [[phab:P40224|some wikis]] that use flagged revisions, [[mw:Special:MyLanguage/Help:Extension:FlaggedRevs#Special:Contributions|a new checkbox will be added]] to Special:Contributions that enables you to see only the [[mw:Special:MyLanguage/Help:Pending changes|pending changes]] by a user. [https://phabricator.wikimedia.org/T321445]

'''Future changes'''
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] How media is structured in the parser's HTML output will change early next week at [https://wikitech.wikimedia.org/wiki/Deployments/Train#Wednesday group1 wikis] (but not Wikimedia Commons or Meta-Wiki). This change improves the accessibility of content, and makes it easier to write related CSS. You may need to update your site-CSS, or userscripts and gadgets. There are [[mw:Special:MyLanguage/Parsoid/Parser_Unification/Media_structure/FAQ|details on what code to check, how to update the code, and where to report any related problems]]. [https://phabricator.wikimedia.org/T314318]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/48|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W48&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:03, 28 November 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24114342 --&gt;

== Tech News: 2022-49 ==

&lt;section begin=&quot;technews-2022-W49&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/49|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] The Wikisources use a tool called ProofreadPage. ProofreadPage uses OpenSeadragon which is an open source tool. The OpenSeadragon JavaScript API has been significantly re-written to support dynamically loading images. The functionality provided by the older version of the API should still work but it is no longer supported. User scripts and gadgets should migrate over to the newer version of the API. The functionality provided by the newer version of the API is [[mw:Extension:Proofread_Page/Page_viewer#JS_API|documented on MediaWiki]]. [https://phabricator.wikimedia.org/T308098][https://www.mediawiki.org/wiki/Extension:Proofread_Page/Edit-in-Sequence]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.13|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-12-06|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-12-07|en}}. It will be on all wikis from {{#time:j xg|2022-12-08|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/49|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W49&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:41, 6 December 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24151590 --&gt;

== Tech News: 2022-50 ==

&lt;section begin=&quot;technews-2022-W50&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/50|Translations]] are available.

'''Recent changes'''
* An [[mw:Special:MyLanguage/Talk pages project/Mobile|A/B test has begun]] at 15 Wikipedias for [[mw:Special:MyLanguage/Help:DiscussionTools#Mobile|DiscussionTools on mobile]].  Half of the editors on the [[mw:Reading/Web/Mobile|mobile web site]] will have access to the {{int:discussiontools-replybutton}} tool and other features. [https://phabricator.wikimedia.org/T321961]
* The character &lt;code&gt;=&lt;/code&gt; cannot be used in new usernames, to make usernames work better with templates. Existing usernames are not affected. [https://phabricator.wikimedia.org/T254045]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.14|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2022-12-13|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2022-12-14|en}}. It will be on all wikis from {{#time:j xg|2022-12-15|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] The HTML markup used by [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] to [[mw:Special:MyLanguage/Talk_pages_project/Usability#Phase_1:_Topic_containers|show discussion metadata below section headings]] will be inserted after these headings, not inside of them. This change improves the accessibility of discussion pages for screen reader software. [https://phabricator.wikimedia.org/T314714]

'''Events'''
* The fourth edition of the [[m:Special:MyLanguage/Coolest_Tool_Award|Coolest Tool Award]] will happen online on [https://zonestamp.toolforge.org/1671210002 Friday 16 December 2022 at 17:00 UTC]! The event will be live-streamed on YouTube in the [https://www.youtube.com/user/watchmediawiki MediaWiki channel] and added to Commons afterwards.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/50|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W50&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:34, 12 December 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24216570 --&gt;

== Tech News: 2022-51 ==

&lt;section begin=&quot;technews-2022-W51&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2022/51|Translations]] are available.

'''Tech News'''
* Because of the [[w:en:Christmas and holiday season|holidays]] the next issue of Tech News will be sent out on 9 January 2023.

'''Recent changes'''
* On a user's contributions page, you can filter it for edits with a tag like 'reverted'. Now, you can also filter for all edits that are not tagged like that. This was part of a Community Wishlist 2022 request. [https://phabricator.wikimedia.org/T119072]
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] A new function has been used for gadget developers to add content underneath the title on article pages. This is considered a stable API that should work across all skins. [[mw:Special:MyLanguage/ResourceLoader/Core_modules#addSubtitle|Documentation is available]]. [https://phabricator.wikimedia.org/T316830]
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] [[test2wiki:|One of our test wikis]] is now being served from a new infrastructure powered by [[w:Kubernetes|Kubernetes]] ([[wikitech:MediaWiki On Kubernetes|read more]]). More Wikis will switch to this new infrastructure in early 2023. Please test and let us know of any issues. [https://phabricator.wikimedia.org/T290536]

'''Problems'''
* Last week, all wikis had no edit access for 9 minutes. This was caused by a database problem. [https://wikitech.wikimedia.org/wiki/Incidents/2022-12-13_sessionstore]

'''Changes later this week'''
* There is no new MediaWiki version this week or next week.
* The word &quot;{{int:discussiontools-replybutton}}&quot; is very short in some languages, such as Arabic (&quot;&lt;bdi lang=&quot;ar&quot;&gt;ردّ&lt;/bdi&gt;&quot;). This makes the {{int:discussiontools-preference-label}} button on talk pages difficult to use. An arrow icon will be added to those languages. This will only be visible to editors who have the [[Special:Preferences#mw-prefsection-betafeatures|Beta Feature]] turned on. [https://www.mediawiki.org/wiki/Talk_pages_project/Usability#Status] [https://phabricator.wikimedia.org/T323537]

'''Future changes'''
* Edits can be automatically &quot;tagged&quot; by the system software or the {{int:Abusefilter}} system. Those tags link to a help page about the tags. Soon they will also link to Recent Changes to let you see other edits tagged this way. This was a Community Wishlist 2022 request. [https://phabricator.wikimedia.org/T301063]
* The Trust &amp; Safety tools team [[m:Special:MyLanguage/Private Incident Reporting System/Timeline and Updates|have shared new plans]] for building the Private Incident Reporting System. The system will make it easier for editors to ask for help if they are harassed or abused. 
* [[m:Special:MyLanguage/Community Wishlist Survey 2021/Real Time Preview for Wikitext|Realtime Preview for Wikitext]] is coming out of beta as an enabled feature for every user of the 2010 Wikitext [[mw:Special:MyLanguage/Editor|editor]] in the week of January 9, 2023. It will be available to use via the toolbar in the 2010 Wikitext editor. The feature was the 4th most popular wish of the Community Wishlist Survey 2021.

'''Events'''
* You can now [[mw:Special:MyLanguage/Wikimedia Hackathon 2023/Participate|register for the Wikimedia Hackathon 2023]], taking place on May 19–21 in Athens, Greece. You can also apply for a scholarship until January 14th.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2022/51|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2022-W51&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:00, 20 December 2022 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24258101 --&gt;

== Tech News: 2023-02 ==

&lt;section begin=&quot;technews-2023-W02&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/02|Translations]] are available.

'''Recent changes'''
* You can use tags to filter edits in the recent changes feed or on your watchlist. You can now use tags to filter out edits you don't want to see. Previously you could only use tags to focus on the edits with those tags. [https://phabricator.wikimedia.org/T174349]
* [[Special:WhatLinksHere|Special:WhatLinksHere]] shows all pages that link to a specific page. There is now a [https://wlh.toolforge.org prototype] for how to sort those pages alphabetically. You can see the discussion in the [[phab:T4306|Phabricator ticket]].
* You can now use the [[mw:Special:MyLanguage/Extension:Thanks|thanks]] function on your watchlist and the user contribution page. [https://phabricator.wikimedia.org/T51541]
* A wiki page can be moved to give it a new name. You can now get a dropdown menu with common reasons when you move a page. This is so you don't have to write the explanation every time. [https://phabricator.wikimedia.org/T325257]
* [[m:Special:MyLanguage/Matrix.org|Matrix]] is a chat tool. You can now use &lt;code&gt;matrix:&lt;/code&gt; to create Matrix links on wiki pages. [https://phabricator.wikimedia.org/T326021]
* You can filter out translations when you look at the recent changes on multilingual wikis. This didn't hide translation pages. You can now also hide subpages which are translation pages. [https://phabricator.wikimedia.org/T233493]

'''Changes later this week'''
* [[m:Special:MyLanguage/Real Time Preview for Wikitext|Realtime preview for wikitext]] is a tool which lets editors preview the page when they edit wikitext. It will be enabled for all users of the 2010 wikitext editor. You will find it in the editor toolbar.
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] Some wikis will be in read-only for a few minutes because of a switch of their main database. It will be performed on {{#time:j xg|2023-01-10|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s5.dblist targeted wikis]) and on {{#time:j xg|2023-01-12|en}} at 07:00 UTC ([https://noc.wikimedia.org/conf/highlight.php?file=dblists/s6.dblist targeted wikis]).
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.18|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-01-10|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-01-11|en}}. It will be on all wikis from {{#time:j xg|2023-01-12|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/02|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W02&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:07, 10 January 2023 (UTC)
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== Tech News: 2023-03 ==

&lt;section begin=&quot;technews-2023-W03&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/03|Translations]] are available.

'''Problems'''
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] The URLs in &quot;{{int:last}}&quot; links on page history now contain &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;diff=prev&amp;oldid=[revision ID]&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; in place of &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;diff=[revision ID]&amp;oldid=[revision ID]&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. This is to fix a problem with links pointing to incorrect diffs when history was filtered by a tag. Some user scripts may break as a result of this change. [https://phabricator.wikimedia.org/T243569]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.19|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-01-17|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-01-18|en}}. It will be on all wikis from {{#time:j xg|2023-01-19|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* Some [[mw:Special:MyLanguage/Talk pages project/Usability|changes to the appearance of talk pages]] have only been available on &lt;code&gt;{{ns:1}}:&lt;/code&gt; and &lt;code&gt;{{ns:3}}:&lt;/code&gt; namespaces.  These will be extended to other talk namespaces, such as &lt;code&gt;{{ns:5}}:&lt;/code&gt;.  They will continue to be unavailable in non-talk namespaces, including &lt;code&gt;{{ns:4}}:&lt;/code&gt; pages (e.g., at the Village Pump).  You can [[Special:Preferences#mw-prefsection-editing-discussion|change your preferences]] ([[Special:Preferences#mw-prefsection-betafeatures|beta feature]]). [https://phabricator.wikimedia.org/T325417]
*On Wikisources, when an image is zoomed or panned in the Page: namespace, the same zoom and pan settings will be remembered for all Page: namespace pages that are linked to a particular Index: namespace page. [https://gerrit.wikimedia.org/r/c/mediawiki/extensions/ProofreadPage/+/868841]
* The Vector 2022 skin will become the default for the English Wikipedia desktop users. The change will take place on January 18 at 15:00 UTC. [[:en:w:Wikipedia:Vector 2022|Learn more]].

'''Future changes'''
* The 2023 edition of the [[m:Special:MyLanguage/Community Wishlist Survey 2023|Community Wishlist Survey]], which invites contributors to make technical proposals and vote for tools and improvements, starts next week on 23 January 2023 at 18:00 UTC. You can start drafting your proposals in [[m:Community Wishlist Survey/Sandbox|the CWS sandbox]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/03|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W03&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:10, 17 January 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24381020 --&gt;

== Tech News: 2023-04 ==

&lt;section begin=&quot;technews-2023-W04&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/04|Translations]] are available.

'''Problems'''
* Last week, for ~15 minutes, all wikis were unreachable for logged-in users and non-cached pages. This was caused by a timing issue. [https://wikitech.wikimedia.org/wiki/Incidents/2023-01-17_MediaWiki]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.20|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-01-24|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-01-25|en}}. It will be on all wikis from {{#time:j xg|2023-01-26|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* If you have the Beta Feature for [[mw:Special:MyLanguage/Talk pages project|DiscussionTools]] enabled, the appearance of talk pages will add more information about discussion activity. [https://www.mediawiki.org/wiki/Special:MyLanguage/Talk_pages_project/Usability#Status][https://phabricator.wikimedia.org/T317907]
* The 2023 edition of the [[m:Special:MyLanguage/Community Wishlist Survey 2023|Community Wishlist Survey]] (CWS), which invites contributors to make technical proposals and vote for tools and improvements, starts on Monday 23 January 2023 at [https://zonestamp.toolforge.org/1674496814 18:00 UTC]. 

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/04|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W04&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:46, 23 January 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24418874 --&gt;

== Tech News: 2023-05 ==

&lt;section begin=&quot;technews-2023-W05&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/05|Translations]] are available.

'''Problems'''
* Last week, for ~15 minutes, some users were unable to log in or edit pages. This was caused by a problem with session storage. [https://wikitech.wikimedia.org/wiki/Incidents/2023-01-24_sessionstore_quorum_issues]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.21|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-01-31|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-02-01|en}}. It will be on all wikis from {{#time:j xg|2023-02-02|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).

'''Future changes'''
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Wikis that use localized numbering schemes for references need to add new CSS. This will help to show citation numbers the same way in all reading and editing modes. If your wiki would prefer to do it yourselves, please see the [[mw:Special:MyLanguage/Parsoid/Parser Unification/Cite CSS|details and example CSS to copy from]], and also add your wiki to the list. Otherwise, the developers will directly help out starting the week of February 5.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/05|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W05&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:05, 31 January 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24455949 --&gt;

== Tech News: 2023-06 ==

&lt;section begin=&quot;technews-2023-W06&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/06|Translations]] are available.

'''Recent changes'''
* In the [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements|Vector 2022 skin]], logged-out users using the full-width toggle will be able to see the setting of their choice even after refreshing pages or opening new ones. This only applies to wikis where Vector 2022 is the default. [https://phabricator.wikimedia.org/T321498]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.22|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-02-07|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-02-08|en}}. It will be on all wikis from {{#time:j xg|2023-02-09|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* Previously, we announced when some wikis would be in read-only for a few minutes because of a switch of their main database. These switches will not be announced any more, as the read-only time has become non-significant. Switches will continue to happen at 7AM UTC on Tuesdays and Thursdays. [https://phabricator.wikimedia.org/T292543#8568433]
* Across all the wikis, in the Vector 2022 skin, logged-in users will see the page-related links such as &quot;What links here&quot; in a [[mw:Special:MyLanguage/Reading/Web/Desktop_Improvements/Features/Page_tools|new side menu]]. It will be displayed on the other side of the screen. This change had previously been made on Czech, English, and Vietnamese Wikipedias. [https://phabricator.wikimedia.org/T328692]
*[[m:Special:MyLanguage/Community Wishlist Survey 2023|Community Wishlist Survey 2023]] will stop receiving new proposals on [https://zonestamp.toolforge.org/1675706431 Monday, 6 February 2023, at 18:00 UTC]. Proposers should complete any edits by then, to give time for [[m:Special:MyLanguage/Community_Wishlist_Survey/Help_us|translations]] and review. Voting will begin on Friday, 10 February.

'''Future changes'''
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] Gadgets and user scripts will be changing to load on desktop and mobile sites. Previously they would only load on the desktop site. It is recommended that wiki administrators audit the [[MediaWiki:Gadgets-definition|gadget definitions]] prior to this change, and add &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;skins=…&lt;/code&gt;&lt;/bdi&gt; for any gadgets which should not load on mobile. [https://phabricator.wikimedia.org/T328610 More details are available].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/06|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W06&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 10:21, 6 February 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24491749 --&gt;

== Tech News: 2023-07 ==

&lt;section begin=&quot;technews-2023-W07&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/07|Translations]] are available.

'''Problems'''
* On wikis where patrolled edits are enabled, changes made to the [[mw:Special:MyLanguage/Growth/Communities/How to configure the mentors' list|mentor list]] by autopatrolled mentors are not correctly marked as patrolled. It will be fixed later this week. [https://phabricator.wikimedia.org/T328444]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.23|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-02-14|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-02-15|en}}. It will be on all wikis from {{#time:j xg|2023-02-16|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* The Reply tool and other parts of [[mw:Special:MyLanguage/Help:DiscussionTools#Mobile|DiscussionTools]] will be deployed for all editors using the mobile site.  You can [[mw:Special:MyLanguage/Talk_pages_project/Mobile#Status_Updates|read more about this decision]]. [https://phabricator.wikimedia.org/T298060]

'''Future changes'''
* All wikis will be read-only for a few minutes on March 1. This is planned for [https://zonestamp.toolforge.org/1677679222 14:00 UTC]. More information will be published in Tech News and will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T328287][https://phabricator.wikimedia.org/T327920][https://wikitech.wikimedia.org/wiki/Deployments]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/07|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W07&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:48, 14 February 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24540832 --&gt;

== Tech News: 2023-08 ==

&lt;section begin=&quot;technews-2023-W08&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/08|Translations]] are available.

'''Problems'''
* Last week, during planned maintenance of Cloud Services, unforeseen complications forced the team to turn off all tools for 2–3 hours to prevent data corruption. Work is ongoing to prevent similar problems in the future. [https://phabricator.wikimedia.org/T329535] 

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.23|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-02-21|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-02-22|en}}. It will be on all wikis from {{#time:j xg|2023-02-23|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
*The voting phase for the [[m:Special:MyLanguage/Community Wishlist Survey 2023|Community Wishlist Survey 2023]] ends on [https://zonestamp.toolforge.org/1677261621 24 February at 18:00 UTC]. The results of the survey will be announced on 28 February.

'''Future changes'''
* All wikis will be read-only for a few minutes on March 1. This is planned for [https://zonestamp.toolforge.org/1677679222 14:00 UTC]. More information will be published in Tech News and will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T328287][https://phabricator.wikimedia.org/T327920][https://wikitech.wikimedia.org/wiki/Deployments]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/08|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W08&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:57, 21 February 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24570514 --&gt;

== Tech News: 2023-09 ==

&lt;section begin=&quot;technews-2023-W09&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/09|Translations]] are available.

'''Problems'''
* Last week, in some areas of the world, there were problems with loading pages for 20 minutes and saving edits for 55 minutes. These issues were caused by a problem with our caching servers due to unforseen events during a routine maintenance task. [https://wikitech.wikimedia.org/wiki/Incidents/2023-02-22_wiki_outage][https://wikitech.wikimedia.org/wiki/Incidents/2023-02-22_read_only] 

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.25|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-02-28|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-03-01|en}}. It will be on all wikis from {{#time:j xg|2023-03-02|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* All wikis will be read-only for a few minutes on March 1. This is planned for [https://zonestamp.toolforge.org/1677679222 14:00 UTC]. [https://meta.wikimedia.org/wiki/Special:MyLanguage/Tech/Server_switch]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/09|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W09&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:47, 27 February 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24634242 --&gt;

== Tech News: 2023-10 ==

&lt;section begin=&quot;technews-2023-W10&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/10|Translations]] are available.

'''Recent changes'''
* The Community Wishlist Survey 2023 edition has been concluded. Community Tech has [[m:Special:MyLanguage/Community Wishlist Survey 2023/Results|published the results]] of the survey and will provide an update on what is next in April 2023.
* On wikis which use [[mw:Special:MyLanguage/Writing_systems|LanguageConverter]] to handle multiple writing systems, articles which used custom conversion rules in the wikitext (primarily on Chinese Wikipedia) would have these rules applied inconsistently in the table of contents, especially in the Vector 2022 skin. This has now been fixed. [https://phabricator.wikimedia.org/T306862]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.26|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-03-07|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-03-08|en}}. It will be on all wikis from {{#time:j xg|2023-03-09|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* A search system has been added to the [[Special:Preferences|Preferences screen]]. This will let you find different options more easily. Making it work on mobile devices will happen soon. [https://phabricator.wikimedia.org/T313804]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/10|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W10&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:49, 6 March 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24676916 --&gt;

== Tech News: 2023-11 ==

&lt;section begin=&quot;technews-2023-W11&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/11|Translations]] are available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.40/wmf.27|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-03-14|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-03-15|en}}. It will be on all wikis from {{#time:j xg|2023-03-16|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* Starting on Wednesday, a new set of Wikipedias will get &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]&quot; ({{int:project-localized-name-cbk_zamwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-cdowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-cewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-cebwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-chwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-chrwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-chywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ckbwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-cowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-csbwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-cuwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-cvwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-cywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-itwiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T304542][https://phabricator.wikimedia.org/T304550] 

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/11|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W11&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:20, 13 March 2023 (UTC)
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== Tech News: 2023-12 ==

&lt;section begin=&quot;technews-2023-W12&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/12|Translations]] are available.

'''Problems'''
* Last week, some users experienced issues loading image thumbnails. This was due to incorrectly cached images. [https://phabricator.wikimedia.org/T331820]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.1|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-03-21|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-03-22|en}}. It will be on all wikis from {{#time:j xg|2023-03-23|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] A link to the user's [[{{#special:CentralAuth}}]] page will appear on [[{{#special:Contributions}}]] — some user scripts which previously added this link may cause conflicts. This feature request was [[:m:Community Wishlist Survey 2023/Admins and patrollers/Add link to CentralAuth on Special:Contributions|voted #17 in the 2023 Community Wishlist Survey]].
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] The [[{{#special:AbuseFilter}}]] edit window will be resizable and larger by default. This feature request was [[:m:Community Wishlist Survey 2023/Anti-harassment/Make the AbuseFilter edit window resizable and larger by default|voted #80 in the 2023 Community Wishlist Survey]].
* There will be a new option for Administrators when they are unblocking a user, to add the unblocked user’s user page to their watchlist. This will work both via [[{{#special:Unblock}}]] and via the API. [https://phabricator.wikimedia.org/T257662]

'''Meetings'''
* You can join the next meeting with the Wikipedia mobile apps teams. During the meeting, we will discuss the current features and future roadmap. The meeting will be on [https://zonestamp.toolforge.org/1679677204 24 March at 17:00 (UTC)]. See [[mw:Special:MyLanguage/Wikimedia Apps/Office Hours|details and how to join]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/12|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W12&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:25, 21 March 2023 (UTC)
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== Tech News: 2023-13 ==

&lt;section begin=&quot;technews-2023-W13&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/13|Translations]] are available.

'''Recent changes'''
* The [[:mw:Special:MyLanguage/Extension:AbuseFilter|AbuseFilter]] condition limit was increased from 1000 to 2000. [https://phabricator.wikimedia.org/T309609]
* [[:m:Special:MyLanguage/Global AbuseFilter#Locally disabled actions|Some Global AbuseFilter]] actions will no longer apply to local projects. [https://phabricator.wikimedia.org/T332521]
* Desktop users are now able to subscribe to talk pages by clicking on the {{int:discussiontools-newtopicssubscription-button-subscribe-label}} link in the {{int:toolbox}} menu. If you subscribe to a talk page, you receive [[mw:Special:MyLanguage/Notifications|notifications]] when new topics are started on that talk page. This is separate from putting the page on your watchlist or subscribing to a single discussion. [https://phabricator.wikimedia.org/T263821]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.2|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-03-28|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-03-29|en}}. It will be on all wikis from {{#time:j xg|2023-03-30|en}} ([[mw:MediaWiki 1.40/Roadmap|calendar]]).

'''Future changes'''
* You will be able to choose [[mw:Special:MyLanguage/VisualEditor/Diffs|visual diffs]] on all [[m:Special:MyLanguage/Help:Page history|history pages]] at the Wiktionaries and Wikipedias. [https://phabricator.wikimedia.org/T314588]
* [[File:Octicons-tools.svg|15px|link=|alt=|Advanced item]] The legacy [[mw:Mobile Content Service|Mobile Content Service]] is going away in July 2023. Developers are encouraged to switch to Parsoid or another API before then to ensure service continuity. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/4MVQQTONJT7FJAXNVOFV3WWVVMCHRINE/]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/13|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W13&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:13, 28 March 2023 (UTC)
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== Tech News: 2023-14 ==

&lt;section begin=&quot;technews-2023-W14&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/14|Translations]] are available.

'''Recent changes'''
* The system for automatically creating categories for the [[mw:Special:MyLanguage/Extension:Babel|Babel]] extension has had several important changes and fixes. One of them allows you to insert templates for automatic category descriptions on creation, allowing you to categorize the new categories. [https://phabricator.wikimedia.org/T211665][https://phabricator.wikimedia.org/T64714][https://phabricator.wikimedia.org/T170654][https://phabricator.wikimedia.org/T184941][https://phabricator.wikimedia.org/T33074]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.3|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-04-04|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-04-05|en}}. It will be on all wikis from {{#time:j xg|2023-04-06|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* Some older [[w:en:Web browser|Web browsers]] will stop being able to use [[w:en:JavaScript|JavaScript]] on Wikimedia wikis from this week. This mainly affects users of Internet Explorer 11. If you have an old web browser on your computer you can try to upgrade to a newer version. [https://phabricator.wikimedia.org/T178356]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The deprecated &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;jquery.hoverIntent&lt;/code&gt;&lt;/bdi&gt; module has been removed. This module could be used by gadgets and user scripts, to create an artificial delay in how JavaScript responds to a hover event. Gadgets and user scripts should now use jQuery &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;hover()&lt;/code&gt;&lt;/bdi&gt; or &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;on()&lt;/code&gt;&lt;/bdi&gt; instead. Examples can be found in the [[mw:Special:MyLanguage/ResourceLoader/Migration_guide_(users)#jquery.hoverIntent|migration guide]]. [https://phabricator.wikimedia.org/T311194]
* Some of the links in [[{{#special:SpecialPages}}]] will be re-arranged. There will be a clearer separation between links that relate to all users, and links related to your own user account. [https://phabricator.wikimedia.org/T333242]
* You will be able to hide the [[mw:Special:MyLanguage/Talk pages project/Replying|Reply button]] in archived discussion pages with a new &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;__ARCHIVEDTALK__&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; magic word.  There will also be a new &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;.mw-archivedtalk&lt;/code&gt;&lt;/bdi&gt; CSS class for hiding the Reply button in individual sections on a page. [https://phabricator.wikimedia.org/T249293][https://phabricator.wikimedia.org/T295553][https://gerrit.wikimedia.org/r/c/mediawiki/extensions/DiscussionTools/+/738221]

'''Future changes'''
* The Vega software that creates data visualizations in pages, such as graphs, will be upgraded to the newest version in the future. Graphs that still use the very old version 1.5 syntax may stop working properly. Most existing uses have been found and updated, but you can help to check, and to update any local documentation. [[phab:T260542|Examples of how to find and fix these graphs are available]]. 

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/14|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W14&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:39, 3 April 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24820268 --&gt;

== Tech News: 2023-15 ==

&lt;section begin=&quot;technews-2023-W15&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/15|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] In the visual editor, it is now possible to edit captions of images in galleries without opening the gallery dialog. This feature request was [[:m:Community Wishlist Survey 2023/Editing/Editable gallery captions in Visual Editor|voted #61 in the 2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T190224]
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] You can now receive notifications when another user edits your user page. See the &quot;{{int:Echo-category-title-edit-user-page}}&quot; option in [[Special:Preferences#mw-prefsection-echo|your Preferences]]. This feature request was [[:m:Community Wishlist Survey 2023/Anti-harassment/Notifications for user page edits|voted #3 in the 2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T3876]

'''Problems'''
* There was a problem with all types of CentralNotice banners still being shown to logged-in users even if they had [[Special:Preferences#mw-prefsection-centralnotice-banners|turned off]] specific banner types. This has now been fixed. [https://phabricator.wikimedia.org/T331671]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.4|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-04-11|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-04-12|en}}. It will be on all wikis from {{#time:j xg|2023-04-13|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* Starting on Wednesday, a new set of Wikipedias will get &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]&quot; ({{int:project-localized-name-arywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-dawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-dinwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-dsbwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-eewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-elwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-emlwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-eowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-etwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-euwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-extwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tumwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ffwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-fiwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-fiu_vrowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-fjwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-fowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-frpwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-frrwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-furwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-gawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-gcrwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-gdwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-glwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-glkwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-gnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-gomwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-gotwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-guwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-gvwiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T304551][https://phabricator.wikimedia.org/T308133]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/15|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W15&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:05, 10 April 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=24851886 --&gt;

== Tech News: 2023-16 ==

&lt;section begin=&quot;technews-2023-W16&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/16|Translations]] are available.

'''Recent changes'''
* You can now see [[mw:Special:MyLanguage/Help:Extension:Kartographer#Show_nearby_articles|nearby articles on a Kartographer map]] with the button for the new feature &quot;{{int:Kartographer-sidebar-nearbybutton}}&quot;. Six wikis have been testing this feature since October. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/Geoinformation/Nearby_articles#Implementation][https://phabricator.wikimedia.org/T334079]
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] The [[m:Special:GlobalWatchlist|Special:GlobalWatchlist]] page now has links for &quot;{{int:globalwatchlist-markpageseen}}&quot; for each entry. This feature request was [[m:Community Wishlist Survey 2023/Notifications, Watchlists and Talk Pages/Button to mark a single change as read in the global watch list|voted #161 in the 2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T334246]

'''Problems'''
* At Wikimedia Commons, some thumbnails have not been getting replaced correctly after a new version of the image is uploaded. This should be fixed later this week. [https://phabricator.wikimedia.org/T331138][https://phabricator.wikimedia.org/T333042]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] For the last few weeks, some external tools had inconsistent problems with logging-in with OAuth. This has now been fixed. [https://phabricator.wikimedia.org/T332650]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.5|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-04-18|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-04-19|en}}. It will be on all wikis from {{#time:j xg|2023-04-20|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/16|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W16&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:54, 18 April 2023 (UTC)
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== Tech News: 2023-17 ==

&lt;section begin=&quot;technews-2023-W17&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/17|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] The date-selection menu on pages such as [[{{#special:Contributions}}]] will now show year-ranges that are in the current and past decade, instead of the current and future decade. This feature request was [[m:Community Wishlist Survey 2023/Miscellaneous/Change year range shown in date selection popup|voted #145 in the 2023 Community Wishlist Survey]].  [https://phabricator.wikimedia.org/T334316]

'''Problems'''
* Due to security issues with the [[mw:Special:MyLanguage/Extension:Graph|Graph extension]], graphs have been disabled in all Wikimedia projects. Wikimedia Foundation teams are working to respond to these vulnerabilities. [https://phabricator.wikimedia.org/T334940]
* For a few days, it was not possible to save some kinds of edits on the mobile version of a wiki. This has been fixed. [https://phabricator.wikimedia.org/T334797][https://phabricator.wikimedia.org/T334799][https://phabricator.wikimedia.org/T334794]

'''Changes later this week'''
* All wikis will be read-only for a few minutes on April 26. This is planned for [https://zonestamp.toolforge.org/1682517653 14:00 UTC]. [https://meta.wikimedia.org/wiki/Special:MyLanguage/Tech/Server_switch]
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.6|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-04-25|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-04-26|en}}. It will be on all wikis from {{#time:j xg|2023-04-27|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).

'''Future changes'''
* The Editing team plans an A/B test for [[mw:Special:MyLanguage/Talk pages project/Usability|a usability analysis of the Talk page project]]. The [[mw:Special:MyLanguage/Talk pages project/Usability/Analysis|planned measurements are available]]. Your wiki [[phab:T332946|may be invited to participate]]. Please suggest improvements to the measurement plan at [[mw:Talk:Talk pages project/Usability|the discussion page]].
* [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2023-2024|The Wikimedia Foundation annual plan 2023-2024 draft is open for comment and input]] until May 19. The final plan will be published in July 2023 on Meta-wiki.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/17|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W17&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:03, 24 April 2023 (UTC)
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== Tech News: 2023-18 ==

&lt;section begin=&quot;technews-2023-W18&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/18|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] The content attribution tools [[mw:Special:MyLanguage/Who Wrote That?|Who Wrote That?]], [[xtools:authorship|XTools Authorship]], and [[xtools:blame|XTools Blame]] now support the French and Italian Wikipedias. More languages will be added in the near future. This is part of the [[m:Community Wishlist Survey 2023/Reading/Extend &quot;Who Wrote That?&quot; tool to more wikis|#7 wish in the 2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T243711][https://phabricator.wikimedia.org/T270490][https://phabricator.wikimedia.org/T334891]
* The [[:commons:Special:MyLanguage/Commons:Video2commons|Video2commons]] tool has been updated. This fixed several bugs related to YouTube uploads. [https://github.com/toolforge/video2commons/pull/162/commits]
* The [[{{#special:Preferences}}]] page has been redesigned on mobile web. The new design makes it easier to browse the different categories and settings at low screen widths. You can also now access the page via a link in the Settings menu in the mobile web sidebar. [https://www.mediawiki.org/wiki/Moderator_Tools/Content_moderation_on_mobile_web/Preferences]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.7|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-05-02|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-05-03|en}}. It will be on all wikis from {{#time:j xg|2023-05-04|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/18|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W18&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:45, 2 May 2023 (UTC)
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== Tech News: 2023-19 ==

&lt;section begin=&quot;technews-2023-W19&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/19|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] Last week, Community Tech released the first update for providing [[m:Special:MyLanguage/Community Wishlist Survey 2022/Better diff handling of paragraph splits|better diffs]], the #1 request in the 2022 Community Wishlist Survey. [[phab:T324759|This update]] adds legends and tooltips to inline diffs so that users unfamiliar with the blue and yellow highlights can better understand the type of edits made.
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] When you close an image that is displayed via MediaViewer, it will now return to the wiki page instead of going back in your browser history. This feature request was [[m:Community Wishlist Survey 2023/Reading/Return to the article when closing the MediaViewer|voted #65 in the 2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T236591]
* The [[mw:Special:MyLanguage/Extension:SyntaxHighlight|SyntaxHighlight]] extension now supports &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;wikitext&lt;/code&gt;&lt;/bdi&gt; as a selected language. Old alternatives that were used to highlight wikitext, such as &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;html5&lt;/code&gt;&lt;/bdi&gt;, &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;moin&lt;/code&gt;&lt;/bdi&gt;, and &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;html+handlebars&lt;/code&gt;&lt;/bdi&gt;, can now be replaced. [https://phabricator.wikimedia.org/T29828]
* [[mw:Special:MyLanguage/Manual:Creating pages with preloaded text|Preloading text to new pages/sections]] now supports preloading from localized MediaWiki interface messages. [https://cs.wikipedia.org/wiki/User_talk:Martin_Urbanec_(WMF)?action=edit&amp;section=new&amp;preload=MediaWiki:July Here is an example] at the {{int:project-localized-name-cswiki/en}} that uses &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;preload=MediaWiki:July&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. [https://phabricator.wikimedia.org/T330337]

'''Problems'''
* Graph Extension update: Foundation developers have completed upgrading the visualization software to Vega5. Existing community graphs based on Vega2 are no longer compatible. Communities need to update local graphs and templates, and shared lua modules like &lt;bdi lang=&quot;de&quot; dir=&quot;ltr&quot;&gt;[[:de:Modul:Graph]]&lt;/bdi&gt;. The [https://vega.github.io/vega/docs/porting-guide/ Vega Porting guide] provides the most comprehensive detail on migration from Vega2 and [https://www.mediawiki.org/w/index.php?title=Template:Graph:PageViews&amp;action=history here is an example migration]. Vega5 has currently just been enabled on mediawiki.org to provide a test environment for communities. [https://phabricator.wikimedia.org/T334940#8813922]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.8|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-05-09|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-05-10|en}}. It will be on all wikis from {{#time:j xg|2023-05-11|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Until now, all new OAuth apps went through manual review. Starting this week, apps using identification-only or basic authorizations will not require review. [https://phabricator.wikimedia.org/T67750]

'''Future changes'''
* During the next year, MediaWiki will stop using IP addresses to identify logged-out users, and will start automatically assigning unique temporary usernames. Read more at [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation/Updates|IP Editing: Privacy Enhancement and Abuse Mitigation/Updates]]. You can [[m:Talk:IP Editing: Privacy Enhancement and Abuse Mitigation#What should it look like?|join the discussion]] about the [[m:Special:MyLanguage/IP Editing: Privacy Enhancement and Abuse Mitigation/Updates#What will temporary usernames look like?|format of the temporary usernames]]. [https://phabricator.wikimedia.org/T332805]
* There will be an [[:w:en:A/B testing|A/B test]] on 10 Wikipedias where the Vector 2022 skin is the default skin. Half of logged-in desktop users will see an interface where the different parts of the page are more clearly separated. You can [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Updates/2023-05 Zebra9 A/B test|read more]]. [https://phabricator.wikimedia.org/T333180][https://phabricator.wikimedia.org/T335972]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] &lt;code&gt;jquery.tipsy&lt;/code&gt; will be removed from the MediaWiki core. This will affect some user scripts. Many lines with &lt;code&gt;.tipsy(&lt;/code&gt; can be commented out. &lt;code&gt;OO.ui.PopupWidget&lt;/code&gt; can be used to keep things working like they are now. You can [[phab:T336019|read more]] and [[:mw:Help:Locating broken scripts|read about how to find broken scripts]]. [https://phabricator.wikimedia.org/T336019]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/19|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W19&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:36, 9 May 2023 (UTC)
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== Tech News: 2023-20 ==

&lt;section begin=&quot;technews-2023-W20&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/20|Translations]] are available.

'''Problems'''
* Citations that are automatically generated based on [[d:Q33057|ISBN]] are currently broken. This affects citations made with the [[mw:Special:MyLanguage/Help:VisualEditor/User_guide/Citations-Full#Automatic|VisualEditor Automatic tab]], and the use of the citoid API in gadgets and user scripts. Work is ongoing to restore this feature. [https://phabricator.wikimedia.org/T336298] 

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.9|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-05-16|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-05-17|en}}. It will be on all wikis from {{#time:j xg|2023-05-18|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* Starting on Wednesday, a new set of Wikipedias will get &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]&quot; ({{int:project-localized-name-gorwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hakwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hawwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hifwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hrwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hsbwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-htwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-iawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-iewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-igwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ilowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-inhwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-iowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-iswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-iuwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-jamwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-jvwiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T308134] 

'''Future changes'''
* There is a recently formed team at the Wikimedia Foundation which will be focusing on experimenting with new tools. Currently they are building [[m:Wikimedia_Foundation_Annual_Plan/2023-2024/Draft/Future_Audiences#FA2.2_Conversational_AI|a prototype ChatGPT plugin that allows information generated by ChatGPT to be properly attributed]] to the Wikimedia projects.
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Gadget and userscript developers should replace &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;jquery.cookie&lt;/code&gt;&lt;/bdi&gt; with &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;mediawiki.cookie&lt;/code&gt;&lt;/bdi&gt;. The &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;jquery.cookie&lt;/code&gt;&lt;/bdi&gt; library will be removed in ~1 month, and staff developers will run a script to replace any remaining uses at that time. [https://phabricator.wikimedia.org/T336018]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/20|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W20&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:45, 15 May 2023 (UTC)
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== Tech News: 2023-21 ==

&lt;section begin=&quot;technews-2023-W21&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/21|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] The &quot;recent edits&quot; time period for page watchers is now 30 days. It used to be 180 days. This was a [[m:Community Wishlist Survey 2023/Notifications, Watchlists and Talk Pages/Change information about the number of watchers on a page|Community Wishlist Survey proposal]]. [https://phabricator.wikimedia.org/T336250]

'''Changes later this week'''
* An [[mw:special:MyLanguage/Growth/Positive reinforcement#Impact|improved impact module]] will be available at Wikipedias. The impact module is a feature available to newcomers [[mw:Special:MyLanguage/Growth/Feature summary#Newcomer homepage|at their personal homepage]]. It will show their number of edits, how many readers their edited pages have, how many thanks they have received and similar things. It is also accessible by accessing Special:Impact. [https://phabricator.wikimedia.org/T336203]
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.10|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-05-23|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-05-24|en}}. It will be on all wikis from {{#time:j xg|2023-05-25|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/21|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W21&quot;/&gt;

16:55, 22 May 2023 (UTC)
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== Tech News: 2023-22 ==

&lt;section begin=&quot;technews-2023-W22&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/22|Translations]] are available.

'''Recent changes'''
* Citations can once again be added automatically from ISBNs, thanks to Zotero's ISBN searches. The current data sources are the Library of Congress (United States), the Bibliothèque nationale de France (French National Library), and K10plus ISBN (German repository). Additional data source searches can be [[mw:Citoid/Creating Zotero translators|proposed to Zotero]]. The ISBN labels in the [[mw:Special:MyLanguage/Help:VisualEditor/User_guide/Citations-Full#Automatic|VisualEditor Automatic tab]] will reappear later this week. [https://phabricator.wikimedia.org/T336298#8859917]
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] The page [[{{#special:EditWatchlist}}]] now has &quot;{{int:watchlistedit-normal-check-all}}&quot; options to select all the pages within a namespace. This feature request was [[m:Community Wishlist Survey 2023/Notifications, Watchlists and Talk Pages/Watchlist edit - &quot;check all&quot; checkbox|voted #161 in the 2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T334252]	

'''Problems'''
* For a few days earlier this month, the &quot;Add interlanguage link&quot; item in the Tools menu did not work properly. This has now been fixed. [https://phabricator.wikimedia.org/T337081]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.11|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-05-30|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-05-31|en}}. It will be on all wikis from {{#time:j xg|2023-06-01|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* VisualEditor will be switched to a new backend on [https://phabricator.wikimedia.org/source/mediawiki-config/browse/master/dblists/small.dblist small] and [https://phabricator.wikimedia.org/source/mediawiki-config/browse/master/dblists/medium.dblist medium] wikis this week. Large wikis will follow in the coming weeks. This is part of the effort to move Parsoid into MediaWiki core. The change should have no noticeable effect on users, but if you experience any slow loading or other strangeness when using VisualEditor, please report it on the phabricator ticket linked here. [https://phabricator.wikimedia.org/T320529]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/22|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W22&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:03, 29 May 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25079963 --&gt;

== Tech News: 2023-23 ==

&lt;section begin=&quot;technews-2023-W23&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/23|Translations]] are available.

'''Recent changes'''
* The [[:mw:Special:MyLanguage/Help:Extension:RealMe|RealMe]] extension allows you to mark URLs on your user page as verified for Mastodon and similar software.
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] Citation and footnote editing can now be started from the reference list when using the visual editor. This feature request was [[m:Community Wishlist Survey 2023/Citations/Allow citations to be edited in the references section with VisualEditor|voted #2 in the 2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T54750]
* Previously, clicking on someone else's link to Recent Changes with filters applied within the URL could unintentionally change your preference for &quot;{{int:Rcfilters-group-results-by-page}}&quot;. This has now been fixed. [https://phabricator.wikimedia.org/T202916#8874081]

'''Problems'''
* For a few days last week, some tools and bots returned outdated information due to database replication problems, and may have been down entirely while it was being fixed. These issues have now been fixed. [https://phabricator.wikimedia.org/T337446]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.12|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-06-06|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-06-07|en}}. It will be on all wikis from {{#time:j xg|2023-06-08|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* Bots will no longer be prevented from making edits because of URLs that match the [[mw:Special:MyLanguage/Extension:SpamBlacklist|spam blacklist]]. [https://phabricator.wikimedia.org/T313107]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/23|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W23&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:52, 5 June 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25114640 --&gt;

== Tech News: 2023-24 ==

&lt;section begin=&quot;technews-2023-W24&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/24|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] The content attribution tools [[mw:Special:MyLanguage/Who Wrote That?|Who Wrote That?]], [[xtools:authorship|XTools Authorship]], and [[xtools:blame|XTools Blame]] now support the Dutch, German, Hungarian, Indonesian, Japanese, Polish and Portuguese Wikipedias. This was the [[m:Community Wishlist Survey 2023/Reading/Extend &quot;Who Wrote That?&quot; tool to more wikis|#7 wish in the 2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T334891]
* The [[mw:Special:MyLanguage/Structured Data Across Wikimedia/Search Improvements#Search Preview panel|Search Preview panel]] has been deployed on four Wikipedias (Catalan, Dutch, Hungarian and Norwegian). The panel will show an image related to the article (if existing), the top sections of the article, related images (coming from MediaSearch on Commons), and eventually the sister projects associated with the article. [https://phabricator.wikimedia.org/T306341]
* The [[:mw:Special:MyLanguage/Help:Extension:RealMe#Verifying_a_link_on_non-user_pages|RealMe]] extension now allows administrators to verify URLs for any page, for Mastodon and similar software. [https://phabricator.wikimedia.org/T324937]
* The default project license [https://lists.wikimedia.org/hyperkitty/list/wikimediaannounce-l@lists.wikimedia.org/thread/7G6XPWZPQFLZ2JANN3ZX6RT4DVUI3HZQ/ has been officially upgraded] to CC BY-SA 4.0. The software interface messages have been updated. Communities should feel free to start updating any mentions of the old CC BY-SA 3.0 licensing within policies and related documentation pages. [https://phabricator.wikimedia.org/T319064]

'''Problems'''
* For three days last month, some Wikipedia pages edited with VisualEditor or DiscussionTools had an unintended &lt;code&gt;&lt;nowiki&gt;__TOC__&lt;/nowiki&gt;&lt;/code&gt; (or its localized form) added during an edit. There is [[mw:Parsoid/Deployments/T336101_followup|a listing of affected pages sorted by wiki]], that may still need to be fixed. [https://phabricator.wikimedia.org/T336101]
* Currently, the &quot;{{int:Visualeditor-dialog-meta-categories-defaultsort-label}}&quot; feature in VisualEditor is broken. Existing &lt;code&gt;&lt;nowiki&gt;{{DEFAULTSORT:...}}&lt;/nowiki&gt;&lt;/code&gt; keywords incorrectly appear as missing templates in VisualEditor. Developers are exploring how to fix this. In the meantime, those wishing to edit the default sortkey of a page are advised to switch to source editing. [https://phabricator.wikimedia.org/T337398]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Last week, an update to the delete form may have broken some gadgets or user scripts. If you need to manipulate (empty) the reason field, replace &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;#wpReason&lt;/code&gt;&lt;/bdi&gt; with &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot; style=&quot;white-space: nowrap;&quot;&gt;&lt;code&gt;#wpReason &gt; input&lt;/code&gt;&lt;/bdi&gt;. See [https://cs.wikipedia.org/w/index.php?title=MediaWiki%3AGadget-CleanDeleteReasons.js&amp;diff=22859956&amp;oldid=12794189 an example fix]. [https://phabricator.wikimedia.org/T337809]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.13|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-06-13|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-06-14|en}}. It will be on all wikis from {{#time:j xg|2023-06-15|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* VisualEditor will be switched to a new backend on English Wikipedia on Monday, and all other [https://phabricator.wikimedia.org/source/mediawiki-config/browse/master/dblists/large.dblist large] wikis on Thursday. The change should have no noticeable effect on users, but if you experience any slow loading or other strangeness when using VisualEditor, please report it on the phabricator ticket linked here. [https://phabricator.wikimedia.org/T320529]

'''Future changes'''
* From 5 June to 17 July, the Foundation's [[:mw:Wikimedia Security Team|Security team]] is holding a consultation with contributors regarding a draft policy to govern the use of third-party resources in volunteer-developed gadgets and scripts. Feedback and suggestions are warmly welcome at [[m:Special:MyLanguage/Third-party resources policy|Third-party resources policy]] on meta-wiki.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/24|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W24&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 14:51, 12 June 2023 (UTC)
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== Tech News: 2023-25 ==

&lt;section begin=&quot;technews-2023-W25&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/25|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Flame graphs are now available in WikimediaDebug. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/JXNQD3EHG5V5QW5UXFDPSHQG4MJ3FWJQ/][https://techblog.wikimedia.org/2023/06/08/flame-graphs-arrive-in-wikimediadebug/]

'''Changes later this week'''
* There is no new MediaWiki version this week.
* There is now a toolbar search popup in the visual editor. You can trigger it by typing &lt;code&gt;\&lt;/code&gt; or pressing &lt;code&gt;ctrl + shift + p&lt;/code&gt;. It can help you quickly access most tools in the editor. [https://commons.wikimedia.org/wiki/File:Visual_editor_toolbar_search_feature.png][https://phabricator.wikimedia.org/T66905]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/25|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W25&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:08, 19 June 2023 (UTC)
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== Tech News: 2023-26 ==

&lt;section begin=&quot;technews-2023-W26&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/26|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The Action API modules and Special:LinkSearch will now add a trailing &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;/&lt;/code&gt;&lt;/bdi&gt; to all &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;prop=extlinks&lt;/code&gt;&lt;/bdi&gt; responses for bare domains. This is part of the work to remove duplication in the &lt;code&gt;externallinks&lt;/code&gt; database table. [https://phabricator.wikimedia.org/T337994]

'''Problems'''
* Last week, search was broken on Commons and Wikidata for 23 hours. [https://phabricator.wikimedia.org/T339810][https://wikitech.wikimedia.org/wiki/Incidents/2023-06-18_search_broken_on_wikidata_and_commons]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.15|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-06-27|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-06-28|en}}. It will be on all wikis from {{#time:j xg|2023-06-29|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The Minerva skin now applies more predefined styles to the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;.mbox-text&lt;/code&gt;&lt;/bdi&gt; CSS class. This enables support for mbox templates that use divs instead of tables. Please make sure that the new styles won't affect other templates in your wiki. [https://gerrit.wikimedia.org/r/c/mediawiki/skins/MinervaNeue/+/930901/][https://phabricator.wikimedia.org/T339040]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Gadgets will now load on both desktop and mobile by default. Previously, gadgets loaded only on desktop by default. Changing this default using the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;|targets=&lt;/code&gt;&lt;/bdi&gt; parameter is also deprecated and should not be used. You should make gadgets work on mobile or disable them based on the skin (with the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;|skins=&lt;/code&gt;&lt;/bdi&gt; parameter in &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;MediaWiki:Gadgets-definition&lt;/bdi&gt;) rather than whether the user uses the mobile or the desktop website. Popular gadgets that create errors on mobile will be disabled by developers on the Minerva skin as a temporary solution. [https://phabricator.wikimedia.org/T127268]
* All namespace tabs now have the same browser [[m:Special:MyLanguage/Help:Keyboard_shortcuts|access key]] by default. Previously, custom and extension-defined namespaces would have to have their access keys set manually on-wiki, but that is no longer necessary. [https://phabricator.wikimedia.org/T22126]
* The review form of the Flagged Revisions extension now uses the standardized [[mw:Special:MyLanguage/Codex|user interface components]]. [https://phabricator.wikimedia.org/T191156]

'''Future changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] How media is structured in the parser's HTML output will change in the coming weeks at [[:wikitech:Deployments/Train#Thursday|group2 wikis]]. This change improves the accessibility of content. You may need to update your site-CSS, or userscripts and gadgets. There are [[mw:Special:MyLanguage/Parsoid/Parser_Unification/Media_structure/FAQ|details on what code to check, how to update the code, and where to report any related problems]]. [https://phabricator.wikimedia.org/T314318]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/26|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W26&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 16:18, 26 June 2023 (UTC)
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== Tech News: 2023-27 ==

&lt;section begin=&quot;technews-2023-W27&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/27|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] As part of the rolling out of the [[m:Community Wishlist Survey 2022/Multimedia and Commons/Audio links that play on click|audio links that play on click]] wishlist proposal, [https://noc.wikimedia.org/conf/highlight.php?file=dblists/small.dblist small wikis] will now be able to use the [[mw:Special:MyLanguage/Help:Extension:Phonos#Inline audio player mode|inline audio player]] that is implemented by the [[mw:Extension:Phonos|Phonos]] extension. [https://phabricator.wikimedia.org/T336763]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] From this week all gadgets automatically load on mobile and desktop sites. If you see any problems with gadgets on your wikis, please adjust the [[mw:Special:MyLanguage/Extension:Gadgets#Options|gadget options]] in your gadget definitions file. [https://phabricator.wikimedia.org/T328610]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.16|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-07-04|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-07-05|en}}. It will be on all wikis from {{#time:j xg|2023-07-06|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/27|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W27&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:51, 3 July 2023 (UTC)
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== Tech News: 2023-28 ==

&lt;section begin=&quot;technews-2023-W28&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/28|Translations]] are available.

'''Recent changes'''
* The [[:mw:Special:MyLanguage/Structured Data Across Wikimedia/Section-level Image Suggestions|Section-level Image Suggestions feature]] has been deployed on seven Wikipedias (Portuguese, Russian, Indonesian, Catalan, Hungarian, Finnish and Norwegian Bokmål). The feature recommends images for articles on contributors' watchlists that are a good match for individual sections of those articles.
* [[:m:Special:MyLanguage/Global AbuseFilter|Global abuse filters]] have been enabled on all Wikimedia projects, except English and Japanese Wikipedias (who opted out). This change was made following a [[:m:Requests for comment/Make global abuse filters opt-out|global request for comments]]. [https://phabricator.wikimedia.org/T341159]
* [[{{#special:BlockedExternalDomains}}]] is a new tool for administrators to help fight spam. It provides a clearer interface for blocking plain domains (and their subdomains), is more easily searchable, and is faster for the software to process for each edit on the wiki. It does not support regex (for complex cases), nor URL path-matching, nor the [[MediaWiki:Spam-whitelist|MediaWiki:Spam-whitelist]], but otherwise it replaces most of the functionalities of the existing [[MediaWiki:Spam-blacklist|MediaWiki:Spam-blacklist]]. There is a Python script to help migrate all simple domains into this tool, and more feature details, within [[mw:Special:MyLanguage/Manual:BlockedExternalDomains|the tool's documentation]]. It is available at all wikis except for Meta-wiki, Commons, and Wikidata. [https://phabricator.wikimedia.org/T337431]
* The WikiEditor extension was updated. It includes some of the most frequently used features of wikitext editing. In the past, many of its messages could only be translated by administrators, but now all regular translators on translatewiki can translate them. Please check [https://translatewiki.net/wiki/Special:MessageGroupStats?group=ext-wikieditor&amp;messages=&amp;x=D#sortable:0=asc the state of WikiEditor localization into your language], and if the &quot;Completion&quot; for your language shows anything less than 100%, please complete the translation. See [https://lists.wikimedia.org/hyperkitty/list/wikitech-ambassadors@lists.wikimedia.org/thread/D4YELU2DXMZ75PGELUOKXXMFF3FH45XA/ a more detailed explanation].

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.17|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-07-11|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-07-12|en}}. It will be on all wikis from {{#time:j xg|2023-07-13|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* The default protocol of [[{{#special:LinkSearch}}]] and API counterparts has changed from http to both http and https. [https://phabricator.wikimedia.org/T14810]
* [[{{#special:LinkSearch}}]] and its API counterparts will now search for all of the URL provided in the query. It used to be only the first 60 characters. This feature was requested fifteen years ago. [https://phabricator.wikimedia.org/T17218]

'''Future changes'''
* There is an experiment with a [[:w:en:ChatGPT|ChatGPT]] plugin. This is to show users where the information is coming from when they read information from Wikipedia. It has been tested by Wikimedia Foundation staff and other Wikimedians. Soon all ChatGPT plugin users can use the Wikipedia plugin. This is the same plugin which was mentioned in [[m:Special:MyLanguage/Tech/News/2023/20|Tech News 2023/20]]. [https://meta.wikimedia.org/wiki/Wikimedia_Foundation_Annual_Plan/2023-2024/Draft/Future_Audiences#FA2.2_Conversational_AI]
* There is an ongoing discussion on a [[m:Special:MyLanguage/Third-party resources policy|proposed Third-party resources policy]]. The proposal will impact the use of third-party resources in gadgets and userscripts. Based on the ideas received so far, policy includes some of the risks related to user scripts and gadgets loading third-party resources, some best practices and exemption requirements such as code transparency and inspectability. Your feedback and suggestions are warmly welcome until July 17, 2023 on [[m:Talk:Third-party resources policy|on the policy talk page]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/28|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W28&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 19:54, 10 July 2023 (UTC)
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== Tech News: 2023-29 ==

&lt;section begin=&quot;technews-2023-W29&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/29|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] We are now serving 1% of all global user traffic from [[w:en:Kubernetes|Kubernetes]] (you can [[wikitech:MediaWiki On Kubernetes|read more technical details]]). We are planning to increment this percentage regularly. You can [[phab:T290536|follow the progress of this work]]. 

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.18|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-07-18|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-07-19|en}}. It will be on all wikis from {{#time:j xg|2023-07-20|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] MediaWiki [[mw:Special:MyLanguage/Help:System_message|system messages]] will now look for available local fallbacks, instead of always using the default fallback defined by software. This means wikis no longer need to override each language on the [[mw:Special:MyLanguage/Manual:Language#Fallback_languages|fallback chain]] separately. For example, English Wikipedia doesn't have to create &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;en-ca&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;en-gb&lt;/code&gt;&lt;/bdi&gt; subpages with a transclusion of the base pages anymore. This makes it easier to maintain local overrides. [https://phabricator.wikimedia.org/T229992]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;action=growthsetmentorstatus&lt;/code&gt;&lt;/bdi&gt; API will be deprecated with the new MediaWiki version. Bots or scripts calling that API should use the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;action=growthmanagementorlist&lt;/code&gt;&lt;/bdi&gt; API now. [https://phabricator.wikimedia.org/T321503]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/29|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W29&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:08, 17 July 2023 (UTC)
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== Tech News: 2023-30 ==

&lt;section begin=&quot;technews-2023-W30&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/30|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]]   On July 18, the Wikimedia Foundation launched a survey about the [[:mw:Technical_decision_making|technical decision making process]] for people who do technical work that relies on software that is maintained by the Foundation or affiliates. If this applies to you, [https://wikimediafoundation.limesurvey.net/885471 please take part in the survey]. The survey will be open for three weeks, until August 7. You can find more information in [[listarchive:list/wikitech-l@lists.wikimedia.org/thread/Q7DUCFA75DXG3G2KHTO7CEWMLCYTSDB2/|the announcement e-mail on wikitech-l]].

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.19|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-07-25|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-07-26|en}}. It will be on all wikis from {{#time:j xg|2023-07-27|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/30|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W30&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 02:20, 25 July 2023 (UTC)
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== Tech News: 2023-31 ==

&lt;section begin=&quot;technews-2023-W31&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/31|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The [[mw:Synchronizer|Synchronizer]] tool is now available to keep Lua modules synced across Wikimedia wikis, along with [[mw:Multilingual Templates and Modules|updated documentation]] to develop global Lua modules and templates.
* The tag filter on [[{{#special:NewPages}}]] and revision history pages can now be inverted. For example, you can hide edits that were made using an automated tool. [https://phabricator.wikimedia.org/T334337][https://phabricator.wikimedia.org/T334338]
* The Wikipedia [[:w:en:ChatGPT|ChatGPT]] plugin experiment can now be used by ChatGPT users who can use plugins. You can participate in a [[:m:Talk:Wikimedia Foundation Annual Plan/2023-2024/Draft/Future Audiences#Announcing monthly Future Audiences open &quot;office hours&quot;|video call]] if you want to talk about this experiment or similar work. [https://meta.wikimedia.org/wiki/Wikimedia_Foundation_Annual_Plan/2023-2024/Draft/Future_Audiences#FA2.2_Conversational_AI]

'''Problems'''
* It was not possible to generate a PDF for pages with non-Latin characters in the title, for the last two weeks. This has now been fixed. [https://phabricator.wikimedia.org/T342442]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.20|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-08-01|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-08-02|en}}. It will be on all wikis from {{#time:j xg|2023-08-03|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* Starting on Tuesday, a new set of Wikipedias will get &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]&quot; ({{int:project-localized-name-kawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kaawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kabwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kbdwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kbpwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kiwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kkwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kmwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-knwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kshwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kuwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kwwiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T308135] 

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/31|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W31&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:54, 31 July 2023 (UTC)
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== Tech News: 2023-32 ==

&lt;section begin=&quot;technews-2023-W32&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/32|Translations]] are available.

'''Recent changes'''
* Mobile Web editors can now [[mw:Special:MyLanguage/Reading/Web/Advanced_mobile_contributions#August_1,_2023_-_Full-page_editing_added_on_mobile|edit a whole page at once]]. To use this feature, turn on &quot;{{int:Mobile-frontend-mobile-option-amc}}&quot; in your settings and use the &quot;{{int:Minerva-page-actions-editfull}}&quot; button in the &quot;{{int:Minerva-page-actions-overflow}}&quot; menu. [https://phabricator.wikimedia.org/T203151]

'''Changes later this week'''
* There is no new MediaWiki version this week.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/32|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W32&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:20, 7 August 2023 (UTC)
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== Tech News: 2023-33 ==

&lt;section begin=&quot;technews-2023-W33&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/33|Translations]] are available.

'''Recent changes'''
* The Content translation system is no longer using Youdao's [[mw:Special:MyLanguage/Help:Content_translation/Translating/Initial_machine_translation|machine translation service]]. The service was in place for several years, but due to no usage, and availability of alternatives, it was deprecated to reduce maintenance overheads. Other services which cover the same languages are still available. [https://phabricator.wikimedia.org/T329137]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.22|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-08-15|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-08-16|en}}. It will be on all wikis from {{#time:j xg|2023-08-17|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* Starting on Wednesday, a new set of Wikipedias will get &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]&quot; ({{int:project-localized-name-lawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ladwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lbwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lbewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lezwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lfnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lgwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-liwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lijwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lmowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ltgwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-lvwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-maiwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-map_bmswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mdfwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mgwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kywiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T308136] &lt;!-- TODO replace wiki codes --&gt;

'''Future changes'''
* A few gadgets/user scripts which add icons to the Minerva skin need to have their CSS updated. There are more details available including a [[phab:T344067|search for all existing instances and how to update them]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/33|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W33&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 05:59, 15 August 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25428668 --&gt;

== Tech News: 2023-34 ==

&lt;section begin=&quot;technews-2023-W34&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/34|Translations]] are available.

'''Recent changes'''
* The [https://gdrive-to-commons.toolforge.org/ GDrive to Commons Uploader] tool is now available. It enables [[m:Special:MyLanguage/GDrive to Commons Uploader|securely selecting and uploading files]] from your Google Drive directly to Wikimedia Commons. [https://phabricator.wikimedia.org/T267868]
* From now on, we will announce new Wikimedia wikis in Tech News, so you can update any tools or pages.
** Since the last edition, two new wikis have been created:
*** a Wiktionary in [[d:Q7121294|Pa'O]] ([[wikt:blk:|&lt;code&gt;wikt:blk:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T343540]
*** a Wikisource in [[d:Q34002|Sundanese]] ([[s:su:|&lt;code&gt;s:su:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T343539]
** To catch up, the next most recent six wikis are:
*** Wikifunctions ([[f:|&lt;code&gt;f:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T275945]
*** a Wiktionary in [[d:Q2891049|Mandailing]] ([[wikt:btm:|&lt;code&gt;wikt:btm:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T335216]
*** a Wikipedia in [[d:Q5555465|Ghanaian Pidgin]] ([[w:gpe:|&lt;code&gt;w:gpe:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T335969]
*** a Wikinews in [[d:Q3111668|Gungbe]] ([[n:guw:|&lt;code&gt;n:guw:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T334394]
*** a Wiktionary in [[d:Q33522|Kabardian]] ([[wikt:kbd:|&lt;code&gt;wikt:kbd:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T333266]
*** a Wikipedia in [[d:Q35570|Fante]] ([[w:fat:|&lt;code&gt;w:fat:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T335016]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.23|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-08-22|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-08-23|en}}. It will be on all wikis from {{#time:j xg|2023-08-24|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).

'''Future changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] There is an existing [[mw:Stable interface policy|stable interface policy]] for MediaWiki backend code. There is a [[mw:User:Jdlrobson/Stable interface policy/frontend|proposed stable interface policy for frontend code]]. This is relevant for anyone who works on gadgets or Wikimedia frontend code. You can read it, discuss it, and let the proposer know if there are any problems. [https://phabricator.wikimedia.org/T344079]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/34|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W34&quot;/&gt;

15:25, 21 August 2023 (UTC)
&lt;!-- Message sent by User:Trizek (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25497111 --&gt;

== Tech News: 2023-35 ==

&lt;section begin=&quot;technews-2023-W35&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/35|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] As part of the changes for the [[m:Community Wishlist Survey 2022/Better diff handling of paragraph splits|better diff handling of paragraph splits]], improved detection of splits is being rolled out. Over the last two weeks, we deployed this support to [[wikitech:Deployments/Train#Groups|group0]] and group1 wikis. This week it will be deployed to group2 wikis. [https://phabricator.wikimedia.org/T341754]
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] All [[{{#special:Contributions}}]] pages now show the user's local edit count and the account's creation date. [https://phabricator.wikimedia.org/T324166]
* Wikisource users can now use the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;prpbengalicurrency&lt;/code&gt;&lt;/bdi&gt; label to denote Bengali currency characters as page numbers inside the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;&lt;pagelist&gt;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; tag.  [https://phabricator.wikimedia.org/T268932]
* Two preferences have been relocated. The preference &quot;{{int:visualeditor-preference-visualeditor}}&quot; is now shown on the [[Special:Preferences#mw-prefsection-editing|&quot;{{int:prefs-editing}}&quot; tab]] at all wikis. Previously it was shown on the &quot;{{int:prefs-betafeatures}}&quot; tab at some wikis. The preference &quot;{{int:visualeditor-preference-newwikitexteditor-enable}}&quot; is now also shown on the &quot;{{int:prefs-editing}}&quot; tab at all wikis, instead of the &quot;{{int:prefs-betafeatures}}&quot; tab. [https://phabricator.wikimedia.org/T335056][https://phabricator.wikimedia.org/T344158]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.24|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-08-29|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-08-30|en}}. It will be on all wikis from {{#time:j xg|2023-08-31|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] New signups for a Wikimedia developer account will start being pushed towards &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[https://idm.wikimedia.org/ idm.wikimedia.org]&lt;/bdi&gt;, rather than going via Wikitech. [[wikitech:IDM|Further information about the new system is available]].
* All right-to-left language wikis, plus Korean, Armenian, Ukrainian, Russian, and Bulgarian Wikipedias, will have a link in the sidebar that provides a short URL of that page, using the [[m:Special:MyLanguage/Wikimedia URL Shortener|Wikimedia URL Shortener]]. This feature will come to more wikis in future weeks. [https://phabricator.wikimedia.org/T267921]

'''Future changes'''
* The removal of the [[mw:Special:MyLanguage/Extension:DoubleWiki|DoubleWiki extension]] is being discussed. This extension currently allows Wikisource users to view articles from multiple language versions side by side when the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;=&gt;&lt;/code&gt;&lt;/bdi&gt; symbol next to a specific language edition is selected. Comments on this are welcomed at [[phab:T344544|the phabricator task]].
* A proposal has been made to merge the second hidden-categories list (which appears below the wikitext editing form) with the main list of categories (which is further down the page). [[phab:T340606|More information is available on Phabricator]]; feedback is welcome!

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/35|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W35&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 14:00, 28 August 2023 (UTC)
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== Tech News: 2023-36 ==

&lt;section begin=&quot;technews-2023-W36&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/36|Translations]] are available.

'''Recent changes'''
* [[m:Wikisource_EditInSequence|EditInSequence]], a feature that allows users to edit pages faster on Wikisource has been moved to a Beta Feature based on community feedback. To enable it, you can navigate to the [[Special:Preferences#mw-prefsection-betafeatures|beta features tab in Preferences]]. [https://phabricator.wikimedia.org/T308098]
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] As part of the changes for the [[m:Special:MyLanguage/Community Wishlist Survey 2022/Generate Audio for IPA|Generate Audio for IPA]] and [[m:Community Wishlist Survey 2022/Multimedia and Commons/Audio links that play on click|Audio links that play on click]] wishlist proposals, the [[mw:Special:MyLanguage/Help:Extension:Phonos#Inline_audio_player_mode|inline audio player mode]] of [[mw:Extension:Phonos|Phonos]] has been deployed to all projects. [https://phabricator.wikimedia.org/T336763]
* There is a new option for Administrators when they are changing the usergroups for a user, to add the user’s user page to their watchlist. This works both via [[{{#special:UserRights}}]] and via the API. [https://phabricator.wikimedia.org/T272294]
* One new wiki has been created:
** a {{int:project-localized-name-group-wikipedia}} in [[d:Q34318|Talysh]] ([[w:tly:|&lt;code&gt;w:tly:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T345166]
'''Problems'''
* The [[mw:Special:MyLanguage/Extension:LoginNotify|LoginNotify extension]] was not sending notifications since January. It has now been fixed, so going forward, you may see notifications for failed login attempts, and successful login attempts from a new device. [https://phabricator.wikimedia.org/T344785]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.25|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-09-05|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-09-06|en}}. It will be on all wikis from {{#time:j xg|2023-09-07|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* Starting on Wednesday, a new set of Wikipedias will get &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]&quot; ({{int:project-localized-name-mhrwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-miwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-minwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mkwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mlwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mrwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mrjwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mtwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mwlwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-myvwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-mznwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nahwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-napwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ndswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nds_nlwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-newiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-newwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-novwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nqowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nrmwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nsowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nvwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ocwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-olowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-omwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-orwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-oswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pagwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pamwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-papwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pcdwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pdcwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pflwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pihwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pmswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pnbwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pntwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-pswiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T308137][https://phabricator.wikimedia.org/T308138]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/36|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W36&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:33, 4 September 2023 (UTC)
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== Tech News: 2023-37 ==

&lt;section begin=&quot;technews-2023-W37&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/37|Translations]] are available.

'''Recent changes'''
* [[mw:Special:MyLanguage/ORES|ORES]], the revision evaluation service, is now using a new open-source infrastructure on all wikis except for English Wikipedia and Wikidata. These two will follow this week. If you notice any unusual results from the Recent Changes filters that are related to ORES (for example, &quot;{{int:ores-rcfilters-damaging-title}}&quot; and &quot;{{int:ores-rcfilters-goodfaith-title}}&quot;), please [[mw:Talk:Machine Learning|report them]]. [https://phabricator.wikimedia.org/T342115]
* When you are logged in on one Wikimedia wiki and visit a different Wikimedia wiki, the system tries to log you in there automatically. This has been unreliable for a long time. You can now visit the login page to make the system try extra hard. If you feel that made logging in better or worse than it used to be, your feedback is appreciated. [https://phabricator.wikimedia.org/T326281]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.26|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-09-12|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-09-13|en}}. It will be on all wikis from {{#time:j xg|2023-09-14|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The [[mw:Special:MyLanguage/Technical decision making|Technical Decision-Making Forum Retrospective]] team invites anyone involved in the technical field of Wikimedia projects to signup to and join [[mw:Technical decision making/Listening Sessions|one of their listening sessions]] on 13 September. Another date will be scheduled later. The goal is to improve the technical decision-making processes.
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] As part of the changes for the [[m:Special:MyLanguage/Community Wishlist Survey 2022/Better diff handling of paragraph splits|Better diff handling of paragraph splits]] wishlist proposal, the inline switch widget in diff pages is being rolled out this week to all wikis. The inline switch will allow viewers to toggle between a unified inline or two-column diff wikitext format. [https://phabricator.wikimedia.org/T336716]

'''Future changes'''
* All wikis will be read-only for a few minutes on 20 September. [[m:Special:MyLanguage/Tech/Server switch|This is planned at 14:00 UTC.]] More information will be published in Tech News and will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T345263]
* The Enterprise API is launching a new feature called &quot;[http://breakingnews-beta.enterprise.wikimedia.com/ breaking news]&quot;. Currently in BETA, this attempts to identify likely &quot;newsworthy&quot; topics as they are currently being written about in any Wikipedia. Your help is requested to improve the accuracy of its detection model, especially on smaller language editions, by recommending templates or identifiable editing patterns. See more information at [[mw:Special:MyLanguage/Wikimedia Enterprise/Breaking news|the documentation page]] on MediaWiki or [[m:Special:MyLanguage/Wikimedia Enterprise/FAQ#What is Breaking News|the FAQ]] on Meta.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/37|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W37&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:07, 11 September 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25589064 --&gt;

== Tech News: 2023-38 ==

&lt;section begin=&quot;technews-2023-W38&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/38|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] MediaWiki now has a [[mw:Stable interface policy/frontend|stable interface policy for frontend code]] that more clearly defines how we deprecate MediaWiki code and wiki-based code (e.g. gadgets and user scripts). Thank you to everyone who contributed to the content and discussions. [https://phabricator.wikimedia.org/T346467][https://phabricator.wikimedia.org/T344079]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.27|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-09-19|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-09-20|en}}. It will be on all wikis from {{#time:j xg|2023-09-21|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* All wikis will be read-only for a few minutes on September 20. [[m:Special:MyLanguage/Tech/Server switch|This is planned at 14:00 UTC.]] [https://phabricator.wikimedia.org/T345263]
* All wikis will have a link in the sidebar that provides a short URL of that page, using the [[m:Special:MyLanguage/Wikimedia URL Shortener|Wikimedia URL Shortener]]. [https://phabricator.wikimedia.org/T267921]

'''Future changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The team investigating the Graph Extension posted [[mw:Extension:Graph/Plans#Proposal|a proposal for reenabling it]] and they need your input.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/38|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W38&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 19:19, 18 September 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25623533 --&gt;

== Tech News: 2023-39 ==

&lt;section begin=&quot;technews-2023-W39&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/39|Translations]] are available.

'''Recent changes'''
* The Vector 2022 skin will now remember the pinned/unpinned status for the Table of Contents for all logged-out users. [https://phabricator.wikimedia.org/T316060]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.28|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-09-26|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-09-27|en}}. It will be on all wikis from {{#time:j xg|2023-09-28|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The ResourceLoader &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;mediawiki.ui&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; modules are now deprecated as part of the move to Vue.js and Codex. There is a [[mw:Codex/Migrating_from_MediaWiki_UI|guide for migrating from MediaWiki UI to Codex]] for any tools that use it. More [[phab:T346468|details are available in the task]] and your questions are welcome there.
* Gadget definitions will have a [[mw:Special:MyLanguage/Extension:Gadgets#Options|new &quot;namespaces&quot; option]]. The option takes a list of namespace IDs. Gadgets that use this option will only load on pages in the given namespaces.

'''Future changes'''
* New variables will be added to [[mw:Special:MyLanguage/Extension:AbuseFilter|AbuseFilter]]: &lt;code&gt;&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;global_account_groups&lt;/bdi&gt;&lt;/code&gt; and &lt;code&gt;&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;global_account_editcount&lt;/bdi&gt;&lt;/code&gt;. They are available only when an account is being created. You can use them to prevent blocking automatic creation of accounts when users with many edits elsewhere visit your wiki for the first time. [https://phabricator.wikimedia.org/T345632][https://www.mediawiki.org/wiki/Special:MyLanguage/Extension:AbuseFilter/Rules_format]

'''Meetings'''
* You can join the next meeting with the Wikipedia mobile apps teams. During the meeting, we will discuss the current features and future roadmap. The meeting will be on [https://zonestamp.toolforge.org/1698426015 27 October at 17:00 (UTC)]. See [[mw:Special:MyLanguage/Wikimedia_Apps/Office_Hours#October_2023|details and how to join]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/39|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W39&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 16:51, 26 September 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25655264 --&gt;

== Tech News: 2023-40 ==

&lt;section begin=&quot;technews-2023-W40&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/40|Translations]] are available.

'''Recent changes'''
* There is a new [[Special:Preferences#mw-prefsection-rendering-advancedrendering|user preference]] for &quot;{{int:tog-forcesafemode}}&quot;. This setting will make pages load without including any on-wiki JavaScript or on-wiki stylesheet pages. It can be useful for debugging broken JavaScript gadgets. [https://phabricator.wikimedia.org/T342347]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Gadget definitions now have a [[mw:Special:MyLanguage/Extension:Gadgets#Options|new &quot;&lt;var&gt;contentModels&lt;/var&gt;&quot; option]]. The option takes a list of page content models, like &lt;code&gt;&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;wikitext&lt;/bdi&gt;&lt;/code&gt; or &lt;code&gt;&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;css&lt;/bdi&gt;&lt;/code&gt;. Gadgets that use this option will only load on pages with the given content models.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.29|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-10-03|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-10-04|en}}. It will be on all wikis from {{#time:j xg|2023-10-05|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).

'''Future changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The Vector 2022 skin will no longer use the custom styles and scripts of Vector legacy (2010). The change will be made later this year or in early 2024. See [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Features/Loading Vector 2010 scripts|how to adjust the CSS and JS pages on your wiki]]. [https://phabricator.wikimedia.org/T331679]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/40|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W40&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:26, 3 October 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25686930 --&gt;

== Tech News: 2023-41 ==

&lt;section begin=&quot;technews-2023-W41&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/41|Translations]] are available.

'''Recent changes'''
* One new wiki has been created: a {{int:project-localized-name-group-wikipedia}} in [[d:Q33291|Fon]] ([[w:fon:|&lt;code&gt;w:fon:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T347935]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.41/wmf.30|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-10-10|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-10-11|en}}. It will be on all wikis from {{#time:j xg|2023-10-12|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).
* Starting on Wednesday, a new set of Wikipedias will get &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]&quot; ({{int:project-localized-name-swwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-wawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-warwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-wowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-xalwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-xhwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-xmfwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-yiwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-yowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-zawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-zeawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-zh_min_nanwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-zuwiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T308139]
* At some wikis, newcomers are suggested images from Commons to add to articles without any images. Starting on Tuesday, newcomers at these wikis will be able to add images to unillustrated article sections. The specific wikis are listed under &quot;Images recommendations&quot; [[mw:Special:MyLanguage/Growth/Deployment table|at the Growth team deployment table]]. You can [[mw:Special:MyLanguage/Help:Growth/Tools/Add an image|learn more about this feature.]] [https://phabricator.wikimedia.org/T345940]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] In the mobile web skin (Minerva) the CSS ID &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;#page-actions&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; will be replaced with &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;#p-views&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. This change is to make it consistent with other skins and to improve support for gadgets and extensions in the mobile skin. A few gadgets may need to be updated; there are [https://phabricator.wikimedia.org/T348267 details and search-links in the task].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/41|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W41&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 14:39, 9 October 2023 (UTC)
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== Tech News: 2023-42 ==

&lt;section begin=&quot;technews-2023-W42&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/42|Translations]] are available.

'''Recent changes'''
* The [[m:Special:MyLanguage/Help:Unified login|Unified login]] system's edge login should now be fixed for some browsers (Chrome, Edge, Opera). This means that if you visit a new sister project wiki, you should be logged in automatically without the need to click &quot;Log in&quot; or reload the page. Feedback on whether it's working for you is welcome. [https://phabricator.wikimedia.org/T347889]
* [[mw:Special:MyLanguage/Manual:Interface/Edit_notice|Edit notices]] are now available within the MobileFrontend/Minerva skin. This feature was inspired by [[w:en:Wikipedia:EditNoticesOnMobile|the gadget on English Wikipedia]]. See more details in [[phab:T316178|T316178]].

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.1|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-10-17|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-10-18|en}}. It will be on all wikis from {{#time:j xg|2023-10-19|en}} ([[mw:MediaWiki 1.41/Roadmap|calendar]]).

'''Future changes'''
* In 3 weeks, in the Vector 2022 skin, code related to &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;addPortletLink&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;#p-namespaces&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; that was deprecated one year ago will be removed. If you notice tools that should appear next to the &quot;Discussion&quot; tab are then missing, please tell the gadget's maintainers to see [[phab:T347907|instructions in the Phabricator task]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/42|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W42&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:47, 16 October 2023 (UTC)
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== Tech News: 2023-43 ==

&lt;section begin=&quot;technews-2023-W43&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/43|Translations]] are available.

'''Recent changes'''
* There is a new [[mw:Special:MyLanguage/Wikimedia Language engineering/Newsletter/2023/October|Language and internationalization newsletter]], written quarterly. It contains updates on new feature development, improvements in various language-related technical projects, and related support work.
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Source map support has been enabled on all wikis. When you open the debugger in your browser's developer tools, you should be able to see the unminified JavaScript source code. [https://phabricator.wikimedia.org/T47514]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.2|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-10-24|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-10-25|en}}. It will be on all wikis from {{#time:j xg|2023-10-26|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]).

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/43|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W43&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:16, 23 October 2023 (UTC)
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== Tech News: 2023-44 ==

&lt;section begin=&quot;technews-2023-W44&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/44|Translations]] are available.

'''Recent changes'''
* The Structured Content team, as part of its project of [[:commons:Commons:WMF support for Commons/Upload Wizard Improvements|improving UploadWizard on Commons]], made some UX improvements to the upload step of choosing own vs not own work ([[phab:T347590|T347590]]), as well as to the licensing step for own work ([[phab:T347756|T347756]]).
* The Design Systems team has released version 1.0.0 of [[wmdoc:codex/latest/|Codex]], the new design system for Wikimedia. See the [[mw:Special:MyLanguage/Design_Systems_Team/Announcing_Codex_1.0|full announcement about the release of Codex 1.0.0]].

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.3|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-10-31|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-11-01|en}}. It will be on all wikis from {{#time:j xg|2023-11-02|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]).
* Listings on category pages are sorted on each wiki for that language using a [[:w:en:International Components for Unicode|library]]. For a brief period on 2 November, changes to categories will not be sorted correctly for many languages. This is because the developers are upgrading to a new version of the library. They will then use a script to fix the existing categories. This will take a few hours or a few days depending on how big the wiki is. You can [[mw:Special:MyLanguage/Wikimedia Technical Operations/ICU announcement|read more]]. [https://phabricator.wikimedia.org/T345561][https://phabricator.wikimedia.org/T267145] 
* Starting November 1, the impact module (Special:Impact) will be upgraded by the Growth team. The new impact module shows newcomers more data regarding their impact on the wiki. It was tested by a few wikis during the last few months. [https://phabricator.wikimedia.org/T336203]

'''Future changes'''
* There is [[mw:Special:MyLanguage/Extension:Graph/Plans#Roadmap|a proposed plan]] for re-enabling the Graph Extension. You can help by reviewing this proposal and [[mw:Extension_talk:Graph/Plans#c-PPelberg_(WMF)-20231020221600-Update:_20_October|sharing what you think about it]].
* The WMF is working on making it possible for administrators to [[mw:Special:MyLanguage/Community_configuration_2.0|edit MediaWiki configuration directly]]. This is similar to previous work on Special:EditGrowthConfig. [[phab:T349757|A technical RfC is running until November 08, where you can provide feedback.]] 

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/44|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W44&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:21, 30 October 2023 (UTC)
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== Tech News: 2023-45 ==

&lt;section begin=&quot;technews-2023-W45&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/45|Translations]] are available.

'''Recent changes'''
* In the Vector 2022 skin, the default font-size of a number of navigational elements (tagline, tools menu, navigational links, and more) has been increased slightly to match the font size used in page content. [https://phabricator.wikimedia.org/T346062]

'''Problems'''
* Last week, there was a problem displaying some recent edits on [https://noc.wikimedia.org/conf/highlight.php?file=dblists/s5.dblist a few wikis], for 1-6 hours. The edits were saved but not immediately shown. This was due to a database problem. [https://phabricator.wikimedia.org/T350443]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.4|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-11-07|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-11-08|en}}. It will be on all wikis from {{#time:j xg|2023-11-09|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]).
* The Growth team will reassign newcomers from former mentors to [[mw:Special:MyLanguage/Growth/Structured mentor list|the currently active mentors]]. They have also changed the notification language to be more user-friendly. [https://phabricator.wikimedia.org/T330071][https://phabricator.wikimedia.org/T327493]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/45|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W45&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:05, 6 November 2023 (UTC)
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== Tech News: 2023-46 ==

&lt;section begin=&quot;technews-2023-W46&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/46|Translations]] are available.

'''Recent changes'''
* Four new wikis have been created:
** a Wikipedia in [[d:Q7598268|Moroccan Amazigh]] ([[w:zgh:|&lt;code&gt;w:zgh:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T350216]
** a Wikipedia in [[d:Q35159|Dagaare]] ([[w:dga:|&lt;code&gt;w:dga:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T350218]
** a Wikipedia in [[d:Q33017|Toba Batak]] ([[w:bbc:|&lt;code&gt;w:bbc:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T350320]
** a Wikiquote in [[d:Q33151|Banjar]] ([[q:bjn:|&lt;code&gt;q:bjn:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T350217]

'''Problems'''
* Last week, users who previously visited Meta-Wiki or Wikimedia Commons and then became logged out on those wikis could not log in again. The problem is now resolved. [https://phabricator.wikimedia.org/T350695]
* Last week, some pop-up dialogs and menus were shown with the wrong font size. The problem is now resolved. [https://phabricator.wikimedia.org/T350544]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.5|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-11-14|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-11-15|en}}. It will be on all wikis from {{#time:j xg|2023-11-16|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]).

'''Future changes'''
* Reference Previews are coming to many wikis as a default feature. They are popups for references, similar to the [[mw:Special:MyLanguage/Page Previews|PagePreviews feature]]. [[m:WMDE Technical Wishes/ReferencePreviews#Opt-out feature|You can opt out]] of seeing them. If you are [[Special:Preferences#mw-prefsection-gadgets|using the gadgets]] Reference Tooltips or Navigation Popups, you won’t see Reference Previews. [[phab:T282999|Deployment]] is planned for November 22, 2023. 
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Canary (also known as heartbeat) events will be produced into [https://stream.wikimedia.org/?doc#/streams Wikimedia event streams] from December 11. Streams users are advised to filter out these events, by discarding all events where &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;meta.domain == &quot;canary&quot;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. Updates to [[mw:Special:MyLanguage/Manual:Pywikibot|Pywikibot]] or [https://github.com/ChlodAlejandro/wikimedia-streams wikimedia-streams] will discard these events by default. [https://phabricator.wikimedia.org/T266798]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/46|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W46&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:52, 13 November 2023 (UTC)
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== Tech News: 2023-47 ==

&lt;section begin=&quot;technews-2023-W47&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/47|Translations]] are available.

'''Changes later this week'''
* There is no new MediaWiki version this week. [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* Starting on Wednesday, a new set of Wikipedias will get &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]&quot; ({{int:project-localized-name-quwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-rmwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-rmywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-rnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-roa_rupwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-roa_tarawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ruewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-rwwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-sawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-sahwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-satwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-scwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-scnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-scowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-sdwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-sewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-sgwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-shwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-siwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-skwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-slwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-smwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-sowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-sqwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-srwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-srnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-sswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-stwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-stqwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-suwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-szlwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tcywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tetwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tgwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-thwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tkwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tlwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-towiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tpiwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-trwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ttwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-twwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tywiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-tyvwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-udmwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ugwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-uzwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-vewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-vecwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-vepwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-vlswiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-vowiki/en}}). This is part of the [[phab:T304110|progressive deployment of this tool to more Wikipedias]]. The communities can [[mw:Special:MyLanguage/Growth/Community configuration|configure how this feature works locally]]. [https://phabricator.wikimedia.org/T308141][https://phabricator.wikimedia.org/T308142][https://phabricator.wikimedia.org/T308143] 
* The Vector 2022 skin will have some minor visual changes to drop-down menus, column widths, and more. These changes were added to four Wikipedias last week. If no issues are found, these changes will proceed to all wikis this week. These changes will make it possible to add new menus for readability and dark mode. [[mw:Special:MyLanguage/Reading/Web/Desktop_Improvements/Updates#November_2023:_Visual_changes,_more_deployments,_and_shifting_focus|Learn more]]. [https://phabricator.wikimedia.org/T347711]

'''Future changes'''
* There is [[mw:Extension talk:Graph/Plans#Update: 15 November|an update on re-enabling the Graph Extension]]. To speed up the process, Vega 2 will not be supported and only [https://phabricator.wikimedia.org/T335325 some protocols] will be available at launch. You can help by sharing what you think about the plan.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/47|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W47&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:55, 21 November 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25884616 --&gt;

== Tech News: 2023-48 ==

&lt;section begin=&quot;technews-2023-W48&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/48|Translations]] are available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.7|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-11-28|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-11-29|en}}. It will be on all wikis from {{#time:j xg|2023-11-30|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). There is no new MediaWiki version next week. [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] MediaWiki's JavaScript system will now allow &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;async&lt;/code&gt;/&lt;code&gt;await&lt;/code&gt;&lt;/bdi&gt; syntax in gadgets and user scripts. Gadget authors should remember that users' browsers may not support it, so it should be used appropriately. [https://phabricator.wikimedia.org/T343499]
* The deployment of &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add_a_link|Add a link]]&quot; announced [[m:Special:MyLanguage/Tech/News/2023/47|last week]] was postponed. It will resume this week.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/48|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W48&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:08, 27 November 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25906379 --&gt;

== Tech News: 2023-49 ==

&lt;section begin=&quot;technews-2023-W49&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/49|Translations]] are available.

'''Recent changes'''
* The spacing between paragraphs on Vector 2022 has been changed from 7px to 14px to match the size of the text. This will make it easier to distinguish paragraphs from sentences. [https://phabricator.wikimedia.org/T351754]
* The &quot;{{int:Visualeditor-dialog-meta-categories-defaultsort-label}}&quot; feature in VisualEditor is working again. You no longer need to switch to source editing to edit &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;{{DEFAULTSORT:...}}&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; keywords. [https://phabricator.wikimedia.org/T337398]

'''Changes later this week'''
* There is no new MediaWiki version this week. [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* On 6 December, people who have the enabled the preference for &quot;{{int:Discussiontools-preference-visualenhancements}}&quot; will notice the [[mw:Special:MyLanguage/Talk pages project/Usability|talk page usability improvements]] appear on pages that include the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;__NEWSECTIONLINK__&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; magic word. If you notice any issues, please [[phab:T352232|share them with the team on Phabricator]].

'''Future changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The Toolforge [[wikitech:News/Toolforge Grid Engine deprecation|Grid Engine shutdown process]] will start on December 14. Maintainers of [[toolforge:grid-deprecation|tools that still use this old system]] should plan to migrate to Kubernetes, or tell the team your plans on Phabricator in the task about your tool, before that date. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/VIWWQKMSQO2ED3TVUR7KPPWRTOBYBVOA/]
* Communities using [[mw:Special:MyLanguage/Structured_Discussions|Structured Discussions]] are being contacted regarding [[mw:Special:MyLanguage/Structured_Discussions/Deprecation|the upcoming deprecation of Structured Discussions]]. You can read more about this project, and share your comments, [[mw:Special:MyLanguage/Structured_Discussions/Deprecation|on the project's page]].

'''Events'''
* Registration &amp; Scholarship applications are now open for the [[mw:Special:MyLanguage/Wikimedia Hackathon 2024|Wikimedia Hackathon 2024]] that will take place from 3–5 May in Tallinn, Estonia. Scholarship applications are open until 5 January 2024.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/49|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W49&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:50, 4 December 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25914435 --&gt;

== Tech News: 2023-50 ==

&lt;section begin=&quot;technews-2023-W50&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/50|Translations]] are available.

'''Recent changes'''
* On Wikimedia Commons, there are some minor user-interface improvements for the &quot;choosing own vs not own work&quot; step in the UploadWizard. This is part of the Structured Content team's project of [[:commons:Commons:WMF support for Commons/Upload Wizard Improvements|improving UploadWizard on Commons]]. [https://phabricator.wikimedia.org/T352707][https://phabricator.wikimedia.org/T352709]

'''Problems'''
* There was a problem showing the [[mw:Special:MyLanguage/Growth/Personalized first day/Newcomer homepage|Newcomer homepage]] feature with the &quot;impact module&quot; and their page-view graphs, for a few days in early December. This has now been fixed. [https://phabricator.wikimedia.org/T352352][https://phabricator.wikimedia.org/T352349]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.9|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-12-12|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-12-13|en}}. It will be on all wikis from {{#time:j xg|2023-12-14|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

'''Future changes'''
* [[File:Octicons-tools.svg|15px|link=]] The [https://wikimediafoundation.limesurvey.net/796964 2023 Developer Satisfaction Survey] is seeking the opinions of the Wikimedia developer community. Please take the survey if you have any role in developing software for the Wikimedia ecosystem. The survey is open until 5 January 2024, and has an associated [[foundation:Legal:December_2023_Developer_Satisfaction_Survey|privacy statement]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/50|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W50&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 02:12, 12 December 2023 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25945501 --&gt;

== Tech News: 2023-51 ==

&lt;section begin=&quot;technews-2023-W51&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2023/51|Translations]] are available.

'''Tech News'''
* The next issue of Tech News will be sent out on 8 January 2024 because of [[w:en:Christmas and holiday season|the holidays]].

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.10|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2023-12-19|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2023-12-20|en}}. It will be on all wikis from {{#time:j xg|2023-12-21|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). There is no new MediaWiki version next week. [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* Starting December 18, it won't be possible to activate Structured Discussions on a user's own talk page using the Beta feature. The Beta feature option remains available for users who want to deactivate Structured Discussions. This is part of [[mw:Structured Discussions/Deprecation|Structured Discussions' deprecation work]]. [https://phabricator.wikimedia.org/T248309] 
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] There will be full support for redirects in the Module namespace. The &quot;Move Page&quot; feature will leave an appropriate redirect behind, and such redirects will be appropriately recognized by the software (e.g. hidden from [[{{#special:UnconnectedPages}}]]). There will also be support for [[mw:Special:MyLanguage/Extension:Scribunto/Lua reference manual#Renaming or moving modules|manual redirects]]. [https://phabricator.wikimedia.org/T120794]

'''Future changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The MediaWiki JavaScript documentation is moving to a new format. During the move, you can read the old docs using [https://doc.wikimedia.org/mediawiki-core/REL1_41/js/ version 1.41]. Feedback about [https://doc.wikimedia.org/mediawiki-core/master/js/ the new site] is welcome on the [[mw:Talk:JSDoc_WMF_theme|project talk page]].
* The Wishathon is a new initiative that encourages collaboration across the Wikimedia community to develop solutions for wishes collected through the [[m:Special:MyLanguage/Community Wishlist Survey|Community Wishlist Survey]]. The first community Wishathon will take place from 15–17 March. If you are interested in a project proposal as a user, developer, designer, or product lead, you can [[m:Special:MyLanguage/Event:WishathonMarch2024|register for the event and read more]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2023/51|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2023-W51&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 16:17, 18 December 2023 (UTC)
&lt;!-- Message sent by User:Johan (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=25959059 --&gt;

== Tech News: 2024-02 ==

&lt;section begin=&quot;technews-2024-W02&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/02|Translations]] are available.

'''Recent changes'''
* [https://mediawiki2latex.wmflabs.org/ mediawiki2latex] is a tool that converts wiki content into the formats of LaTeX, PDF, ODT, and EPUB. The code now runs many times faster due to recent improvements. There is also an optional Docker container you can [[b:de:Benutzer:Dirk_Hünniger/wb2pdf/install#Using_Docker|install]] on your local machine.
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The way that Random pages are selected has been updated. This will slowly reduce the problem of some pages having a lower chance of appearing. [https://phabricator.wikimedia.org/T309477]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.13|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-01-09|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-01-10|en}}. It will be on all wikis from {{#time:j xg|2024-01-11|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/02|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W02&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:19, 9 January 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26026251 --&gt;

== Tech News: 2024-03 ==

&lt;section begin=&quot;technews-2024-W03&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/03|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Pages that use the JSON [[mw:Special:MyLanguage/Manual:ContentHandler|contentmodel]] will now use tabs instead of spaces for auto-indentation. This will significantly reduce the page size. [https://phabricator.wikimedia.org/T326065]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] [[mw:Special:MyLanguage/Extension:Gadgets|Gadgets]] and personal user scripts may now use JavaScript syntax introduced in ES6 (also known as &quot;ES2015&quot;) and ES7 (&quot;ES2016&quot;). MediaWiki validates the source code to protect other site functionality from syntax errors, and to ensure scripts are valid in all [[mw:Special:MyLanguage/Compatibility#Browsers|supported browsers]]. Previously, Gadgets could use the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;requiresES6&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; option. This option is no longer needed and will be removed in the future. [https://phabricator.wikimedia.org/T75714]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] [[mw:Special:MyLanguage/Manual:Bot passwords|Bot passwords]] and [[mw:Special:MyLanguage/OAuth/Owner-only consumers|owner-only OAuth consumers]] can now be restricted to allow editing only specific pages. [https://phabricator.wikimedia.org/T349957]
* You can now [[mw:Special:MyLanguage/Extension:Thanks|thank]] edits made by bots. [https://phabricator.wikimedia.org/T341388]
* An update on the status of the Community Wishlist Survey for 2024 [[m:Special:MyLanguage/Community Wishlist Survey/Future Of The Wishlist/January 4, 2024 Update|has been published]]. Please read and give your feedback.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.14|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-01-16|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-01-17|en}}. It will be on all wikis from {{#time:j xg|2024-01-18|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* Starting on January 17, it will not be possible to login to Wikimedia wikis from some specific old versions of the Chrome browser (versions 51–66, released between 2016 and 2018). Additionally, users of iOS 12, or Safari on Mac OS 10.14, may need to login to each wiki separately. [https://phabricator.wikimedia.org/T344791]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;jquery.cookie&lt;/code&gt;&lt;/bdi&gt; module was deprecated and replaced with the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;mediawiki.cookie&lt;/code&gt;&lt;/bdi&gt; module last year. A script has now been run to replace any remaining uses, and this week the temporary alias will be removed. [https://phabricator.wikimedia.org/T354966]

'''Future changes'''
* Wikimedia Deutschland is working to [[m:WMDE Technical Wishes/Reusing references|make reusing references easier]]. They are looking for people who are interested in participating in [https://wikimedia.sslsurvey.de/User-research-into-Reusing-References-Sign-up-Form-2024/en/ individual video calls for user research in January and February].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/03|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W03&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:13, 16 January 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26074460 --&gt;

== Tech News: 2024-04 ==

&lt;section begin=&quot;technews-2024-W04&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/04|Translations]] are available.

'''Problems'''
* A bug in UploadWizard prevented linking to the userpage of the uploader when uploading. It has now been fixed. [https://phabricator.wikimedia.org/T354529]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.15|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-01-23|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-01-24|en}}. It will be on all wikis from {{#time:j xg|2024-01-25|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/04|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W04&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:03, 23 January 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26096197 --&gt;

== Tech News: 2024-05 ==

&lt;section begin=&quot;technews-2024-W05&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/05|Translations]] are available.

'''Recent changes'''
* Starting Monday January 29, all talk pages messages' timestamps will become a link. This link is a permanent link to the comment. It allows users to find the comment they are looking for, even if this comment was moved elsewhere. This will affect all wikis except for the English Wikipedia. You can read more about this change [https://diff.wikimedia.org/2024/01/29/talk-page-permalinks-dont-lose-your-threads/ on Diff] or [[mw:Special:MyLanguage/Help:DiscussionTools#Talk_pages_permalinking|on Mediawiki.org]].&lt;!-- The Diff post will be published on Monday morning UTC--&gt; [https://phabricator.wikimedia.org/T302011]
* There are some improvements to the CAPTCHA to make it harder for spam bots and scripts to bypass it. If you have feedback on this change, please comment on [[phab:T141490|the task]]. Staff are monitoring metrics related to the CAPTCHA, as well as secondary metrics such as account creations and edit counts.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.16|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-01-30|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-01-31|en}}. It will be on all wikis from {{#time:j xg|2024-02-01|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] On February 1, a link will be added to the &quot;Tools&quot; menu to download a [[w:en:QR code|QR code]] that links to the page you are viewing. There will also be a new [[{{#special:QrCode}}]] page to create QR codes for any Wikimedia URL. This addresses the [[m:Community Wishlist Survey 2023/Mobile and apps/Add ability to share QR code for a page in any Wikimedia project|#19 most-voted wish]] from the [[m:Community Wishlist Survey 2023/Results|2023 Community Wishlist Survey]]. [https://phabricator.wikimedia.org/T329973]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] [[mw:Special:MyLanguage/Extension:Gadgets|Gadgets]] which only work in some skins have sometimes used the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;targets&lt;/code&gt;&lt;/bdi&gt; option to limit where you can use them. This will stop working this week. You should use the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;skins&lt;/code&gt;&lt;/bdi&gt; option instead. [https://phabricator.wikimedia.org/T328497]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/05|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W05&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 19:31, 29 January 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26137870 --&gt;

== Tech News: 2024-06 ==

&lt;section begin=&quot;technews-2024-W06&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/06|Translations]] are available.

'''Recent changes'''
*The mobile site history pages now use the same HTML as the desktop history pages. If you hear of any problems relating to mobile history usage please point them to [[phab:T353388|the phabricator task]].
*On most wikis, admins can now block users from making specific actions. These actions are: uploading files, creating new pages, moving (renaming) pages, and sending thanks. The goal of this feature is to allow admins to apply blocks that are adequate to the blocked users' activity. [[m:Special:MyLanguage/Community health initiative/Partial blocks#action-blocks|Learn more about &quot;action blocks&quot;]]. [https://phabricator.wikimedia.org/T242541][https://phabricator.wikimedia.org/T280531]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.17|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-02-06|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-02-07|en}}. It will be on all wikis from {{#time:j xg|2024-02-08|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* Talk pages permalinks that included diacritics and non-Latin script were malfunctioning. This issue is fixed. [https://phabricator.wikimedia.org/T356199]

'''Future changes'''
* [[m:WMDE Technical Wishes/ReferencePreviews#24WPs|24 Wikipedias]] with [[mw:Special:MyLanguage/Reference_Tooltips|Reference Tooltips]] as a default gadget are encouraged to remove that default flag. This would make [[mw:Special:MyLanguage/Help:Reference_Previews|Reference Previews]] the new default for reference popups, leading to a more consistent experience across wikis. For [[m:WMDE Technical Wishes/ReferencePreviews#46WPs|46 Wikipedias]] with less than 4 interface admins, the change is already scheduled for mid-February, [[m:Talk:WMDE Technical Wishes/ReferencePreviews#Reference Previews to become the default for previewing references on more wikis.|unless there are concerns]]. The older Reference Tooltips gadget will still remain usable and will override this feature, if it is available on your wiki and you have enabled it in your settings. [https://meta.wikimedia.org/wiki/WMDE_Technical_Wishes/ReferencePreviews#Reference_Previews_to_become_the_default_for_previewing_references_on_more_wikis][https://phabricator.wikimedia.org/T355312]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/06|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W06&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 19:22, 5 February 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26180971 --&gt;

== Tech News: 2024-07 ==

&lt;section begin=&quot;technews-2024-W07&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/07|Translations]] are available.

'''Recent changes'''
*  The [[d:Wikidata:SPARQL query service/WDQS graph split|WDQS Graph Split experiment]] is working and loaded onto 3 test servers. The team in charge is testing the split's impact and requires feedback from WDQS users through the UI or programmatically in different channels. [https://www.wikidata.org/wiki/Wikidata_talk:SPARQL_query_service/WDQS_graph_split][https://phabricator.wikimedia.org/T356773][https://www.wikidata.org/wiki/User:Sannita_(WMF)] Users' feedback will validate the impact of various use cases and workflows around the Wikidata Query service. [https://www.wikidata.org/wiki/Wikidata:SPARQL_query_service/WDQS_backend_update/October_2023_scaling_update][https://www.mediawiki.org/wiki/Wikidata_Query_Service/User_Manual#Federation]

'''Problems'''
*There was a bug that affected the appearance of visited links when using mobile device to access wiki sites. It made the links appear black; [[phab:T356928|this issue]] is fixed.  

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.18|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-02-13|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-02-14|en}}. It will be on all wikis from {{#time:j xg|2024-02-15|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] As work continues on the grid engine deprecation,[https://wikitech.wikimedia.org/wiki/News/Toolforge_Grid_Engine_deprecation] tools on the grid engine will be stopped starting on February 14th, 2024. If you have tools actively migrating you can ask for an extension so they are not stopped. [https://wikitech.wikimedia.org/wiki/Portal:Toolforge/About_Toolforge#Communication_and_support] 

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/07|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W07&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 05:48, 13 February 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26223994 --&gt;

== Tech News: 2024-08 ==

&lt;section begin=&quot;technews-2024-W08&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/08|Translations]] are available.

'''Recent changes'''
* If you have the &quot;{{int:Tog-enotifwatchlistpages}}&quot; option enabled, edits by bot accounts no longer trigger notification emails. Previously, only minor edits would not trigger the notification emails. [https://phabricator.wikimedia.org/T356984]
* There are changes to how user and site scripts load for [[mw:Special:MyLanguage/Skin:Vector/2022| Vector 2022]] on specific wikis. The changes impacted the following Wikis: all projects with [[mw:Special:MyLanguage/Skin:Vector|Vector legacy]] as the default skin, Wikivoyage, and Wikibooks. Other wikis will be affected over the course of the next three months. Gadgets are not impacted. If you have been affected or want to minimize the impact on your project, see [[Phab:T357580| this ticket]]. Please coordinate and take action proactively.
*Newly auto-created accounts (the accounts you get when you visit a new wiki) now have the same local notification preferences as users who freshly register on that wiki. It is effected in four notification types listed in the [[phab:T353225|task's description]].
*The maximum file size when using [[c:Special:MyLanguage/Commons:Upload_Wizard|Upload Wizard]] is now 5 GiB. [https://phabricator.wikimedia.org/T191804]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.19|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-02-20|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-02-21|en}}. It will be on all wikis from {{#time:j xg|2024-02-22|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Selected tools on the grid engine have been [[wikitech:News/Toolforge_Grid_Engine_deprecation|stopped]] as we prepare to shut down the grid on March 14th, 2024. The tool's code and data have not been deleted. If you are a maintainer and you want your tool re-enabled reach out to the [[wikitech:Portal:Toolforge/About_Toolforge#Communication_and_support|team]]. Only tools that have asked for extension are still running on the grid.
* The CSS &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;[https://developer.mozilla.org/en-US/docs/Web/CSS/filter filter]&lt;/code&gt;&lt;/bdi&gt; property can now be used in HTML &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;style&lt;/code&gt;&lt;/bdi&gt; attributes in wikitext. [https://phabricator.wikimedia.org/T308160]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/08|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W08&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 15:36, 19 February 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26254282 --&gt;

== Tech News: 2024-09 ==

&lt;section begin=&quot;technews-2024-W09&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/09|Translations]] are available.

'''Recent changes'''
* The [[mw:Special:MyLanguage/VisualEditor_on_mobile|mobile visual editor]] is now the default editor for users who never edited before, at a small group of wikis. [[mw:Special:MyLanguage/VisualEditor_on_mobile/VE_mobile_default#A/B_test_results| Research ]] shows that users using this editor are slightly more successful publishing the edits they started, and slightly less successful publishing non-reverted edits. Users who defined the wikitext editor as their default on desktop will get the wikitext editor on mobile for their first edit on mobile as well. [https://phabricator.wikimedia.org/T352127]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The [[mw:Special:MyLanguage/ResourceLoader/Core modules#mw.config|mw.config]] value &lt;code&gt;wgGlobalGroups&lt;/code&gt; now only contains groups that are active in the wiki. Scripts no longer have to check whether the group is active on the wiki via an API request. A code example of the above is: &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;if (/globalgroupname/.test(mw.config.get(&quot;wgGlobalGroups&quot;)))&lt;/code&gt;&lt;/bdi&gt;. [https://phabricator.wikimedia.org/T356008]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.20|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-02-27|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-02-28|en}}. It will be on all wikis from {{#time:j xg|2024-02-29|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

'''Future changes'''
* The right to change [[mw:Special:MyLanguage/Manual:Tags|edit tags]] (&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;changetags&lt;/code&gt;&lt;/bdi&gt;) will be removed from users in Wikimedia sites, keeping it by default for admins and bots only. Your community can ask to retain the old configuration on your wiki before this change happens. Please indicate in [[phab:T355639|this ticket]] to keep it for your community before the end of March 2024.
'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/09|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W09&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 19:23, 26 February 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26294125 --&gt;

== Tech News: 2024-10 ==

&lt;section begin=&quot;technews-2024-W10&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/10|Translations]] are available.

'''Recent changes'''
* The &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;Special:Book&lt;/code&gt;&lt;/bdi&gt; page (as well as the associated &quot;Create a book&quot; functionality) provided by the old [[mw:Special:MyLanguage/Extension:Collection|Collection extension]] has been removed from all Wikisource wikis, as it was broken. This does not affect the ability to download normal books, which is provided by the [[mw:Special:MyLanguage/Extension:Wikisource|Wikisource extension]]. [https://phabricator.wikimedia.org/T358437]
* [[m:Wikitech|Wikitech]] now uses the next-generation [[mw:Special:MyLanguage/Parsoid|Parsoid]] wikitext parser by default to generate all pages in the Talk namespace.  Report any problems on the [[mw:Talk:Parsoid/Parser_Unification/Known_Issues|Known Issues discussion page]]. You can use the [[mw:Special:MyLanguage/Extension:ParserMigration|ParserMigration]] extension to control the use of Parsoid; see the [[mw:Special:MyLanguage/Help:Extension:ParserMigration|ParserMigration help documentation]] for more details.
* Maintenance on [https://etherpad.wikimedia.org etherpad] is completed. If you encounter any issues, please indicate in [[phab:T316421|this ticket]].
* [[File:Octicons-tools.svg|12px|link=|alt=| Advanced item]] [[mw:Special:MyLanguage/Extension:Gadgets|Gadgets]] allow interface admins to create custom features with CSS and JavaScript. The &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;Gadget&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;Gadget_definition&lt;/code&gt;&lt;/bdi&gt; namespaces and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;gadgets-definition-edit&lt;/code&gt;&lt;/bdi&gt; user right were reserved for an experiment in 2015, but were never used. These were visible on Special:Search and Special:ListGroupRights. The unused namespaces and user rights are now removed. No pages are moved, and no changes need to be made. [https://phabricator.wikimedia.org/T31272]
* A usability improvement to the &quot;Add a citation&quot; in Wikipedia workflow has been made, the insert button was moved to the popup header. [https://phabricator.wikimedia.org/T354847]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.21|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-03-05|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-03-06|en}}. It will be on all wikis from {{#time:j xg|2024-03-07|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

'''Future changes'''
*  All wikis will be read-only for a few minutes on March 20. This is planned at 14:00 UTC. More information will be published in Tech News and will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T358233]
* The HTML markup of headings and section edit links will be changed later this year to improve accessibility. See [[mw:Special:MyLanguage/Heading_HTML_changes|Heading HTML changes]] for details. The new markup will be the same as in the new Parsoid wikitext parser. You can test your gadget or stylesheet with the new markup if you add &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;?useparsoid=1&lt;/code&gt;&lt;/bdi&gt; to your URL ([[mw:Special:MyLanguage/Help:Extension:ParserMigration#Selecting_a_parser_using_a_URL_query_string|more info]]) or turn on Parsoid read views in your user options ([[mw:Special:MyLanguage/Help:Extension:ParserMigration#Enabling_via_user_preference|more info]]).
*

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/10|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W10&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 19:47, 4 March 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26329807 --&gt;

== Tech News: 2024-11 ==

&lt;section begin=&quot;technews-2024-W11&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/11|Translations]] are available.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.22|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-03-12|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-03-13|en}}. It will be on all wikis from {{#time:j xg|2024-03-14|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* After consulting with various communities, the line height of the text on the [[mw:Special:MyLanguage/Skin:Minerva Neue|Minerva skin]] will be increased to its previous value of 1.65. Different options for typography can also be set using the options in the menu, as needed. [https://phabricator.wikimedia.org/T358498]
*The active link color in [[mw:Special:MyLanguage/Skin:Minerva Neue|Minerva]] will be changed to provide more consistency with our other platforms and best practices. [https://phabricator.wikimedia.org/T358516]
* [[c:Special:MyLanguage/Commons:Structured data|Structured data on Commons]] will no longer ask whether you want to leave the page without saving. This will prevent the “information you’ve entered may not be saved” popups from appearing when no information have been entered. It will also make file pages on Commons load faster in certain cases. However, the popups will be hidden even if information has indeed been entered. If you accidentally close the page before saving the structured data you entered, that data will be lost. [https://phabricator.wikimedia.org/T312315]

'''Future changes'''
* All wikis will be read-only for a few minutes on March 20. This is planned at 14:00 UTC. More information will be published in Tech News and will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T358233][https://meta.wikimedia.org/wiki/Special:MyLanguage/Tech/Server_switch]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/11|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W11&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:04, 11 March 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26374013 --&gt;

== Tech News: 2024-12 ==

&lt;section begin=&quot;technews-2024-W12&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/12|Translations]] are available.

'''Recent changes'''
* The notice &quot;Language links are at the top of the page&quot; that appears in the [[mw:Special:MyLanguage/Skin:Vector/2022|Vector 2022 skin]] main menu has been removed now that users have learned the new location of the Language switcher. [https://phabricator.wikimedia.org/T353619]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] [[m:Special:MyLanguage/IP_Editing:_Privacy_Enhancement_and_Abuse_Mitigation/IP_Info_feature|IP info feature]] displays data from Spur, an IP addresses database. Previously, the only data source for this feature was MaxMind. Now, IP info is more useful for patrollers. [https://phabricator.wikimedia.org/T341395]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The Toolforge Grid Engine services have been shut down after the final migration process from Grid Engine to Kubernetes. [https://wikitech.wikimedia.org/wiki/Obsolete:Toolforge/Grid][https://wikitech.wikimedia.org/wiki/News/Toolforge_Grid_Engine_deprecation][https://techblog.wikimedia.org/2022/03/14/toolforge-and-grid-engine/]
* Communities can now customize the default reasons for undeleting a page by creating [[MediaWiki:Undelete-comment-dropdown]]. [https://phabricator.wikimedia.org/T326746]

'''Problems'''
* [[m:Special:MyLanguage/WMDE_Technical_Wishes/RevisionSlider|RevisionSlider]] is an interface to interactively browse a page's history. Users in [[mw:Special:MyLanguage/Extension:RevisionSlider/Developing_a_RTL-accessible_feature_in_MediaWiki_-_what_we%27ve_learned_while_creating_the_RevisionSlider|right-to-left]] languages reported RevisionSlider reacting wrong to mouse clicks. This should be fixed now. [https://phabricator.wikimedia.org/T352169]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.23|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-03-19|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-03-20|en}}. It will be on all wikis from {{#time:j xg|2024-03-21|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* All wikis will be read-only for a few minutes on March 20. This is planned at [https://zonestamp.toolforge.org/1710943200 14:00 UTC]. [https://phabricator.wikimedia.org/T358233][https://meta.wikimedia.org/wiki/Special:MyLanguage/Tech/Server_switch]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/12|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W12&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 17:39, 18 March 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26410165 --&gt;

== Tech News: 2024-13 ==

&lt;section begin=&quot;technews-2024-W13&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/13|Translations]] are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] An update was made on March 18th 2024 to how various projects load site, user JavaScript and CSS in [[mw:Special:MyLanguage/Skin:Vector/2022|Vector 2022 skin]]. A [[phab:T360384|checklist]] is provided for site admins to follow.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.24|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-03-26|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-03-27|en}}. It will be on all wikis from {{#time:j xg|2024-03-28|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/13|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W13&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 18:56, 25 March 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26446209 --&gt;

== Tech News: 2024-14 ==

&lt;section begin=&quot;technews-2024-W14&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/14|Translations]] are available.

'''Recent changes'''
* Users of the [[mw:Special:MyLanguage/Reading/Web/Accessibility_for_reading|reading accessibility]] beta feature will notice that the default line height for the standard and large text options has changed. [https://phabricator.wikimedia.org/T359030]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.25|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-04-02|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-04-03|en}}. It will be on all wikis from {{#time:j xg|2024-04-04|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

'''Future changes'''
* The Wikimedia Foundation has an annual plan. The annual plan decides what the Wikimedia Foundation will work on. You can now read [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2024-2025/Product &amp; Technology OKRs#Draft Key Results|the draft key results]] for the Product and Technology department. They are suggestions for what results the Foundation wants from big technical changes from July 2024 to June 2025. You can [[m:Talk:Wikimedia Foundation Annual Plan/2024-2025/Product &amp; Technology OKRs|comment on the talk page]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/14|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W14&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 03:36, 2 April 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26462933 --&gt;

== Tech News: 2024-15 ==

&lt;section begin=&quot;technews-2024-W15&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/15|Translations]] are available.

'''Recent changes'''
* Web browsers can use tools called [[:w:en:Browser extension|extensions]]. There is now a Chrome extension called [[m:Future Audiences/Experiment:Citation Needed|Citation Needed]] which you can use to see if an online statement is supported by a Wikipedia article. This is a small experiment to see if Wikipedia can be used this way. Because it is a small experiment, it can only be used in Chrome in English.
* [[File:Octicons-gift.svg|12px|link=|alt=|Wishlist item]] A new [[mw:Special:MyLanguage/Help:Edit Recovery|Edit Recovery]] feature has been added to all wikis, available as a [[Special:Preferences#mw-prefsection-editing|user preference]]. Once you enable it, your in-progress edits will be stored in your web browser, and if you accidentally close an editing window or your browser or computer crashes, you will be prompted to recover the unpublished text. Please leave any feedback on the [[m:Special:MyLanguage/Talk:Community Wishlist Survey 2023/Edit-recovery feature|project talk page]]. This was the #8 wish in the 2023 Community Wishlist Survey.
* Initial results of [[mw:Special:MyLanguage/Edit check|Edit check]] experiments [[mw:Special:MyLanguage/Edit_check#4_April_2024|have been published]]. Edit Check is now deployed as a default feature at [[phab:T342930#9538364|the wikis that tested it]]. [[mw:Talk:Edit check|Let us know]] if you want your wiki to be part of the next deployment of Edit check. [https://phabricator.wikimedia.org/T342930][https://phabricator.wikimedia.org/T361727]
* Readers using the [[mw:Special:MyLanguage/Skin:Minerva Neue|Minerva skin]] on mobile will notice there has been an improvement in the line height across all typography settings. [https://phabricator.wikimedia.org/T359029]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.42/wmf.26|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-04-09|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-04-10|en}}. It will be on all wikis from {{#time:j xg|2024-04-11|en}} ([[mw:MediaWiki 1.42/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* New accounts and logged-out users will get the [[mw:Special:MyLanguage/VisualEditor|visual editor]] as their default editor on mobile. This deployment is made at all wikis except for the English Wikipedia. [https://phabricator.wikimedia.org/T361134]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/15|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W15&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:37, 8 April 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26564838 --&gt;

== Tech News: 2024-16 ==

&lt;section begin=&quot;technews-2024-W16&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/16|Translations]] are available.

'''Problems'''
* Between 2 April and 8 April, on wikis using [[mw:Special:MyLanguage/Extension:FlaggedRevs|Flagged Revisions]], the &quot;{{Int:tag-mw-reverted}}&quot; tag was not applied to undone edits. In addition, page moves, protections and imports were not autoreviewed. This problem is now fixed. [https://phabricator.wikimedia.org/T361918][https://phabricator.wikimedia.org/T361940]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.1|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-04-16|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-04-17|en}}. It will be on all wikis from {{#time:j xg|2024-04-18|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* [[mw:Special:MyLanguage/Help:Magic words#DEFAULTSORT|Default category sort keys]] will now affect categories added by templates placed in [[mw:Special:MyLanguage/Help:Cite|footnotes]]. Previously footnotes used the page title as the default sort key even if a different default sort key was specified (category-specific sort keys already worked). [https://phabricator.wikimedia.org/T40435]
* A new variable &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;page_last_edit_age&lt;/code&gt;&lt;/bdi&gt; will be added to [[Special:AbuseFilter|abuse filters]]. It tells how many seconds ago the last edit to a page was made. [https://phabricator.wikimedia.org/T269769]

'''Future changes'''
* Volunteer developers are kindly asked to update the code of their tools and features to handle [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|temporary accounts]]. [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/For developers/2024-04 CTA|Learn more]].
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Four database fields will be removed from database replicas (including [[quarry:|Quarry]]). This affects only the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;abuse_filter&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;abuse_filter_history&lt;/code&gt;&lt;/bdi&gt; tables. Some queries might need to be updated. [https://phabricator.wikimedia.org/T361996]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/16|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W16&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:29, 15 April 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26564838 --&gt;

== Tech News: 2024-17 ==

&lt;section begin=&quot;technews-2024-W17&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/17|Translations]] are available.

'''Recent changes'''
* Starting this week, newcomers editing Wikipedia [[mw:Special:MyLanguage/Growth/Positive reinforcement#Leveling up 3|will be encouraged]] to try structured tasks. [[mw:Special:MyLanguage/Growth/Feature summary#Newcomer tasks|Structured tasks]] have been shown to [[mw:Special:MyLanguage/Growth/Personalized first day/Structured tasks/Add a link/Experiment analysis, December 2021|improve newcomer activation and retention]]. [https://phabricator.wikimedia.org/T348086]
* You can [[m:Special:MyLanguage/Coolest Tool Award|nominate your favorite tools]] for the fifth edition of the Coolest Tool Award. Nominations will be open until May 10. 

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.2|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-04-23|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-04-24|en}}. It will be on all wikis from {{#time:j xg|2024-04-25|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

'''Future changes'''
* This is the last warning that by the end of May 2024 the Vector 2022 skin will no longer share site and user scripts/styles with old Vector. For user-scripts that you want to keep using on Vector 2022, copy the contents of [[{{#special:MyPage}}/vector.js]] to [[{{#special:MyPage}}/vector-2022.js]]. There are [[mw:Special:MyLanguage/Reading/Web/Desktop Improvements/Features/Loading Vector 2010 scripts|more technical details]] available. Interface administrators who foresee this leading to lots of technical support questions may wish to send a mass message to your community, as was done on French Wikipedia. [https://phabricator.wikimedia.org/T362701]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/17|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W17&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:28, 22 April 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26647188 --&gt;

== Tech News: 2024-18 ==

&lt;section begin=&quot;technews-2024-W18&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/18|Translations]] are available.

'''Recent changes'''

[[File:Talk_pages_default_look_(April_2023).jpg|thumb|alt=Screenshot of the visual improvements made on talk pages|Example of a talk page with the new design, in French.]]
* The appearance of talk pages changed for the following wikis: {{int:project-localized-name-azwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-bnwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-dewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-fawiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hewiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-hiwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-idwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-kowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-nlwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ptwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-rowiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-thwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-trwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-ukwiki/en}}{{int:comma-separator/en}}{{int:project-localized-name-viwiki/en}}. These wikis participated to a test, where 50% of users got the new design, for one year. As this test [[Mw:Special:MyLanguage/Talk pages project/Usability/Analysis|gave positive results]], the new design is deployed on these wikis as the default design. It is possible to opt-out these changes [[Special:Preferences#mw-prefsection-editing|in user preferences]] (&quot;{{int:discussiontools-preference-visualenhancements}}&quot;). The deployment will happen at all wikis in the coming weeks. [https://phabricator.wikimedia.org/T341491]
* Seven new wikis have been created: 
** a {{int:project-localized-name-group-wikipedia}} in [[d:Q33014|Betawi]] ([[w:bew:|&lt;code&gt;w:bew:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T357866]
** a {{int:project-localized-name-group-wikipedia}} in [[d:Q35708|Kusaal]] ([[w:kus:|&lt;code&gt;w:kus:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T359757]
** a {{int:project-localized-name-group-wikipedia}} in [[d:Q35513|Igala]] ([[w:igl:|&lt;code&gt;w:igl:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T361644]
** a {{int:project-localized-name-group-wiktionary}} in [[d:Q33541|Karakalpak]] ([[wikt:kaa:|&lt;code&gt;wikt:kaa:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T362135]
** a {{int:project-localized-name-group-wikisource}} in [[d:Q9228|Burmese]] ([[s:my:|&lt;code&gt;s:my:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T361085]
** a {{int:project-localized-name-group-wikisource}} in [[d:Q9237|Malay]] ([[s:ms:|&lt;code&gt;s:ms:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T363039]
** a {{int:project-localized-name-group-wikisource}} in [[d:Q8108|Georgian]] ([[s:ka:|&lt;code&gt;s:ka:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T363085]
* You can now [https://translatewiki.net/wiki/Support#Early_access:_Watch_Message_Groups_on_Translatewiki.net watch message groups/projects] on [[m:Special:MyLanguage/translatewiki.net|Translatewiki.net]]. Initially, this feature will notify you of added or deleted messages in these groups. [https://phabricator.wikimedia.org/T348501]
* Dark mode is now available on all wikis, on mobile web for logged-in users who opt into the [[Special:MobileOptions|advanced mode]]. This is the early release of the feature. Technical editors are invited to [https://night-mode-checker.wmcloud.org/ check for accessibility issues on wikis]. See [[mw:Special:MyLanguage/Reading/Web/Accessibility for reading/Updates/2024-04|more detailed guidelines]].

'''Problems'''
* [[mw:Special:MyLanguage/Help:Extension:Kartographer|Kartographer]] maps can use an alternative visual style without labels, by using &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;mapstyle=&quot;osm&quot;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt;. This wasn't working in previews, creating the wrong impression that it wasn't supported. This has now been fixed. [https://phabricator.wikimedia.org/T362531]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.3|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-04-30|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-05-01|en}}. It will be on all wikis from {{#time:j xg|2024-05-02|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/18|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W18&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 03:33, 30 April 2024 (UTC)
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== Tech News: 2024-19 ==

&lt;section begin=&quot;technews-2024-W19&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/19|Translations]] are available.

'''Recent changes'''
[[File:Talk_pages_default_look_(April_2023).jpg|thumb|alt=Screenshot of the visual improvements made on talk pages|Example of a talk page with the new design, in French.]]
* The appearance of talk pages changed for all wikis, except for Commons, Wikidata and most Wikipedias ([[m:Special:MyLanguage/Tech/News/2024/18|a few]] have already received this design change). You can read the detail of the changes [[diffblog:2024/05/02/making-talk-pages-better-for-everyone/|on ''Diff'']]. It is possible to opt-out these changes [[Special:Preferences#mw-prefsection-editing|in user preferences]] (&quot;{{int:discussiontools-preference-visualenhancements}}&quot;). The deployment will happen at remaining wikis in the coming weeks. [https://phabricator.wikimedia.org/T352087][https://phabricator.wikimedia.org/T319146]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Interface admins now have greater control over the styling of article components on mobile with the introduction of the &lt;code&gt;SiteAdminHelper&lt;/code&gt;. More information on how styles can be disabled can be found [[mw:Special:MyLanguage/Extension:WikimediaMessages#Site_admin_helper|at the extension's page]]. [https://phabricator.wikimedia.org/T363932]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] [[m:Special:MyLanguage/Wikimedia Enterprise|Wikimedia Enterprise]] has added article body sections in JSON format and a curated short description field to the existing parsed Infobox. This expansion to the API is also available via Wikimedia Cloud Services. [https://enterprise.wikimedia.com/blog/article-sections-and-description/]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.4|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-05-07|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-05-08|en}}. It will be on all wikis from {{#time:j xg|2024-05-09|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* When you look at the Special:Log page, the first view is labelled &quot;All public logs&quot;, but it only shows some logs. This label will now say &quot;Main public logs&quot;. [https://phabricator.wikimedia.org/T237729]

'''Future changes'''
* A new service will be built to replace [[mw:Special:MyLanguage/Extension:Graph|Extension:Graph]]. Details can be found in [[mw:Special:MyLanguage/Extension:Graph/Plans|the latest update]] regarding this extension.
* Starting May 21, English Wikipedia and German Wikipedia will get the possibility to activate &quot;[[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add a link]]&quot;. This is part of the [[phab:T304110|progressive deployment of this tool to all Wikipedias]]. These communities can [[mw:Special:MyLanguage/Growth/Community configuration|activate and configure the feature locally]]. [https://phabricator.wikimedia.org/T308144]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/19|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W19&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 16:44, 6 May 2024 (UTC)
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== Tech News: 2024-20 ==

&lt;section begin=&quot;technews-2024-W20&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/20|Translations]] are available.

'''Recent changes'''
* On Wikisource there is a special page listing pages of works without corresponding scan images. Now you can use the new magic word &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;__EXPECTWITHOUTSCANS__&lt;/code&gt;&lt;/bdi&gt; to exclude certain pages (list of editions or translations of works) from that list. [https://phabricator.wikimedia.org/T344214]
* If you use the [[Special:Preferences#mw-prefsection-editing|user-preference]] &quot;{{int:tog-uselivepreview}}&quot;, then the template-page feature &quot;{{int:Templatesandbox-editform-legend}}&quot; will now also work without reloading the page. [https://phabricator.wikimedia.org/T136907]
* [[mw:Special:Mylanguage/Extension:Kartographer|Kartographer]] maps can now specify an alternative text via the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;alt=&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; attribute. This is identical in usage to the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;alt=&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; attribute in the [[mw:Special:MyLanguage/Help:Images#Syntax|image and gallery syntax]]. An exception for this feature is wikis like Wikivoyage where the miniature maps are interactive. [https://phabricator.wikimedia.org/T328137]
* The old [[mw:Special:MyLanguage/Extension:GuidedTour|Guided Tour]] for the &quot;[[mw:Special:MyLanguage/Edit Review Improvements/New filters for edit review|New Filters for Edit Review]]&quot; feature has been removed. It was created in 2017 to show people with older accounts how the interface had changed, and has now been seen by most of the intended people. [https://phabricator.wikimedia.org/T217451]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.5|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-05-14|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-05-15|en}}. It will be on all wikis from {{#time:j xg|2024-05-16|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The [[{{#special:search}}]] results page will now use CSS flex attributes, for better accessibility, instead of a table. If you have a gadget or script that adjusts search results, you should update your script to the new HTML structure. [https://phabricator.wikimedia.org/T320295]

'''Future changes'''
* In the Vector 2022 skin, main pages will be displayed at full width (like special pages). The goal is to keep the number of characters per line large enough. This is related to the coming changes to typography in Vector 2022. [[mw:Special:MyLanguage/Reading/Web/Accessibility for reading/Updates|Learn more]]. [https://phabricator.wikimedia.org/T357706]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Two columns of the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;[[mw:Special:MyLanguage/Manual:pagelinks table|pagelinks]]&lt;/code&gt;&lt;/bdi&gt; database table (&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;pl_namespace&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;pl_title&lt;/code&gt;&lt;/bdi&gt;) are being dropped soon. Users must use two columns of the new &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;[[mw:special:MyLanguage/Manual:linktarget table|linktarget]]&lt;/code&gt;&lt;/bdi&gt; table instead (&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;lt_namespace&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;lt_title&lt;/code&gt;&lt;/bdi&gt;). In your existing SQL queries:
*# Replace &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;JOIN pagelinks&lt;/code&gt;&lt;/bdi&gt; with &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;JOIN linktarget&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;pl_&lt;/code&gt;&lt;/bdi&gt; with &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;lt_&lt;/code&gt;&lt;/bdi&gt; in the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;ON&lt;/code&gt;&lt;/bdi&gt; statement
*# Below that add &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;JOIN pagelinks ON lt_id = pl_target_id&lt;/code&gt;&lt;/bdi&gt;
** See &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[phab:T222224]]&lt;/bdi&gt; for technical reasoning. [https://phabricator.wikimedia.org/T222224][https://phabricator.wikimedia.org/T299947]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/20|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W20&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:58, 13 May 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26762074 --&gt;

== Tech News: 2024-21 ==

&lt;section begin=&quot;technews-2024-W21&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/21|Translations]] are available.

'''Recent changes'''
* The [[mw:Special:MyLanguage/Extension:Nuke|Nuke]] feature, which enables administrators to mass delete pages, will now correctly delete pages which were moved to another title. [https://phabricator.wikimedia.org/T43351]
* New changes have been made to the UploadWizard in Wikimedia Commons: the overall layout has been improved, by following new styling and spacing for the form and its fields; the headers and helper text for each of the fields was changed; the Caption field is now a required field, and there is an option for users to copy their caption into the media description. [https://commons.wikimedia.org/wiki/Commons:WMF_support_for_Commons/Upload_Wizard_Improvements#Changes_to_%22Describe%22_workflow][https://phabricator.wikimedia.org/T361049]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.6|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-05-21|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-05-22|en}}. It will be on all wikis from {{#time:j xg|2024-05-23|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]]  The HTML used to render all headings [[mw:Heading_HTML_changes|is being changed to improve accessibility]]. It will change on 22 May in some skins (Timeless, Modern, CologneBlue, Nostalgia, and Monobook). Please test gadgets on your wiki on these skins and [[phab:T13555|report any related problems]] so that they can be resolved before this change is made in all other skins. The developers are also considering the introduction of a [[phab:T337286|Gadget API for adding buttons to section titles]] if that would be helpful to tool creators, and would appreciate any input you have on that.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/21|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W21&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:04, 20 May 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26786311 --&gt;

== Tech News: 2024-22 ==

&lt;section begin=&quot;technews-2024-W22&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/22|Translations]] are available.

'''Recent changes'''
* Several bugs related to the latest updates to the UploadWizard on Wikimedia Commons have been fixed. For more information, see [[:phab:T365107|T365107]] and [[:phab:T365119|T365119]].
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] In March 2024 a new [[mw:ResourceLoader/Core_modules#addPortlet|addPortlet]] API was added to allow gadgets to create new portlets (menus) in the skin. In certain skins this can be used to create dropdowns. Gadget developers are invited to try it and [[phab:T361661|give feedback]].
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Some CSS in the Minerva skin has been removed to enable easier community configuration. Interface editors should check the rendering on mobile devices for aspects related to the classes: &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;.collapsible&lt;/code&gt;&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;.multicol&lt;/code&gt;&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;.reflist&lt;/code&gt;&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;.coordinates&lt;/code&gt;&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;.topicon&lt;/code&gt;&lt;/bdi&gt;. [[phab:T361659|Further details are available on replacement CSS]] if it is needed.

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.7|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-05-28|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-05-29|en}}. It will be on all wikis from {{#time:j xg|2024-05-30|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* When you visit a wiki where you don't yet have a local account, local rules such as edit filters can sometimes prevent your account from being created. Starting this week, MediaWiki takes your global rights into account when evaluating whether you can override such local rules. [https://phabricator.wikimedia.org/T316303]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/22|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W22&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:15, 28 May 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26832205 --&gt;

== Tech News: 2024-23 ==

&lt;section begin=&quot;technews-2024-W23&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/23|Translations]] are available.

'''Recent changes'''
* It is now possible for local administrators to add new links to the bottom of the site Tools menu without JavaScript. [[mw:Manual:Interface/Sidebar#Add or remove toolbox sections|Documentation is available]]. [https://phabricator.wikimedia.org/T6086]
* The message name for the definition of the tracking category of WikiHiero has changed from &quot;&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;MediaWiki:Wikhiero-usage-tracking-category&lt;/code&gt;&lt;/bdi&gt;&quot; to &quot;&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;MediaWiki:Wikihiero-usage-tracking-category&lt;/code&gt;&lt;/bdi&gt;&quot;. [https://gerrit.wikimedia.org/r/c/mediawiki/extensions/wikihiero/+/1035855]
* One new wiki has been created: a {{int:project-localized-name-group-wikipedia}} in [[d:Q5317225|Kadazandusun]] ([[w:dtp:|&lt;code&gt;w:dtp:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T365220]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.8|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-06-04|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-06-05|en}}. It will be on all wikis from {{#time:j xg|2024-06-06|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]

'''Future changes'''
* Next week, on wikis with the Vector 2022 skin as the default, logged-out desktop users will be able to choose between different font sizes. The default font size will also be increased for them. This is to make Wikimedia projects easier to read. [[mw:Special:MyLanguage/Reading/Web/Accessibility for reading/Updates/2024-06 deployments|Learn more]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/23|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W23&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:35, 3 June 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=26844397 --&gt;

== Tech News: 2024-24 ==

&lt;section begin=&quot;technews-2024-W24&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/24|Translations]] are available.

'''Recent changes'''
* The software used to render SVG files has been updated to a new version, fixing many longstanding bugs in SVG rendering. [https://phabricator.wikimedia.org/T265549]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The HTML used to render all headings [[mw:Heading HTML changes|is being changed to improve accessibility]]. It was changed last week in some skins (Vector legacy and Minerva). Please test gadgets on your wiki on these skins and [[phab:T13555|report any related problems]] so that they can be resolved before this change is made in Vector-2022. The developers are still considering the introduction of a [[phab:T337286|Gadget API for adding buttons to section titles]] if that would be helpful to tool creators, and would appreciate any input you have on that.
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The HTML markup used for citations by [[mw:Special:MyLanguage/Parsoid|Parsoid]] changed last week. In places where Parsoid previously added the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;mw-reference-text&lt;/code&gt;&lt;/bdi&gt; class, Parsoid now also adds the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;reference-text&lt;/code&gt;&lt;/bdi&gt; class for better compatibility with the legacy parser. [[mw:Specs/HTML/2.8.0/Extensions/Cite/Announcement|More details are available]]. [https://gerrit.wikimedia.org/r/1036705]

'''Problems'''
* There was a bug with the Content Translation interface that caused the tools menus to appear in the wrong location. This has now been fixed. [https://phabricator.wikimedia.org/T366374]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.9|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-06-11|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-06-12|en}}. It will be on all wikis from {{#time:j xg|2024-06-13|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] The new version of MediaWiki includes another change to the HTML markup used for citations: [[mw:Special:MyLanguage/Parsoid|Parsoid]] will now generate a &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;&lt;span class=&quot;mw-cite-backlink&quot;&gt;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; wrapper for both named and unnamed references for better compatibility with the legacy parser. Interface administrators should verify that gadgets that interact with citations are compatible with the new markup. [[mw:Specs/HTML/2.8.0/Extensions/Cite/Announcement|More details are available]]. [https://gerrit.wikimedia.org/r/1035809] 
* On multilingual wikis that use the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;&lt;translate&gt;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; system, there is a feature that shows potentially-outdated translations with a pink background until they are updated or confirmed. From this week, confirming translations will be logged, and there is a new user-right that can be required for confirming translations if the community [[m:Special:MyLanguage/Requesting wiki configuration changes|requests it]]. [https://phabricator.wikimedia.org/T49177]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/24|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W24&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:20, 10 June 2024 (UTC)
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== Tech News: 2024-25 ==

&lt;section begin=&quot;technews-2024-W25&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/25|Translations]] are available.

'''Recent changes'''
* People who attempt to add an external link in the visual editor will now receive immediate feedback if they attempt to link to a domain that a project has decided to block. Please see [[mw:Special:MyLanguage/Edit_check#11_June_2024|Edit check]] for more details. [https://phabricator.wikimedia.org/T366751]
* The new [[mw:Special:MyLanguage/Extension:CommunityConfiguration|Community Configuration extension]] is available [[testwiki:Special:CommunityConfiguration|on Test Wikipedia]]. This extension allows communities to customize specific features to meet their local needs. Currently only Growth features are configurable, but the extension will support other [[mw:Special:MyLanguage/Community_configuration#Use_cases|Community Configuration use cases]] in the future. [https://phabricator.wikimedia.org/T323811][https://phabricator.wikimedia.org/T360954]
* The dark mode [[Special:Preferences#mw-prefsection-betafeatures|beta feature]] is now available on category and help pages, as well as more special pages. There may be contrast issues. Please report bugs on the [[mw:Talk:Reading/Web/Accessibility_for_reading|project talk page]]. [https://phabricator.wikimedia.org/T366370]

'''Problems'''
* [[File:Octicons-tools.svg|12px|link=|alt=|Advanced item]] Cloud Services tools were not available for 25 minutes last week. This was caused by a faulty hardware cable in the data center. [https://wikitech.wikimedia.org/wiki/Incidents/2024-06-11_WMCS_Ceph]
* Last week, styling updates were made to the Vector 2022 skin. This caused unforeseen issues with templates, hatnotes, and images. Changes to templates and hatnotes were reverted. Most issues with images were fixed. If you still see any, [[phab:T367463|report them here]]. [https://phabricator.wikimedia.org/T367480]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.10|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-06-18|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-06-19|en}}. It will be on all wikis from {{#time:j xg|2024-06-20|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* Starting June 18, the [[mw:Special:MyLanguage/Help:Edit check#ref|Reference Edit Check]] will be deployed to [[phab:T361843|a new set of Wikipedias]]. This feature is intended to help newcomers and to assist edit-patrollers by inviting people who are adding new content to a Wikipedia article to add a citation when they do not do so themselves. During [[mw:Special:MyLanguage/Edit_check#Reference_Check_A/B_Test|a test at 11 wikis]], the number of citations added [https://diff.wikimedia.org/?p=127553 more than doubled] when Reference Check was shown to people. Reference Check is [[mw:Special:MyLanguage/Edit check/Configuration|community configurable]]. [https://phabricator.wikimedia.org/T361843]&lt;!-- NOTE: THE DIFF BLOG WILL BE PUBLISHED ON MONDAY --&gt;
* [[m:Special:MyLanguage/Mailing_lists|Mailing lists]] will be unavailable for roughly two hours on Tuesday 10:00–12:00 UTC. This is to enable migration to a new server and upgrade its software. [https://phabricator.wikimedia.org/T367521]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/25|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W25&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:48, 17 June 2024 (UTC)
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== Tech News: 2024-26 ==

&lt;section begin=&quot;technews-2024-W26&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/26|Translations]] are available.

'''Recent changes'''
* Editors will notice that there have been some changes to the background color of text in the diff view, and the color of the byte-change numbers, last week. These changes are intended to make text more readable in both light mode and dark mode, and are part of a larger effort to increase accessibility. You can share your comments or questions [[mw:Talk:Reading/Web/Accessibility for reading|on the project talkpage]]. [https://phabricator.wikimedia.org/T361717]
* The text colors that are used for visited-links, hovered-links, and active-links, were also slightly changed last week to improve their accessibility in both light mode and dark mode. [https://phabricator.wikimedia.org/T366515]

'''Problems'''
* You can [[mw:Special:MyLanguage/Help:DiscussionTools#Talk pages permalinking|copy permanent links to talk page comments]] by clicking on a comment's timestamp. [[mw:Talk pages project/Permalinks|This feature]] did not always work when the topic title was very long and the link was used as a wikitext link. This has been fixed. Thanks to Lofhi for submitting the bug. [https://phabricator.wikimedia.org/T356196]

'''Changes later this week'''
* [[File:Octicons-sync.svg|12px|link=|alt=|Recurrent item]] The [[mw:MediaWiki 1.43/wmf.11|new version]] of MediaWiki will be on test wikis and MediaWiki.org from {{#time:j xg|2024-06-25|en}}. It will be on non-Wikipedia wikis and some Wikipedias from {{#time:j xg|2024-06-26|en}}. It will be on all wikis from {{#time:j xg|2024-06-27|en}} ([[mw:MediaWiki 1.43/Roadmap|calendar]]). [https://wikitech.wikimedia.org/wiki/Deployments/Train][https://wikitech.wikimedia.org/wiki/Deployments/Yearly_calendar]
* Starting 26 June, all talk pages messages' timestamps will become a link at English Wikipedia, making this feature available for you to use at all wikis. This link is a permanent link to the comment. It allows users to find the comment they were linked to, even if this comment has since been moved elsewhere. You can read more about this feature [[DiffBlog:/2024/01/29/talk-page-permalinks-dont-lose-your-threads/|on Diff]] or [[mw:Special:MyLanguage/Help:DiscussionTools#Talk pages permalinking|on Mediawiki.org]]. [https://phabricator.wikimedia.org/T365974]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/26|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W26&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:32, 24 June 2024 (UTC)
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== Tech News: 2024-27 ==

&lt;section begin=&quot;technews-2024-W27&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/27|Translations]] are available.

'''Recent changes'''
* Over the next three weeks, dark mode will become available for all users, both logged-in and logged-out, starting with the mobile web version. This fulfils one of the [[m:Special:MyLanguage/Community_Wishlist_Survey_2023/Reading/Dark_mode|top-requested community wishes]], and improves low-contrast reading and usage in low-light settings. As part of these changes, dark mode will also work on User-pages and Portals. There is more information in [[mw:Special:MyLanguage/Reading/Web/Accessibility_for_reading/Updates#June_2024:_Typography_and_dark_mode_deployments,_new_global_preferences|the latest Web team update]]. [https://phabricator.wikimedia.org/T366364]
* Logged-in users can now set [[m:Special:GlobalPreferences#mw-prefsection-rendering-skin-skin-prefs|global preferences for the text-size and dark-mode]], thanks to a combined effort across Foundation teams. This allows Wikimedians using multiple wikis to set up a consistent reading experience easily, for example by switching between light and dark mode only once for all wikis. [https://phabricator.wikimedia.org/T341278] 
* If you use a very old web browser some features might not work on the Wikimedia wikis. This affects Internet Explorer 11 and versions of Chrome, Firefox and Safari older than 2016. This change makes it possible to use new [[d:Q46441|CSS]] features and to send less code to all readers. [https://phabricator.wikimedia.org/T288287][https://www.mediawiki.org/wiki/Special:MyLanguage/Manual:How_to_make_a_MediaWiki_skin#Using_CSS_variables_for_supporting_different_themes_e.g._dark_mode]
* Wikipedia Admins can customize local wiki configuration options easily using [[mw:Special:MyLanguage/Community Configuration|Community Configuration]]. Community Configuration was created to allow communities to customize how some features work, because each language wiki has unique needs. At the moment, admins can configure [[mw:Special:MyLanguage/Growth/Feature_summary|Growth features]] on their home wikis, in order to better recruit and retain new editors. More options will be provided in the coming months. [https://phabricator.wikimedia.org/T366458]
* Editors interested in language issues that are related to [[w:en:Unicode|Unicode standards]], can now discuss those topics at [[mw:Talk:WMF membership with Unicode Consortium|a new conversation space in MediaWiki.org]]. The Wikimedia Foundation is now a [[mw:Special:MyLanguage/WMF membership with Unicode Consortium|member of the Unicode Consortium]], and the coordination group can collaboratively review the issues discussed and, where appropriate, bring them to the attention of the Unicode Consortium.
* One new wiki has been created: a {{int:project-localized-name-group-wikipedia}} in [[d:Q2891049|Mandailing]] ([[w:btm:|&lt;code&gt;w:btm:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T368038] 

'''Problems'''
* Editors can once again click on links within the visual editor's citation-preview, thanks to a bug fix by the Editing Team. [https://phabricator.wikimedia.org/T368119]

'''Future changes'''
* Please [https://wikimediafoundation.limesurvey.net/758713?lang=en help us to improve Tech News by taking this short survey]. The goal is to better meet the needs of the various types of people who read Tech News. The survey will be open for 2 weeks. The survey is covered by [https://foundation.wikimedia.org/wiki/Legal:Tech_News_Survey_2024_Privacy_Statement this privacy statement]. Some translations are available.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/27|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W27&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:59, 1 July 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27038456 --&gt;

== Tech News: 2024-28 ==

&lt;section begin=&quot;technews-2024-W28&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/28|Translations]] are available.

'''Recent changes'''
* At the Wikimedia Foundation a new task force was formed to replace the disabled Graph with [[mw:Special:MyLanguage/Extension:Chart/Project|more secure, easy to use, and extensible Chart]]. You can [[mw:Special:MyLanguage/Newsletter:Chart Project|subscribe to the newsletter]] to get notified about new project updates and other news about Chart.
* The [[m:Special:MyLanguage/CampaignEvents|CampaignEvents]] extension is now available on Meta-wiki, Igbo Wikipedia, and Swahili Wikipedia, and can be requested on your wiki. This extension helps in managing and making events more visible, giving Event organizers the ability to use tools like the Event registration tool. To learn more about the deployment status and how to request this extension for your wiki, visit the [[m:Special:MyLanguage/CampaignEvents/Deployment_status|CampaignEvents page on Meta-wiki]]. 
* Editors using the iOS Wikipedia app who have more than 50 edits can now use the [[mw:Special:MyLanguage/Wikimedia Apps/iOS Suggested edits#Add an image|Add an Image]] feature. This feature presents opportunities for small but useful contributions to Wikipedia.
* Thank you to [[mw:MediaWiki Product Insights/Contributor retention and growth/Celebration|all of the authors]] who have contributed to MediaWiki Core. As a result of these contributions, the [[mw:MediaWiki Product Insights/Contributor retention and growth|percentage of authors contributing more than 5 patches has increased by 25% since last year]], which helps ensure the sustainability of the platform for the Wikimedia projects.

'''Problems'''
* A problem with the color of the talkpage tabs always showing as blue, even for non-existent pages which should have been red, affecting the Vector 2022 skin, [[phab:T367982|has been fixed]]. 

'''Future changes'''
* The Trust and Safety Product team wants to introduce [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|temporary accounts]] with as little disruption to tools and workflows as possible. Volunteer developers, including gadget and user-script maintainers, are kindly asked to update the code of their tools and features to handle temporary accounts. The team has [[mw:Trust and Safety Product/Temporary Accounts/For developers|created documentation]] explaining how to do the update. [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/For developers/2024-04 CTA|Learn more]].

'''Tech News survey'''
* Please [https://wikimediafoundation.limesurvey.net/758713?lang=en help us to improve Tech News by taking this short survey]. The goal is to better meet the needs of the various types of people who read Tech News. The survey will be open for 1 more week. The survey is covered by [https://foundation.wikimedia.org/wiki/Legal:Tech_News_Survey_2024_Privacy_Statement this privacy statement]. Some translations are available.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/28|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W28&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:31, 8 July 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27080357 --&gt;

== Tech News: 2024-29 ==

&lt;section begin=&quot;technews-2024-W29&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/29|Translations]] are available.

'''Tech News survey'''
* Please [https://wikimediafoundation.limesurvey.net/758713?lang=en help us to improve Tech News by taking this short survey]. The goal is to better meet the needs of the various types of people who read Tech News. The survey will be open for 3 more days. The survey is covered by [https://foundation.wikimedia.org/wiki/Legal:Tech_News_Survey_2024_Privacy_Statement this privacy statement]. Some translations are available.

'''Recent changes'''
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Wikimedia developers can now officially continue to use both [[mw:Special:MyLanguage/Gerrit|Gerrit]] and [[mw:Special:MyLanguage/GitLab|GitLab]], due to a June 24 decision by the Wikimedia Foundation to support software development on both platforms. Gerrit and GitLab are both code repositories used by developers to write, review, and deploy the software code that supports the MediaWiki software that the wiki projects are built on, as well as the tools used by editors to create and improve content. This decision will safeguard the productivity of our developers and prevent problems in code review from affecting our users. More details are available in the [[mw:GitLab/Migration status|Migration status]] page.
* The Wikimedia Foundation seeks applicants for the [[m:Special:MyLanguage/Product and Technology Advisory Council/Proposal|Product and Technology Advisory Council]] (PTAC). This group will bring technical contributors and Wikimedia Foundation together to co-define a more resilient, future-proof technological platform. Council members will evaluate and consult on the movement's product and technical activities, so that we develop multi-generational projects. We are looking for a range of technical contributors across the globe, from a variety of Wikimedia projects. [[m:Special:MyLanguage/Product and Technology Advisory Council/Proposal#Joining the PTAC as a technical volunteer|Please apply here by August 10]].
* Editors with rollback user-rights who use the Wikipedia App for Android can use the new [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android/Anti Vandalism|Edit Patrol]] features. These features include a new feed of Recent Changes, related links such as Undo and Rollback, and the ability to create and save a personal library of user talk messages to use while patrolling. If your wiki wants to make these features available to users who do not have rollback rights but have reached a certain edit threshold, [[mw:Special:MyLanguage/Wikimedia Apps/Team/Android#Contact us|you can contact the team]]. You can [[diffblog:2024/07/10/ِaddressing-vandalism-with-a-tap-the-journey-of-introducing-the-patrolling-feature-in-the-mobile-app/|read more about this project on Diff blog]].
* Editors who have access to [[m:Special:MyLanguage/The_Wikipedia_Library|The Wikipedia Library]] can once again use non-open access content in SpringerLinks, after the Foundation [[phab:T368865|contacted]] them to restore access. You can read more about [[m:Tech/News/Recently_resolved_community_tasks|this and 21 other community-submitted tasks that were completed last week]]. 

'''Changes later this week'''
* This week, [[mw:Special:MyLanguage/Reading/Web/Accessibility for reading/Updates/2024-07 deployments|dark mode will be available on a number of Wikipedias]], both desktop and mobile, for logged-in and logged-out users. Interface admins and user script maintainers are encouraged to check gadgets and user scripts in the dark mode, to find any hard-coded colors and fix them. There are some [[mw:Special:MyLanguage/Recommendations for night mode compatibility on Wikimedia wikis|recommendations for dark mode compatibility]] to help.

'''Future changes'''
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Next week, functionaries, volunteers maintaining tools, and software development teams are invited to test the [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|temporary accounts]] feature on testwiki. Temporary accounts is a feature that will help improve privacy on the wikis. No further temporary account deployments are scheduled yet. Please [[mw:Talk:Trust and Safety Product/Temporary Accounts|share your opinions and questions on the project talk page]]. [https://phabricator.wikimedia.org/T348895]
* Editors who upload files cross-wiki, or teach other people how to do so, may wish to join a Wikimedia Commons discussion. The Commons community is discussing limiting who can upload files through the cross-wiki upload/Upload dialog feature to users auto-confirmed on Wikimedia Commons. This is due to the large amount of copyright violations uploaded this way. There is a short summary at [[c:Special:MyLanguage/Commons:Cross-wiki upload|Commons:Cross-wiki upload]] and [[c:Commons:Village pump/Proposals#Deactivate cross-wiki uploads for new users|discussion at Commons:Village Pump]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/29|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' You can also get other news from the [[m:Special:MyLanguage/Wikimedia Foundation Bulletin|Wikimedia Foundation Bulletin]].
&lt;/div&gt;&lt;section end=&quot;technews-2024-W29&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:31, 16 July 2024 (UTC)
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== Tech News: 2024-30 ==

&lt;section begin=&quot;technews-2024-W30&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/30|Translations]] are available.

'''Feature News'''
* Stewards can now [[:m:Special:MyLanguage/Global_blocks|globally block]] accounts. Before [[phab:T17294|the change]] only IP addresses and IP ranges could be blocked globally. Global account blocks are useful when the blocked user should not be logged out. [[:m:Special:MyLanguage/Global_locks|Global locks]] (a similar tool logging the user out of their account) are unaffected by this change. The new global account block feature is related to the [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|Temporary Accounts]] project, which is a new type of user account that replaces IP addresses of unregistered editors that are no longer made public.
* Later this week, Wikimedia site users will notice that the Interface of [[mw:Special:MyLanguage/Extension:FlaggedRevs|FlaggedRevs]] (also known as &quot;Pending Changes&quot;) is improved and consistent with the rest of the MediaWiki interface and [[mw:Special:MyLanguage/Codex|Wikimedia's design system]]. The FlaggedRevs interface experience on mobile and [[mw:Special:MyLanguage/Skin:MinervaNeue|Minerva skin]] was inconsistent before it was fixed and ported to [[mw:Special:MyLanguage/Codex|Codex]] by the WMF Growth team and some volunteers. [https://phabricator.wikimedia.org/T191156]
*  Wikimedia site users can now submit account vanishing requests via [[m:Special:GlobalVanishRequest|GlobalVanishRequest]]. This feature is used when a contributor wishes to stop editing forever. It helps you hide your past association and edit to protect your privacy. Once processed, the account will be locked and renamed. [https://phabricator.wikimedia.org/T367329]
* Have you tried monitoring and addressing vandalism in Wikipedia using your phone? [https://diff.wikimedia.org/2024/07/10/%d9%90addressing-vandalism-with-a-tap-the-journey-of-introducing-the-patrolling-feature-in-the-mobile-app/ A Diff blog post on Patrolling features in the Mobile App] highlights some of the new capabilities of the feature, including swiping through a feed of recent changes and a personal library of user talk messages for use when patrolling from your phone. 
* Wikimedia contributors and GLAM (galleries, libraries, archives, and museums) organisations can now learn and measure the impact Wikimedia Commons is having towards creating quality encyclopedic content using the [https://doc.wikimedia.org/generated-data-platform/aqs/analytics-api/reference/commons.html Commons Impact Metrics] analytics dashboard. The dashboard offers organizations analytics on things like monthly edits in a category, the most viewed files, and which Wikimedia articles are using Commons images. As a result of these new data dumps, GLAM organisation can more reliably measure their return on investment for programs bringing content into the digital Commons. [https://diff.wikimedia.org/2024/07/19/commons-impact-metrics-now-available-via-data-dumps-and-api/]

'''Project Updates'''
*  Come share your ideas for improving the wikis on the newly reopened [[m:Special:MyLanguage/Community Wishlist|Community Wishlist]]. The Community Wishlist is Wikimedia’s forum for volunteers to share ideas (called wishes) to improve how the wikis work. The new version of the wishlist is always open, works with both wikitext and Visual Editor, and allows wishes in any language.

'''Learn more'''
* Have you ever wondered how Wikimedia software works across over 300 languages? This is 253 languages more than the Google Chrome interface, and it's no accident. The Language and Product Localization Team at the Wikimedia Foundation supports your work by adapting all the tools and interfaces in the MediaWiki software so that contributors in our movement who translate pages and strings can translate them and have the sites in all languages. Read more about the team and their upcoming work on [https://diff.wikimedia.org/2024/07/17/building-towards-a-robust-multilingual-knowledge-ecosystem-for-the-wikimedia-movement/ Diff]. 
* How can Wikimedia build innovative and experimental products while maintaining such heavily used websites? A recent [https://diff.wikimedia.org/2024/07/09/on-the-value-of-experimentation/ blog post] by WMF staff Johan Jönsson highlights the work of the [[m:Future Audiences#Objectives and Key Results|WMF Future Audience initiative]], where the goal is not to build polished products but test out new ideas, such as a [[m:Future_Audiences/Experiments: conversational/generative AI|ChatGPT plugin]] and [[m:Future_Audiences/Experiment:Add a Fact|Add a Fact]], to help take Wikimedia into the future. 

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/30|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].'' You can also get other news from the [[m:Special:MyLanguage/Wikimedia Foundation Bulletin|Wikimedia Foundation Bulletin]].
&lt;/div&gt;&lt;section end=&quot;technews-2024-W30&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:04, 23 July 2024 (UTC)
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== Tech News: 2024-31 ==

&lt;section begin=&quot;technews-2024-W31&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/31|Translations]] are available.

'''Feature news'''
* Editors using the Visual Editor in languages that use non-Latin characters for numbers, such as Hindi, Manipuri and Eastern Arabic, may notice some changes in the formatting of reference numbers. This is a side effect of preparing a new sub-referencing feature, and will also allow fixing some general numbering issues in Visual Editor. If you notice any related problems on your wiki, please share details at the [[m:Talk:WMDE Technical Wishes/Sub-referencing|project talkpage]].

'''Bugs status'''
* Some logged-in editors were briefly unable to edit or load pages last week. [[phab:T370304|These errors]] were mainly due to the addition of new [[mw:Special:MyLanguage/Help:Extension:Linter|linter]] rules which led to caching problems. Fixes have been applied and investigations are continuing.
* Editors can use the [[mw:Special:MyLanguage/Trust and Safety Product/IP Info|IP Information tool]] to get information about IP addresses. This tool is available as a Beta Feature in your preferences. The tool was not available for a few days last week, but is now working again. Thank you to Shizhao for filing the bug report. You can read about that, and [[m:Tech/News/Recently resolved community tasks#2024-07-25|28 other community-submitted tasks]] that were resolved last week.

'''Project updates'''
* There are new features and improvements to Phabricator from the Release Engineering and Collaboration Services teams, and some volunteers, including: the search systems, the new task creation system, the login systems, the translation setup which has resulted in support for more languages (thanks to Pppery), and fixes for many edge-case errors. You can [[phab:phame/post/view/316/iterative_improvements/|read details about these and other improvements in this summary]].
* There is an [[mw:Special:MyLanguage/Extension:Chart/Project/Updates|update on the Charts project]]. The team has decided which visualization library to use, which chart types to start focusing on, and where to store chart definitions.
* One new wiki has been created: a {{int:project-localized-name-group-wikivoyage}} in [[d:Q9056|Czech]] ([[voy:cs:|&lt;code&gt;voy:cs:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T370905] 

'''Learn more'''
* There is a [[diffblog:2024/07/26/the-journey-to-open-our-first-data-center-in-south-america/|new Wikimedia Foundation data center]] in São Paulo, Brazil which helps to reduce load times.
* There is new [[diffblog:2024/07/22/the-perplexing-process-of-uploading-images-to-wikipedia/|user research]] on problems with the process of uploading images.
* Commons Impact Metrics are [[diffblog:2024/07/19/commons-impact-metrics-now-available-via-data-dumps-and-api/|now available]] via data dumps and API.
* The latest quarterly [[mw:Technical Community Newsletter/2024/July|Technical Community Newsletter]] is now available.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/31|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W31&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:10, 29 July 2024 (UTC)
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== Tech News: 2024-32 ==

&lt;section begin=&quot;technews-2024-W32&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/32|Translations]] are available.

'''Feature news'''
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Two new parser functions will be available this week: &lt;code&gt;&lt;nowiki&gt;{{&lt;/nowiki&gt;[[mw:Special:MyLanguage/Help:Magic_words#dir|#dir]]&lt;nowiki&gt;}}&lt;/nowiki&gt;&lt;/code&gt; and &lt;code&gt;&lt;nowiki&gt;{{&lt;/nowiki&gt;[[mw:Special:MyLanguage/Help:Magic_words#bcp47|#bcp47]]&lt;nowiki&gt;}}&lt;/nowiki&gt;&lt;/code&gt;. These will reduce the need for &lt;code&gt;Template:Dir&lt;/code&gt; and &lt;code&gt;Template:BCP47&lt;/code&gt; on Commons and allow us to [[phab:T343131|drop 100 million rows]] from the &quot;what links here&quot; database. Editors at any wiki that use these templates, can help by replacing the templates with these new functions. The templates at Commons will be updated during the Hackathon at Wikimania. [https://phabricator.wikimedia.org/T359761][https://phabricator.wikimedia.org/T366623]
* Communities can request the activation of the visual editor on entire namespaces where discussions sometimes happen (for instance ''Wikipedia:'' or ''Wikisource:'' namespaces) if they understand the [[mw:Special:MyLanguage/Help:VisualEditor/FAQ#WPNS|known limitations]]. For discussions, users can already use [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]] in these namespaces.
* The tracking category &quot;Pages using Timeline&quot; has been renamed to &quot;Pages using the EasyTimeline extension&quot; [https://translatewiki.net/wiki/Special:Translations?message=MediaWiki%3ATimeline-tracking-category&amp;namespace=8 in TranslateWiki]. Wikis that have created the category locally should rename their local creation to match.

'''Project updates'''
* Editors who help to organize WikiProjects and similar on-wiki collaborations, are invited to share ideas and examples of successful collaborations with the Campaigns and Programs teams. You can fill out [[m:Special:MyLanguage/Campaigns/WikiProjects|a brief survey]] or share your thoughts [[m:Talk:Campaigns/WikiProjects|on the talkpage]]. The teams are particularly looking for details about successful collaborations on non-English wikis.
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] The new parser is being rolled out on {{int:project-localized-name-group-wikivoyage}} wikis over the next few months. The {{int:project-localized-name-enwikivoyage}} and {{int:project-localized-name-hewikivoyage}} were [[phab:T365367|switched]] to Parsoid last week. For more information, see [[mw:Parsoid/Parser_Unification|Parsoid/Parser Unification]].

'''Learn more'''
* There will be more than 200 sessions at Wikimania this week. Here is a summary of some of the [[diffblog:2024/08/05/interested-in-product-and-tech-here-are-some-wikimania-sessions-you-dont-want-to-miss/|key sessions related to the product and technology area]].
* The latest [[m:Special:MyLanguage/Wikimedia Foundation Bulletin/2024/07-02|Wikimedia Foundation Bulletin]] is available.
* The latest quarterly [[mw:Special:MyLanguage/Wikimedia Language and Product Localization/Newsletter/2024/July|Language and Internationalization newsletter]] is available. It includes: New design previews for Translatable pages; Updates about MinT for Wiki Readers; the release of Translation dumps; and more.
* The latest quarterly [[mw:Special:MyLanguage/Growth/Newsletters/31|Growth newsletter]] is available.
* The latest monthly [[mw:Special:MyLanguage/MediaWiki Product Insights/Reports/July 2024|MediaWiki Product Insights newsletter]] is available.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/32|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W32&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:43, 5 August 2024 (UTC)
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== Tech News: 2024-33 ==

&lt;section begin=&quot;technews-2024-W33&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/33|Translations]] are available.

'''Feature news'''
* [[mw:Special:MyLanguage/Extension:AbuseFilter|AbuseFilter]] editors and maintainers can now [[mw:Special:MyLanguage/Extension:AbuseFilter/Actions#Show a CAPTCHA|make a CAPTCHA show if a filter matches an edit]]. This allows communities to quickly respond to spamming by automated bots. [https://phabricator.wikimedia.org/T20110]
* [[m:Special:MyLanguage/Stewards|Stewards]] can now specify if global blocks should prevent account creation. Before [[phab:T17273|this change]] by the [[mw:Special:MyLanguage/Trust and Safety Product|Trust and Safety Product]] Team, all global blocks would prevent account creation. This will allow stewards to reduce the unintended side-effects of global blocks on IP addresses.

'''Project updates'''
* [[wikitech:Help talk:Toolforge/Toolforge standards committee#August_2024_committee_nominations|Nominations are open on Wikitech]] for new members to refresh the [[wikitech:Help:Toolforge/Toolforge standards committee|Toolforge standards committee]]. The committee oversees the Toolforge [[wikitech:Help:Toolforge/Right to fork policy|Right to fork policy]] and [[wikitech:Help:Toolforge/Abandoned tool policy|Abandoned tool policy]] among other duties. Nominations will remain open until at least 2024-08-26.
* One new wiki has been created: a {{int:project-localized-name-group-wikipedia}} in [[d:Q2880037|West Coast Bajau]] ([[w:bdr:|&lt;code&gt;w:bdr:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T371757]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/33|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W33&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:21, 12 August 2024 (UTC)
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== Tech News: 2024-34 ==

&lt;section begin=&quot;technews-2024-W34&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/34|Translations]] are available.

'''Feature news'''
* Editors who want to re-use references but with different details such as page numbers, will be able to do so by the end of 2024, using a new [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing#Sub-referencing in a nutshell|sub-referencing]] feature. You can read more [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing|about the project]] and [[m:Special:MyLanguage/WMDE Technical Wishes/Sub-referencing#Test|how to test the prototype]].
* Editors using tracking categories to identify which pages use specific extensions may notice that six of the categories have been renamed to make them more easily understood and consistent. These categories are automatically added to pages that use specialized MediaWiki extensions. The affected names are for: [https://translatewiki.net/wiki/Special:Translations?message=MediaWiki%3Aintersection-category&amp;namespace=8 DynamicPageList], [https://translatewiki.net/wiki/Special:Translations?message=MediaWiki%3Akartographer-tracking-category&amp;namespace=8 Kartographer], [https://translatewiki.net/wiki/Special:Translations?message=MediaWiki%3Aphonos-tracking-category&amp;namespace=8 Phonos], [https://translatewiki.net/wiki/Special:Translations?message=MediaWiki%3Arss-tracking-category&amp;namespace=8 RSS], [https://translatewiki.net/wiki/Special:Translations?message=MediaWiki%3Ascore-use-category&amp;namespace=8 Score], [https://translatewiki.net/wiki/Special:Translations?message=MediaWiki%3Awikihiero-usage-tracking-category&amp;namespace=8 WikiHiero]. Wikis that have created the category locally should rename their local creation to match. Thanks to Pppery for these improvements. [https://phabricator.wikimedia.org/T347324]
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Technical volunteers who edit modules and want to get a list of the categories used on a page, can now do so using the &lt;code&gt;&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;categories&lt;/bdi&gt;&lt;/code&gt; property of &lt;code&gt;&lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;[[mediawikiwiki:Special:MyLanguage/Extension:Scribunto/Lua reference manual#Title objects|mw.title objects]]&lt;/bdi&gt;&lt;/code&gt;. This enables wikis to configure workflows such as category-specific edit notices. Thanks to SD001 for these improvements. [https://phabricator.wikimedia.org/T50175][https://phabricator.wikimedia.org/T85372]

'''Bugs status'''
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Your help is needed to check if any pages need to be moved or deleted. A maintenance script was run to clean up unreachable pages (due to Unicode issues or introduction of new namespaces/namespace aliases). The script tried to find appropriate names for the pages (e.g. by following the Unicode changes or by moving pages whose titles on Wikipedia start with &lt;code&gt;Talk:WP:&lt;/code&gt; so that their titles start with &lt;code&gt;Wikipedia talk:&lt;/code&gt;), but it may have failed for some pages, and moved them to &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;[[Special:PrefixIndex/T195546/]]&lt;/bdi&gt; instead. Your community should check if any pages are listed there, and move them to the correct titles, or delete them if they are no longer needed. A full log (including pages for which appropriate names could be found) is available in [[phab:P67388]].
* Editors who volunteer as [[mw:Special:MyLanguage/Help:Growth/Mentorship|mentors]] to newcomers on their wiki are once again able to access lists of potential mentees who they can connect with to offer help and guidance. This functionality was restored thanks to [[phab:T372164|a bug fix]]. Thank you to Mbch331 for filing the bug report. You can read about that, and 18 other community-submitted tasks that were [[m:Tech/News/Recently resolved community tasks|resolved last week]].

'''Project updates'''
* The application deadline for the [[m:Special:MyLanguage/Product and Technology Advisory Council/Proposal|Product &amp; Technology Advisory Council]] (PTAC) has been extended to September 16. Members will help by providing advice to Foundation Product and Technology leadership on short and long term plans, on complex strategic problems, and help to get feedback from more contributors and technical communities. Selected members should expect to spend roughly 5 hours per month for the Council, during the one year pilot. Please consider applying, and spread the word to volunteers you think would make a positive contribution to the committee.

'''Learn more'''
* The [[m:Special:MyLanguage/Coolest Tool Award#2024 Winners|2024 Coolest Tool Awards]] were awarded at Wikimania, in seven categories. For example, one award went to the ISA Tool, used for adding structured data to files on Commons, which was recently improved during the [[m:Event:Wiki Mentor Africa ISA Hackathon 2024|Wiki Mentor Africa Hackathon]]. You can see video demonstrations of each tool at the awards page. Congratulations to this year's recipients, and thank you to all tool creators and maintainers.
* The latest [[m:Special:MyLanguage/Wikimedia Foundation Bulletin/2024/08-01|Wikimedia Foundation Bulletin]] is available, and includes some highlights from Wikimania, an upcoming Language community meeting, and other news from the movement.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/34|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W34&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:54, 20 August 2024 (UTC)
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== Tech News: 2024-35 ==

&lt;section begin=&quot;technews-2024-W35&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/35|Translations]] are available.

'''Feature news'''
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Administrators can now test the [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|temporary accounts]] feature on test2wiki. This was done to allow cross-wiki testing of temporary accounts, for when temporary accounts switch between projects. The feature was enabled on testwiki a few weeks ago. No further temporary account deployments are scheduled yet. Temporary Accounts is a project to create a new type of user account that replaces IP addresses of unregistered editors which are no longer made public. Please [[mw:Talk:Trust and Safety Product/Temporary Accounts|share your opinions and questions on the project talk page]].
* Later this week, editors at wikis that use [[mw:Special:MyLanguage/Extension:FlaggedRevs|FlaggedRevs]] (also known as &quot;Pending Changes&quot;) may notice that the indicators at the top of articles have changed. This change makes the system more consistent with the rest of the MediaWiki interface. [https://phabricator.wikimedia.org/T191156]

'''Bugs status'''
* Editors who use the 2010 wikitext editor, and use the Character Insert buttons, will [[phab:T361465|no longer]] experience problems with the buttons adding content into the edit-summary instead of the edit-window. You can read more about that, and 26 other community-submitted tasks that were [[m:Tech/News/Recently resolved community tasks|resolved last week]].

'''Project updates'''
* [[File:Octicons-gift.svg|12px|link=|class=skin-invert|Wishlist item]] Please review and vote on [[m:Special:MyLanguage/Community Wishlist/Focus areas|Focus Areas]], which are groups of wishes that share a problem. Focus Areas were created for the newly reopened Community Wishlist, which is now open year-round for submissions. The first batch of focus areas are specific to moderator workflows, around welcoming newcomers, minimizing repetitive tasks, and prioritizing tasks. Once volunteers have reviewed and voted on focus areas, the Foundation will then review and select focus areas for prioritization.
* Do you have a project and are willing to provide a three (3) month mentorship for an intern? [[mw:Special:MyLanguage/Outreachy|Outreachy]] is a twice a year program for people to participate in a paid internship that will start in December 2024 and end in early March 2025, and they need mentors and projects to work on. Projects can be focused on coding or non-coding (design, documentation, translation, research). See the Outreachy page for more details, and a list of past projects since 2013.

'''Learn more'''
* If you're curious about the product and technology improvements made by the Wikimedia Foundation last year, read [[diffblog:2024/08/21/wikimedia-foundation-product-technology-improving-the-user-experience/|this recent highlights summary on Diff]].
* To learn more about the technology behind the Wikimedia projects, you can now watch sessions from the technology track at Wikimania 2024 on Commons. This week, check out:
** [[c:File:Wikimania 2024 - Ohrid - Day 2 - Community Configuration - Shaping On-Wiki Functionality Together.webm|Community Configuration - Shaping On-Wiki Functionality Together]] (55 mins) - about the [[mw:Special:MyLanguage/Community Configuration|Community Configuration]] project.
** [[c:File:Wikimania 2024 - Belgrade - Day 1 - Future of MediaWiki. A sustainable platform to support a collaborative user base and billions of page views.webm|Future of MediaWiki. A sustainable platform to support a collaborative user base and billions of page views]] (30 mins) - an overview for both technical and non technical audiences, covering some of the challenges and open questions, related to the [[mw:MediaWiki Product Insights|platform evolution, stewardship and developer experiences]] research.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/35|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W35&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:33, 26 August 2024 (UTC)
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== Tech News: 2024-36 ==

&lt;section begin=&quot;technews-2024-W36&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/36|Translations]] are available.

'''Weekly highlight'''
* Editors and volunteer developers interested in data visualisation can now test the new software for charts. Its early version is available on beta Commons and beta Wikipedia. This is an important milestone before making charts available on regular wikis. You can [[mw:Special:MyLanguage/Extension:Chart/Project/Updates|read more about this project update]] and help to test the charts.

'''Feature news'''
* Editors who use the [[{{#special:Unusedtemplates}}]] page can now filter out pages which are expected to be there permanently, such as sandboxes, test-cases, and templates that are always substituted. Editors can add the new magic word [[mw:Special:MyLanguage/Help:Magic words#EXPECTUNUSEDTEMPLATE|&lt;code dir=&quot;ltr&quot;&gt;&lt;nowiki&gt;__EXPECTUNUSEDTEMPLATE__&lt;/nowiki&gt;&lt;/code&gt;]] to a template page to hide it from the listing. Thanks to Sophivorus and DannyS712 for these improvements. [https://phabricator.wikimedia.org/T184633]
* Editors who use the New Topic tool on discussion pages, will [[phab:T334163|now be reminded]] to add a section header, which should help reduce the quantity of newcomers who add sections without a header. You can read more about that, and {{formatnum:28}} other community-submitted tasks that were [[m:Tech/News/Recently resolved community tasks|resolved last week]].
* Last week, some Toolforge tools had occasional connection problems. The cause is still being investigated, but the problems have been resolved for now. [https://phabricator.wikimedia.org/T373243]
* Translation administrators at multilingual wikis, when editing multiple translation units, can now easily mark which changes require updates to the translation. This is possible with the [[phab:T298852#10087288|new dropdown menu]].

'''Project updates'''
* A new draft text of a policy discussing the use of Wikimedia's APIs [[m:Special:MyLanguage/API Policy Update 2024|has been published on Meta-Wiki]]. The draft text does not reflect a change in policy around the APIs; instead, it is an attempt to codify existing API rules. Comments, questions, and suggestions are welcome on [[m:Talk:API Policy Update 2024|the proposed update’s talk page]] until September 13 or until those discussions have concluded.

'''Learn more'''
* To learn more about the technology behind the Wikimedia projects, you can now watch sessions from the technology track at Wikimania 2024 on Commons. This week, check out:
** [[c:File:Wikimania 2024 - Ohrid - Day 2 - Charts, the successor of Graphs - A secure and extensible tool for data visualization.webm|Charts, the successor of Graphs - A secure and extensible tool for data visualization]] (25 mins) – about the above-mentioned Charts project.
** [[c:File:Wikimania 2024 - Ohrid - Day 3 - State of Language Technology and Onboarding at Wikimedia.webm|State of Language Technology and Onboarding at Wikimedia]] (90 mins) – about some of the language tools that support Wikimedia sites, such as [[mw:Special:MyLanguage/Content translation|Content]]/[[mw:Special:MyLanguage/Content translation/Section translation|Section Translation]], [[mw:Special:MyLanguage/MinT|MinT]], and LanguageConverter; also the current state and future of languages onboarding. [https://phabricator.wikimedia.org/T368772]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/36|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W36&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 01:07, 3 September 2024 (UTC)
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== Tech News: 2024-37 ==

&lt;section begin=&quot;technews-2024-W37&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/37|Translations]] are available.

'''Feature news'''
* Starting this week, the standard [[mw:Special:MyLanguage/Extension:CodeMirror|syntax highlighter]] will receive new colors that make them compatible in dark mode. This is the first of many changes to come as part of a major upgrade to syntax highlighting. You can learn more about what's to come on the [[mw:Special:MyLanguage/Help:Extension:CodeMirror|help page]]. [https://phabricator.wikimedia.org/T365311][https://phabricator.wikimedia.org/T259059]
* Editors of wikis using Wikidata will now be notified of only relevant Wikidata changes in their watchlist. This is because the Lua functions &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;entity:getSitelink()&lt;/code&gt;&lt;/bdi&gt; and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;mw.wikibase.getSitelink(qid)&lt;/code&gt;&lt;/bdi&gt; will have their logic unified for tracking different aspects of sitelinks to reduce junk notifications from [[m:Wikidata For Wikimedia Projects/Projects/Watchlist Wikidata Sitelinks Tracking|inconsistent sitelinks tracking]]. [https://phabricator.wikimedia.org/T295356]

'''Project updates'''
*  Users of all Wikis will have access to Wikimedia sites as read-only for a few minutes on September 25, starting at 15:00 UTC. This is a planned datacenter switchover for maintenance purposes. More information will be published in Tech News and will also be posted on individual wikis in the coming weeks. [https://phabricator.wikimedia.org/T370962]
* Contributors of [[phab:T363538#10123348|11 Wikipedias]], including English will have a new &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;MOS&lt;/code&gt;&lt;/bdi&gt; namespace added to their Wikipedias. This improvement ensures that links beginning with &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;MOS:&lt;/code&gt;&lt;/bdi&gt; (usually shortcuts to the [[w:en:Wikipedia:Manual of Style|Manual of Style]]) are not broken by [[w:en:Mooré|Mooré]] Wikipedia (language code &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;mos&lt;/code&gt;&lt;/bdi&gt;). [https://phabricator.wikimedia.org/T363538]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/37|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W37&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 18:52, 9 September 2024 (UTC)
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== Tech News: 2024-38 ==

&lt;section begin=&quot;technews-2024-W38&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/38|Translations]] are available.

'''Improvements and Maintenance'''
* [[File:Octicons-gift.svg|12px|link=|class=skin-invert|Wishlist item]] Editors interested in templates can help by reading the latest Wishlist focus area, [[m:Special:MyLanguage/Community Wishlist/Focus areas/Template recall and discovery|Template recall and discovery]], and share your feedback on the talkpage. This input helps the Community Tech team to decide the right technical approach to build. Everyone is also encouraged to continue adding [[m:Special:MyLanguage/Community Wishlist|new wishes]].
* The new automated [[{{#special:NamespaceInfo}}]] page helps editors understand which [[mw:Special:MyLanguage/Help:Namespaces|namespaces]] exist on each wiki, and some details about how they are configured. Thanks to DannyS712 for these improvements. [https://phabricator.wikimedia.org/T263513]
* [[mw:Special:MyLanguage/Help:Edit check#Reference check|References Check]] is a feature that encourages editors to add a citation when they add a new paragraph to a Wikipedia article. For a short time, the corresponding tag &quot;Edit Check (references) activated&quot; was erroneously being applied to some edits outside of the main namespace. This has been fixed. [https://phabricator.wikimedia.org/T373692]
* It is now possible for a wiki community to change the order in which a page’s categories are displayed on their wiki. By default, categories are displayed in the order they appear in the wikitext. Now, wikis with a consensus to do so can [[m:Special:MyLanguage/Requesting wiki configuration changes|request]] a configuration change to display them in alphabetical order. [https://phabricator.wikimedia.org/T373480]
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Tool authors can now access ToolsDB's [[wikitech:Portal:Data Services#ToolsDB|public databases]] from both [[m:Special:MyLanguage/Research:Quarry|Quarry]] and [[wikitech:Superset|Superset]]. Those databases have always been accessible to every [[wikitech:Portal:Toolforge|Toolforge]] user, but they are now more broadly accessible, as Quarry can be accessed by anyone with a Wikimedia account. In addition, Quarry's internal database can now be [[m:Special:MyLanguage/Research:Quarry#Querying Quarry's own database|queried from Quarry itself]]. This database contains information about all queries that are being run and starred by users in Quarry. This information was already public through the web interface, but you can now query it using SQL. You can read more about that, and {{formatnum:20}} other community-submitted tasks that were [[m:Tech/News/Recently resolved community tasks|resolved last week]].
* Any pages or tools that still use the very old CSS classes &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;mw-message-box&lt;/code&gt;&lt;/bdi&gt; need to be updated. These old classes will be removed next week or soon afterwards. Editors can use a [https://global-search.toolforge.org/?q=mw-message-box&amp;regex=1&amp;namespaces=&amp;title= global-search] to determine what needs to be changed. It is possible to use the newer &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;cdx-message&lt;/code&gt;&lt;/bdi&gt; group of classes as a replacement (see [https://doc.wikimedia.org/codex/latest/components/demos/message.html#css-only-version the relevant Codex documentation], and [https://meta.wikimedia.org/w/index.php?title=Tech/Header&amp;diff=prev&amp;oldid=27449042 an example update]), but using locally defined onwiki classes would be best. [https://phabricator.wikimedia.org/T374499]

'''Technical project updates'''
* Next week, all Wikimedia wikis will be read-only for a few minutes. This will start on September 25 at [https://zonestamp.toolforge.org/1727276400 15:00 UTC]. This is a planned datacenter switchover for maintenance purposes. [[m:Special:MyLanguage/Tech/Server switch|This maintenance process also targets other services.]] The previous switchover took 3 minutes, and the Site Reliability Engineering teams use many tools to make sure that this essential maintenance work happens as quickly as possible. [https://phabricator.wikimedia.org/T370962]

'''Tech in depth'''
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] The latest monthly [[mw:Special:MyLanguage/MediaWiki Product Insights/Reports/August 2024|MediaWiki Product Insights newsletter]] is available. This edition includes details about: research about [[mw:Special:MyLanguage/Manual:Hooks|hook]] handlers to help simplify development, research about performance improvements, work to improve the REST API for end-users, and more.
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] To learn more about the technology behind the Wikimedia projects, you can now watch sessions from the technology track at Wikimania 2024 on Commons. This week, check out: 
** [[c:File:Wikimania 2024 - Auditorium Kyiv - Day 4 - Hackathon Showcase.webm|Hackathon Showcase]] (45 mins) - 19 short presentations by some of the Hackathon participants, describing some of the projects they worked on, such as automated testing of maintenance scripts, a video-cutting command line tool, and interface improvements for various tools. There are [[phab:T369234|more details and links available]] in the Phabricator task.
** [[c:File:Co-Creating a Sustainable Future for the Toolforge Ecosystem.webm|Co-Creating a Sustainable Future for the Toolforge Ecosystem]] (40 mins) - a roundtable discussion for tool-maintainers, users, and supporters of Toolforge about how to make the platform sustainable and how to evaluate the tools available there.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/38|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W38&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:02, 17 September 2024 (UTC)
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== Tech News: 2024-39 ==

&lt;section begin=&quot;technews-2024-W39&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/39|Translations]] are available.

'''Weekly highlight'''
* All wikis will be [[m:Special:MyLanguage/Tech/Server switch|read-only]] for a few minutes on Wednesday September 25 at [https://zonestamp.toolforge.org/1727276400 15:00 UTC]. Reading the wikis will not be interrupted, but editing will be paused. These twice-yearly processes allow WMF's site reliability engineering teams to remain prepared to keep the wikis functioning even in the event of a major interruption to one of our data centers.

'''Updates for editors'''
[[File:Add alt text from a halfsheet, with the article behind.png|thumb|A screenshot of the interface for the Alt Text suggested-edit feature]]
* Editors who use the iOS Wikipedia app in Spanish, Portuguese, French, or Chinese, may see the [[mw:Special:MyLanguage/Wikimedia Apps/iOS Suggested edits project/Alt Text Experiment|Alt Text suggested-edit experiment]] after editing an article, or completing a suggested edit using &quot;[[mw:Special:MyLanguage/Wikimedia Apps/iOS Suggested edits project#Hypothesis 2 Add an Image Suggested Edit|Add an image]]&quot;. Alt-text helps people with visual impairments to read Wikipedia articles. The team aims to learn if adding alt-text to images is a task that editors can be successful with. Please share any feedback on [[mw:Talk:Wikimedia Apps/iOS Suggested edits project/Alt Text Experiment|the discussion page]].
* The Codex color palette has been updated with new and revised colors for the MediaWiki user interfaces. The [[mw:Special:MyLanguage/Design System Team/Color/Design documentation#Updates|most noticeable changes]] for editors include updates for: dark mode colors for Links and for quiet Buttons (progressive and destructive), visited Link colors for both light and dark modes, and background colors for system-messages in both light and dark modes.
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] It is now possible to include clickable wikilinks and external links inside code blocks. This includes links that are used within &lt;code&gt;&lt;nowiki&gt;&lt;syntaxhighlight&gt;&lt;/nowiki&gt;&lt;/code&gt; tags and on code pages (JavaScript, CSS, Scribunto and Sanitized CSS). Uses of template syntax &lt;code&gt;&lt;nowiki&gt;{{…}}&lt;/nowiki&gt;&lt;/code&gt; are also linked to the template page. Thanks to SD0001 for these improvements. [https://phabricator.wikimedia.org/T368166]
* Two bugs were fixed in the [[m:Special:MyLanguage/Account vanishing|GlobalVanishRequest]] system by improving the logging and by removing an incorrect placeholder message. [https://phabricator.wikimedia.org/T370595][https://phabricator.wikimedia.org/T372223]
* View all {{formatnum:25}} community-submitted {{PLURAL:25|task|tasks}} that were [[m:Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] From [[m:Special:MyLanguage/Wikimedia Enterprise|Wikimedia Enterprise]]: 
** The API now enables 5,000 on-demand API requests per month and twice-monthly HTML snapshots freely (gratis and libre). More information on the updates and also improvements to the software development kits (SDK) are explained on [https://enterprise.wikimedia.com/blog/enhanced-free-api/ the project's blog post]. While Wikimedia Enterprise APIs are designed for high-volume commercial reusers, this change enables many more community use-cases to be built on the service too.
** The Snapshot API (html dumps) have added beta Structured Contents endpoints ([https://enterprise.wikimedia.com/blog/structured-contents-snapshot-api/ blog post on that]) as well as released two beta datasets (English and French Wikipedia) from that endpoint to Hugging Face for public use and feedback ([https://enterprise.wikimedia.com/blog/hugging-face-dataset/ blog post on that]). These pre-parsed data sets enable new options for researchers, developers, and data scientists to use and study the content.

'''In depth'''
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] The Wikidata Query Service (WDQS) is used to get answers to questions using the Wikidata data set. As Wikidata grows, we had to make a major architectural change so that WDQS could remain performant. As part of the [[d:Special:MyLanguage/Wikidata:SPARQL query service/WDQS graph split|WDQS Graph Split project]], we have new SPARQL endpoints available for serving the &quot;[https://query-scholarly.wikidata.org scholarly]&quot; and &quot;[https://query-main.wikidata.org main]&quot; subgraphs of Wikidata. The [http://query.wikidata.org query.wikidata.org endpoint] will continue to serve the full Wikidata graph until March 2025. After this date, it will only serve the main graph. For more information, please see [[d:Special:MyLanguage/Wikidata:SPARQL query service/WDQS backend update/September 2024 scaling update|the announcement on Wikidata]].

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/39|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W39&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:36, 23 September 2024 (UTC)
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== Tech News: 2024-40 ==

&lt;section begin=&quot;technews-2024-W40&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/40|Translations]] are available.

'''Updates for editors'''
* Readers of [[phab:T375401|42 more wikis]] can now use Dark Mode. If the option is not yet available for logged-out users of your wiki, this is likely because many templates do not yet display well in Dark Mode. Please use the [https://night-mode-checker.wmcloud.org/ night-mode-checker tool] if you are interested in helping to reduce the number of issues. The [[mw:Special:MyLanguage/Recommendations for night mode compatibility on Wikimedia wikis|recommendations page]] provides guidance on this. Dark Mode is enabled on additional wikis once per month.
* Editors using the 2010 wikitext editor as their default can access features from the 2017 wikitext editor by adding &lt;code dir=ltr&gt;?veaction=editsource&lt;/code&gt; to the URL. If you would like to enable the 2017 wikitext editor as your default, it can be set in [[Special:Preferences#mw-input-wpvisualeditor-newwikitext|your preferences]]. [https://phabricator.wikimedia.org/T239796]
* For logged-out readers using the Vector 2022 skin, the &quot;donate&quot; link has been moved from a collapsible menu next to the content area into a more prominent top menu, next to &quot;Create an account&quot;. This restores the link to the level of prominence it had in the Vector 2010 skin. [[mw:Readers/2024 Reader and Donor Experiences#Donor Experiences (Key Result WE 3.2 and the related hypotheses)|Learn more]] about the changes related to donor experiences. [https://phabricator.wikimedia.org/T373585]
* The CampaignEvents extension provides tools for organizers to more easily manage events, communicate with participants, and promote their events on the wikis. The extension has been [[m:Special:MyLanguage/CampaignEvents/Deployment status|enabled]] on Arabic Wikipedia, Igbo Wikipedia, Swahili Wikipedia, and Meta-Wiki. [[w:zh:Wikipedia:互助客栈/其他#引進CampaignEvents擴充功能|Chinese Wikipedia has decided]] to enable the extension, and discussions on the extension are in progress [[w:es:Wikipedia:Votaciones/2024/Sobre la política de Organizadores de Eventos|on Spanish Wikipedia]] and [[d:Wikidata:Project chat#Enabling the CampaignEvents Extention on Wikidata|on Wikidata]]. To learn how to enable the extension on your wiki, you can visit [[m:Special:MyLanguage/CampaignEvents|the CampaignEvents page on Meta-Wiki]].
* View all {{formatnum:22}} community-submitted {{PLURAL:22|task|tasks}} that were [[m:Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* Developers with an account on Wikitech-wiki should [[wikitech:Wikitech/SUL-migration|check if any action is required]] for their accounts. The wiki is being changed to use the single-user-login (SUL) system, and other configuration changes. This change will help reduce the overall complexity for the weekly software updates across all our wikis.

'''In depth'''
* The [[m:Special:MyLanguage/Tech/Server switch|server switch]] was completed successfully last week with a read-only time of [[wikitech:Switch Datacenter#Past Switches|only 2 minutes 46 seconds]]. This periodic process makes sure that engineers can switch data centers and keep all of the wikis available for readers, even if there are major technical issues. It also gives engineers a chance to do maintenance and upgrades on systems that normally run 24 hours a day, and often helps to reveal weaknesses in the infrastructure. The process involves dozens of software services and hundreds of hardware servers, and requires multiple teams working together. Work over the past few years has reduced the time from 17 minutes down to 2–3 minutes. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/66ZW7B2MG63AESQVTXDIFQBDBS766JGW/]

'''Meetings and events'''
* October 4–6: [[m:Special:MyLanguage/WikiIndaba conference 2024|WikiIndaba Conference's Hackathon]] in Johannesburg, South Africa
* November 4–6: [[mw:Special:MyLanguage/MediaWiki Users and Developers Conference Fall 2024|MediaWiki Users and Developers Conference Fall 2024]] in Vienna, Austria

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/40|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W40&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:20, 30 September 2024 (UTC)
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== Tech News: 2024-41 ==

&lt;section begin=&quot;technews-2024-W41&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/41|Translations]] are available.

'''Weekly highlight'''
* Communities can now request installation of [[mw:Special:MyLanguage/Moderator Tools/Automoderator|Automoderator]] on their wiki. Automoderator is an automated anti-vandalism tool that reverts bad edits based on scores from the new &quot;Revert Risk&quot; machine learning model. You can [[mw:Special:MyLanguage/Extension:AutoModerator/Deploying|read details about the necessary steps]] for installation and configuration. [https://phabricator.wikimedia.org/T336934]

'''Updates for editors'''
* Translators in wikis where [[mw:Special:MyLanguage/Content translation/Section translation#Try the tool|the mobile experience of Content Translation is available]], can now customize their articles suggestion list from 41 filtering options when using the tool. This topic-based article suggestion feature makes it easy for translators to self-discover relevant articles based on their area of interest and translate them. You can [https://test.wikipedia.org/w/index.php?title=Special:ContentTranslation&amp;active-list=suggestions try it with your mobile device]. [https://phabricator.wikimedia.org/T368422]
* View all {{formatnum:12}} community-submitted {{PLURAL:12|task|tasks}} that were [[m:Tech/News/Recently resolved community tasks|resolved last week]].

'''Updates for technical contributors'''
* It is now possible for &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;&lt;syntaxhighlight&gt;&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; code blocks to offer readers a &quot;Copy&quot; button if the &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;&lt;nowiki&gt;copy=1&lt;/nowiki&gt;&lt;/code&gt;&lt;/bdi&gt; attribute is [[mw:Special:MyLanguage/Extension:SyntaxHighlight#copy|set on the tag]]. Thanks to SD0001 for these improvements. [https://phabricator.wikimedia.org/T40932]
* Customized copyright footer messages on all wikis will be updated. The new versions will use wikitext markup instead of requiring editing raw HTML. [https://phabricator.wikimedia.org/T375789]
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Later this month, [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|temporary accounts]] will be rolled out on several pilot wikis. The final list of the wikis will be published in the second half of the month. If you maintain any tools, bots, or gadgets on [[phab:T376499|these 11 wikis]], and your software is using data about IP addresses or is available for logged-out users, please check if it needs to be updated to work with temporary accounts. [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/For developers|Guidance on how to update the code is available]].
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Rate limiting has been enabled for the code review tools [[Wikitech:Gerrit|Gerrit]] and [[Wikitech:GitLab|GitLab]] to address ongoing issues caused by malicious traffic and scraping. Clients that open too many concurrent connections will be restricted for a few minutes. This rate limiting is managed through [[Wikitech:nftables|nftables]] firewall rules. For more details, see Wikitech's pages on [[Wikitech:Firewall#Throttling with nftables|Firewall]], [[Wikitech:GitLab/Abuse and rate limiting|GitLab limits]] and [[Wikitech:Gerrit/Operations#Throttling IPs|Gerrit operations]].
* Five new wikis have been created: 
** a {{int:project-localized-name-group-wikipedia}} in [[d:Q49224|Komering]] ([[w:kge:|&lt;code&gt;w:kge:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T374813]
** a {{int:project-localized-name-group-wikipedia}} in [[d:Q36096|Mooré]] ([[m:mos:|&lt;code&gt;m:mos:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T374641]
** a {{int:project-localized-name-group-wiktionary}} in [[d:Q36213|Madurese]] ([[wikt:mad:|&lt;code&gt;wikt:mad:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T374968]
** a {{int:project-localized-name-group-wikiquote}} in [[d:Q2501174|Gorontalo]] ([[q:gor:|&lt;code&gt;q:gor:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T375088]
** a {{int:project-localized-name-group-wikinews}} in [[d:Q56482|Shan]] ([[n:shn:|&lt;code&gt;n:shn:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T375430]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/41|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W41&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 23:42, 7 October 2024 (UTC)
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== Tech News: 2024-42 ==

&lt;section begin=&quot;technews-2024-W42&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/42|Translations]] are available.

'''Updates for editors'''
* The Structured Discussion extension (also known as Flow) is starting to be removed. This extension is unmaintained and causes issues. It will be replaced by [[mw:Special:MyLanguage/Help:DiscussionTools|DiscussionTools]], which is used on any regular talk page. [[mw:Special:MyLanguage/Structured Discussions/Deprecation#Deprecation timeline|A first set of wikis]] are being contacted. These wikis are invited to stop using Flow, and to move all Flow boards to sub-pages, as archives. At these wikis, a script will move all Flow pages that aren't a sub-page to a sub-page automatically, starting on 22 October 2024. On 28 October 2024, all Flow boards at these wikis will be set in read-only mode. [https://www.mediawiki.org/wiki/Structured_Discussions/Deprecation][https://phabricator.wikimedia.org/T370722]
* WMF's Search Platform team is working on making it easier for readers to perform text searches in their language. A [[phab:T332342|change last week]] on over 30 languages makes it easier to find words with accents and other diacritics. This applies to both full-text search and to types of advanced search such as the &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;''hastemplate''&lt;/bdi&gt; and &lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;''incategory''&lt;/bdi&gt; keywords. More technical details (including a few other minor search upgrades) are available. [https://www.mediawiki.org/wiki/User:TJones_%28WMF%29/Notes/Language_Analyzer_Harmonization_Notes#ASCII-folding/ICU-folding_%28T332342%29]
* View all {{formatnum:20}} community-submitted {{PLURAL:20|task|tasks}} that were [[m:Tech/News/Recently resolved community tasks|resolved last week]]. For example, [[mw:Special:MyLanguage/Help:Edit check|EditCheck]] was installed at Russian Wikipedia, and fixes were made for some missing user interface styles.

'''Updates for technical contributors'''
* Editors who use the Toolforge tool [[toolforge:copyvios|Earwig's Copyright Violation Detector]] will now be required to log in with their Wikimedia account before running checks using the &quot;search engine&quot; option. This change is needed to help prevent external bots from misusing the system. Thanks to Chlod for these improvements. [https://en.wikipedia.org/wiki/Wikipedia_talk:New_pages_patrol/Reviewers#Authentication_is_now_required_for_search_engine_checks_on_Earwig's_Copyvio_Tool]
* [[m:Special:MyLanguage/Phabricator|Phabricator]] users can create tickets and add comments on existing tickets via Email again. [[mw:Special:MyLanguage/Phabricator/Help#Using email|Sending email to Phabricator]] has been fixed. [https://phabricator.wikimedia.org/T356077]
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Some HTML elements in the interface are now wrapped with a &lt;code&gt;&lt;nowiki&gt;&lt;bdi&gt;&lt;/nowiki&gt;&lt;/code&gt; element, to make our HTML output more aligned with Web standards. More changes like this will be coming in future weeks. This change might break some tools that rely on the previous HTML structure of the interface. Note that relying on the HTML structure of the interface is [[mw:Special:MyLanguage/Stable interface policy/Frontend#What is not stable?|not recommended]] and might break at any time. [https://phabricator.wikimedia.org/T375975]

'''In depth'''
* The latest monthly [[mw:Special:MyLanguage/MediaWiki Product Insights/Reports/September 2024|MediaWiki Product Insights newsletter]] is available. This edition includes: updates on Wikimedia's authentication system, research to simplify feature development in the MediaWiki platform, updates on Parser Unification and MathML rollout, and more.
* The latest quarterly [[mw:Technical Community Newsletter/2024/October|Technical Community Newsletter]] is now available. This edition include: research about improving topic suggestions related to countries, improvements to PHPUnit tests, and more.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/42|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W42&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 21:21, 14 October 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27597254 --&gt;

== Tech News: 2024-43 ==

&lt;section begin=&quot;technews-2024-W43&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/43|Translations]] are available.

'''Weekly highlight'''
* The Mobile Apps team has released an [[mw:Special:MyLanguage/Wikimedia Apps/Team/iOS/Navigation Refresh#Phase 1: Creating a user Profile Menu (T373714)|update]] to the iOS app's navigation, and it is now available in the latest App store version. The team added a new Profile menu that allows for easy access to editor features like Notifications and Watchlist from the Article view, and brings the &quot;Donate&quot; button into a more accessible place for users who are reading an article. This is the first phase of a larger planned [[mw:Special:MyLanguage/Wikimedia Apps/Team/iOS/Navigation Refresh|navigation refresh]] to help the iOS app transition from a primarily reader-focused app, to an app that fully supports reading and editing. The Wikimedia Foundation has added more editing features and support for on-wiki communication based on volunteer requests in recent years.
[[File:IOS App Navigation refresh first phase 05.png|thumb|iOS Wikipedia App's profile menu and contents]]

'''Updates for editors'''
* Wikipedia readers can now download a browser extension to experiment with some early ideas on potential features that recommend articles for further reading, automatically summarize articles, and improve search functionality. For more details and to stay updated, check out the Web team's [[mw:Special:MyLanguage/Reading/Web/Content Discovery Experiments|Content Discovery Experiments page]] and [[mw:Special:MyLanguage/Newsletter:Web team's projects|subscribe to their newsletter]].
* Later this month, logged-out editors of [[phab:T376499|these 12 wikis]] will start to have [[mw:Special:Mylanguage/Trust and Safety Product/Temporary Accounts|temporary accounts]] created. The list may slightly change - some wikis may be removed but none will be added. Temporary account is a new [[mw:Special:MyLanguage/User account types|type of user account]]. It enhances the logged-out editors' privacy and makes it easier for community members to communicate with them. If you maintain any tools, bots, or gadgets on these 12 wikis, and your software is using data about IP addresses or is available for logged-out users, please check if it needs to be updated to work with temporary accounts. [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/For developers|Guidance on how to update the code is available]]. Read more about the [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/Updates|deployment plan across all wikis]].
* View all {{formatnum:33}} community-submitted {{PLURAL:33|task|tasks}} that were [[m:Tech/News/Recently resolved community tasks|resolved last week]]. For example, the [[w:nr:Main Page|South Ndebele]], [[w:rsk:Главни бок|Pannonian Rusyn]], [[w:ann:Uwu|Obolo]], [[w:iba:Lambar Keterubah|Iban]] and [[w:tdd:ᥞᥨᥝᥴ ᥘᥣᥲ ᥖᥥᥰ|Tai Nüa]] Wikipedia languages were created last week. [https://www.wikidata.org/wiki/Q36785][https://www.wikidata.org/wiki/Q35660][https://www.wikidata.org/wiki/Q36614][https://www.wikidata.org/wiki/Q33424][https://www.wikidata.org/wiki/Q36556]
* It is now possible to create functions on Wikifunctions using Wikidata lexemes, through the new [[f:Z6005|Wikidata lexeme type]] launched last week. When you go to one of these functions, the user interface provides a lexeme selector that helps you pick a lexeme from Wikidata that matches the word you type. After hitting run, your selected lexeme is retrieved from Wikidata, transformed into a Wikidata lexeme type, and passed into the selected function. Read more about this in [[f:Special:MyLanguage/Wikifunctions:Status updates/2024-10-17#Function of the Week: select representation from lexeme|the latest Wikifunctions newsletter]].

'''Updates for technical contributors'''
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Users of the Wikimedia sites can now format dates more easily in different languages with the new &lt;code dir=&quot;ltr&quot;&gt;{{[[mw:Special:MyLanguage/Help:Extension:ParserFunctions##timef|#timef]]:…}}&lt;/code&gt; parser function. For example, &lt;code dir=&quot;ltr&quot;&gt;&lt;nowiki&gt;{{#timef:now|date|en}}&lt;/nowiki&gt;&lt;/code&gt; will show as &quot;&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;{{#timef:now|date|en}}&lt;/bdi&gt;&quot;. Previously, &lt;code dir=&quot;ltr&quot;&gt;&lt;nowiki&gt;{{#time:…}}&lt;/nowiki&gt;&lt;/code&gt; could be used to format dates, but this required knowledge of the order of the time and date components and their intervening punctuation. &lt;code dir=&quot;ltr&quot;&gt;#timef&lt;/code&gt; (or &lt;code dir=&quot;ltr&quot;&gt;#timefl&lt;/code&gt; for local time) provides access to the standard date formats that MediaWiki uses in its user interface. This may help to simplify some templates on multi-lingual wikis like Commons and Meta. [https://phabricator.wikimedia.org/T223772][https://www.mediawiki.org/wiki/Special:MyLanguage/Help:Extension:ParserFunctions##timef]
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Commons and Meta users can now efficiently [[mw:Special:MyLanguage/Help:Magic words#Localization|retrieve the user's language]] using &lt;code dir=&quot;ltr&quot;&gt;&lt;nowiki&gt;{{USERLANGUAGE}}&lt;/nowiki&gt;&lt;/code&gt; instead of using &lt;code dir=&quot;ltr&quot;&gt;&lt;nowiki&gt;{{int:lang}}&lt;/nowiki&gt;&lt;/code&gt;. [https://phabricator.wikimedia.org/T4085]
* The [[m:Special:MyLanguage/Product and Technology Advisory Council|Product and Tech Advisory Council]] (PTAC) now has its pilot members with representation across Africa, Asia, Europe, North America and South America. They will work to address the [[Special:MyLanguage/Movement Strategy/Initiatives/Technology Council|Movement Strategy's Technology Council]] initiative of having a co-defined and more resilient technological platform. [https://meta.wikimedia.org/wiki/Movement_Strategy/Initiatives/Technology_Council]

'''In depth'''
* The latest quarterly [[mw:Special:MyLanguage/Growth/Newsletters/32|Growth newsletter]] is available. It includes: an upcoming Newcomer Homepage Community Updates module, new Community Configuration options, and details on new projects.
* The Wikimedia Foundation is [[mw:Special:MyLanguage/Wikimedia Security Team#CNA Partnership|now an official partner of the CVE program]], which is an international effort to catalog publicly disclosed cybersecurity vulnerabilities. This partnership will allow the Security Team to instantly publish [[w:en:Common Vulnerabilities and Exposures|common vulnerabilities and exposures]] (CVE) records that are affecting MediaWiki core, extensions, and skins, along with any other code the Foundation is a steward of.
* The [[m:Special:MyLanguage/Community Wishlist|Community Wishlist]] is now [[m:Community Wishlist/Updates#October 16, 2024: Conversations Made Easier: Machine-Translated Wishes Are Here!|testing machine translations]] for Wishlist content. Volunteers can now read machine-translated versions of wishes and dive into discussions even before translators arrive to translate content.

'''Meetings and events'''
* 24 October - Wiki Education Speaker Series Webinar - [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/N4XTB4G55BUY3M3PNGUAKQWJ7A4UOPAK/ Open Source Tech: Building the Wiki Education Dashboard], featuring Wikimedia interns and a Web developer in the panel.
* 20–22 December 2024 - [[m:Special:MyLanguage/Indic Wikimedia Hackathon Bhubaneswar 2024|Indic Wikimedia Hackathon Bhubaneswar 2024]] in Odisha, India. A hackathon for community members, including developers, designers and content editors, to build technical solutions that improve contributors' experiences.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/43|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W43&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:52, 21 October 2024 (UTC)
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== Tech News: 2024-44 ==

&lt;section begin=&quot;technews-2024-W44&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/44|Translations]] are available.

'''Updates for editors'''
* Later in November, the Charts extension will be deployed to the test wikis in order to help identify and fix any issue. A security review is underway to then enable deployment to pilot wikis for broader testing. You can read [[mw:Special:MyLanguage/Extension:Chart/Project/Updates#October 2024: Working towards production deployment|the October project update]] and see the [https://en.wikipedia.beta.wmflabs.org/wiki/Charts latest documentation and examples on Beta Wikipedia].
* View all {{formatnum:32}} community-submitted {{PLURAL:32|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, [[w:en:PediaPress|Pediapress.com]], an external service that creates books from Wikipedia, can now use [[mw:Special:MyLanguage/Wikimedia Maps|Wikimedia Maps]] to include existing pre-rendered infobox map images in their printed books on Wikipedia. [https://phabricator.wikimedia.org/T375761]

'''Updates for technical contributors'''
* Wikis can use [[:mw:Special:MyLanguage/Extension:GuidedTour|the Guided Tour extension]] to help newcomers understand how to edit. The Guided Tours extension now works with [[mw:Special:MyLanguage/Manual:Dark mode|dark mode]]. Guided Tour maintainers can check their tours to see that nothing looks odd. They can also set &lt;code&gt;emitTransitionOnStep&lt;/code&gt; to &lt;code&gt;true&lt;/code&gt; to fix an old bug. They can use the new flag &lt;code&gt;allowAutomaticBack&lt;/code&gt; to avoid back-buttons they don't want. [https://phabricator.wikimedia.org/T73927#10241528]
*  Administrators in the Wikimedia projects who use the [[mw:Special:MyLanguage/Help:Extension:Nuke|Nuke Extension]] will notice that mass deletions done with this tool have the &quot;Nuke&quot; tag. This change will make reviewing and analyzing deletions performed with the tool easier. [https://phabricator.wikimedia.org/T366068]

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/44|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W44&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:56, 28 October 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27668811 --&gt;

== Tech News: 2024-45 ==

&lt;section begin=&quot;technews-2024-W45&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/45|Translations]] are available.

'''Updates for editors'''
* Stewards can now make [[m:Special:MyLanguage/Global blocks|global account blocks]] cause global [[mw:Special:MyLanguage/Autoblock|autoblocks]]. This will assist stewards in preventing abuse from users who have been globally blocked. This includes preventing globally blocked temporary accounts from exiting their session or switching browsers to make subsequent edits for 24 hours. Previously, temporary accounts could exit their current session or switch browsers to continue editing. This is an anti-abuse tool improvement for the [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|Temporary Accounts]] project. You can read more about the [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/Updates|progress on key features for temporary accounts]]. [https://phabricator.wikimedia.org/T368949]
* Wikis that have the [[m:Special:MyLanguage/CampaignEvents/Deployment status|CampaignEvents extension enabled]] can now use the [[m:Special:MyLanguage/Campaigns/Foundation Product Team/Event list#October 29, 2024: Collaboration List launched|Collaboration List]] feature. This list provides a new, easy way for contributors to learn about WikiProjects on their wikis. Thanks to the Campaign team for this work that is part of [[m:Special:MyLanguage/Wikimedia Foundation Annual Plan/2024-2025/Product %26 Technology OKRs#WE KRs|the 2024/25 annual plan]]. If you are interested in bringing the CampaignEvents extension to your wiki, you can [[m:Special:MyLanguage/CampaignEvents/Deployment status#How to Request the CampaignEvents Extension for your wiki|follow these steps]] or you can reach out to User:Udehb-WMF for help.
* The text color for red links will be slightly changed later this week to improve their contrast in light mode. [https://phabricator.wikimedia.org/T370446]
* View all {{formatnum:32}} community-submitted {{PLURAL:32|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, on multilingual wikis, users [[phab:T216368|can now]] hide translations from the WhatLinksHere special page.

'''Updates for technical contributors'''
* XML [[m:Special:MyLanguage/Data dumps|data dumps]] have been temporarily paused whilst a bug is investigated. [https://lists.wikimedia.org/hyperkitty/list/xmldatadumps-l@lists.wikimedia.org/message/BXWJDPO5QI2QMBCY7HO36ELDCRO6HRM4/]

'''In depth'''
* Temporary Accounts have been deployed to six wikis; thanks to the Trust and Safety Product team for [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts|this work]], you can read about [[phab:T340001|the deployment plans]]. Beginning next week, Temporary Accounts will also be enabled on [[phab:T378336|seven other projects]]. If you are active on these wikis and need help migrating your tools, please reach out to [[m:User:Udehb-WMF|User:Udehb-WMF]] for assistance.
*  The latest quarterly [[mw:Special:MyLanguage/Wikimedia Language and Product Localization/Newsletter/2024/October|Language and Internationalization newsletter]] is available.  It includes: New languages supported in translatewiki or in MediaWiki; New keyboard input methods for some languages; details about recent and upcoming meetings, and more.

'''Meetings and events'''
* [[mw:Special:MyLanguage/MediaWiki Users and Developers Conference Fall 2024|MediaWiki Users and Developers Conference Fall 2024]] is happening in Vienna, Austria and online from 4 to 6 November 2024. The conference will feature discussions around the usage of MediaWiki software by and within companies in different industries and will inspire and onboard new users.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/45|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W45&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 20:50, 4 November 2024 (UTC)
&lt;!-- Message sent by User:UOzurumba (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27693917 --&gt;

== Tech News: 2024-46 ==

&lt;section begin=&quot;technews-2024-W46&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/46|Translations]] are available.

'''Updates for editors'''
* On wikis with the [[mw:Special:MyLanguage/Help:Extension:Translate|Translate extension]] enabled, users will notice that the FuzzyBot will now automatically create translated versions of categories used on translated pages. [https://phabricator.wikimedia.org/T285463]
* View all {{formatnum:29}} community-submitted {{PLURAL:29|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, the submitted task to use the [[mw:Special:MyLanguage/Extension:SecurePoll|SecurePoll extension]] for English Wikipedia's special [[w:en:Wikipedia:Administrator elections|administrator election]] was resolved on time. [https://phabricator.wikimedia.org/T371454] 

'''Updates for technical contributors'''
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] In &lt;code dir=&quot;ltr&quot;&gt;[[mw:MediaWiki_1.44/wmf.2|1.44.0-wmf-2]]&lt;/code&gt;, the logic of Wikibase function &lt;code&gt;getAllStatements&lt;/code&gt; changed to behave like &lt;code&gt;getBestStatements&lt;/code&gt;. Invoking the function now returns a copy of values which are immutable. [https://phabricator.wikimedia.org/T270851]
* [https://en.wikipedia.org/api/rest_v1/ Wikimedia REST API] users, such as bot operators and tool maintainers, may be affected by ongoing upgrades. The API will be rerouting some page content endpoints from RESTbase to the newer [[mw:Special:MyLanguage/API:REST API|MediaWiki REST API]] endpoints. The [[phab:T374683|impacted endpoints]] include getting page/revision metadata and rendered HTML content. These changes will be available on testwiki later this week, with other projects to follow. This change should not affect existing functionality, but active users of the impacted endpoints should verify behavior on testwiki, and raise any concerns on the related [[phab:T374683|Phabricator ticket]].

'''In depth'''
* Admins and users of the Wikimedia projects [[mw:Special:MyLanguage/Moderator_Tools/Automoderator#Usage|where Automoderator is enabled]] can now monitor and evaluate important metrics related to Automoderator's actions. [https://superset.wmcloud.org/superset/dashboard/unified-automoderator-activity-dashboard/ This Superset dashboard] calculates and aggregates metrics about Automoderator's behaviour on the projects in which it is deployed. Thanks to the Moderator Tools team for this Dashboard; you can visit [[mw:Special:MyLanguage/Moderator Tools/Automoderator/Unified Activity Dashboard|the documentation page]] for more information about this work. [https://phabricator.wikimedia.org/T369488]

'''Meetings and events'''
* 21 November 2024 ([[m:Special:MyLanguage/Event:Commons community discussion - 21 November 2024 8:00 UTC|8:00 UTC]] &amp; [[m:Special:MyLanguage/Event:Commons community discussion - 21 November 2024 16:00 UTC|16:00 UTC]]) - [[c:Commons:WMF support for Commons/Commons community calls|Community call]] with Wikimedia Commons volunteers and stakeholders to help prioritize support efforts for 2025-2026 Fiscal Year. The theme of this call is how content should be organised on Wikimedia Commons.
'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/46|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W46&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 00:07, 12 November 2024 (UTC)
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== Tech News: 2024-47 ==

&lt;section begin=&quot;technews-2024-W47&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/47|Translations]] are available.

'''Updates for editors'''
* Users of Wikimedia sites will now be warned when they create a [[mw:Special:MyLanguage/Help:Redirects|redirect]] to a page that doesn't exist. This will reduce the number of broken redirects to red links in our projects. [https://phabricator.wikimedia.org/T326057]
* View all {{formatnum:42}} community-submitted {{PLURAL:42|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, [[mw:Special:MyLanguage/Manual:Pywikibot/Overview|Pywikibot]], which automates work on MediaWiki sites, was upgraded to 9.5.0 on Toolforge. [https://phabricator.wikimedia.org/T378676]

'''Updates for technical contributors'''
* On wikis that use the [[mw:Special:MyLanguage/Extension:FlaggedRevs|FlaggedRevs extension]], pages created or moved by users with the appropriate permissions are marked as flagged automatically. This feature has not been working recently, and changes fixing it should be deployed this week. Thanks to Daniel and Wargo for working on this. [https://phabricator.wikimedia.org/T379218][https://phabricator.wikimedia.org/T368380]

'''In depth'''
* There is a new [https://diff.wikimedia.org/2024/11/05/say-hi-to-temporary-accounts-easier-collaboration-with-logged-out-editors-with-better-privacy-protection Diff post] about Temporary Accounts, available in more than 15 languages. Read it to learn about what Temporary Accounts are, their impact on different groups of users, and the plan to introduce the change on all wikis.

'''Meetings and events'''
* Technical volunteers can now register for the [[mw:Special:MyLanguage/Wikimedia Hackathon 2025|2025 Wikimedia Hackathon]], which will take place in Istanbul, Turkey. [https://pretix.eu/wikimedia/hackathon2025/ Application for travel and accommodation scholarships] is open from '''November 12 to December 10 2024'''. The registration for the event will close in mid-April 2025. The Wikimedia Hackathon is an annual gathering that unites the global technical community to collaborate on existing projects and explore new ideas.
* Join the [[C:Special:MyLanguage/Commons:WMF%20support%20for%20Commons/Commons%20community%20calls|Wikimedia Commons community calls]] this week to help prioritize support for Commons which will be planned for 2025–2026. The theme will be how content should be organised on Wikimedia Commons. This is an opportunity for volunteers who work on different things to come together and talk about what matters for the future of the project. The calls will take place '''November 21, 2024, [[m:Special:MyLanguage/Event:Commons community discussion - 21 November 2024 8:00 UTC|8:00 UTC]] and [[m:Special:MyLanguage/Event:Commons community discussion - 21 November 2024 16:00 UTC|16:00 UTC]]'''.
* A [[mw:Special:MyLanguage/Wikimedia_Language_and_Product_Localization/Community meetings#29 November 2024|Language community meeting]] will take place '''November 29, 16:00 UTC''' to discuss updates and technical problem-solving.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/47|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W47&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 02:00, 19 November 2024 (UTC)
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== Tech News: 2024-48 ==

&lt;section begin=&quot;technews-2024-W48&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/48|Translations]] are available.

'''Updates for editors'''
* [[File:Octicons-gift.svg|12px|link=|class=skin-invert|Wishlist item]] A new version of the standard wikitext editor-mode [[mw:Special:MyLanguage/Extension:CodeMirror|syntax highlighter]] will be available as a [[Special:Preferences#mw-prefsection-betafeatures|beta feature]] later this week. This brings many new features and bug fixes, including right-to-left support, [[mw:Special:MyLanguage/Help:Extension:CodeMirror#Template folding|template folding]], [[mw:Special:MyLanguage/Help:Extension:CodeMirror#Autocompletion|autocompletion]], and an improved search panel. You can learn more on the [[mw:Special:MyLanguage/Help:Extension:CodeMirror|help page]].
* The 2010 wikitext editor now supports common keyboard shortcuts such &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;Ctrl&lt;/code&gt;+&lt;code&gt;B&lt;/code&gt;&lt;/bdi&gt; for bold and &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;Ctrl&lt;/code&gt;+&lt;code&gt;I&lt;/code&gt;&lt;/bdi&gt; for italics. A full [[mw:Help:Extension:WikiEditor#Keyboard shortcuts|list of all six shortcuts]] is available. Thanks to SD0001 for this improvement. [https://phabricator.wikimedia.org/T62928]
* Starting November 28, Flow/Structured Discussions pages will be automatically archived and set to read-only at the following wikis: &lt;bdi&gt;bswiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;elwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;euwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;fawiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;fiwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;frwikiquote&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;frwikisource&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;frwikiversity&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;frwikivoyage&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;idwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;lvwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;plwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;ptwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;urwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;viwikisource&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;zhwikisource&lt;/bdi&gt;. This is done as part of [[mw:Special:MyLanguage/Structured_Discussions/Deprecation|StructuredDiscussions deprecation work]]. If you need any assistance to archive your page in advance, please contact [[m:User:Trizek (WMF)|Trizek (WMF)]]. 
* View all {{formatnum:25}} community-submitted {{PLURAL:25|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, a user creating a new AbuseFilter can now only set the filter to &quot;protected&quot; [[phab:T377765|if it includes a protected variable]].

'''Updates for technical contributors'''
* The [[mw:Special:MyLanguage/Extension:CodeEditor|CodeEditor]], which can be used in JavaScript, CSS, JSON, and Lua pages, [[phab:T377663|now offers]] live autocompletion. Thanks to SD0001 for this improvement. The feature can be temporarily disabled on a page by pressing &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;Ctrl&lt;/code&gt;+&lt;code&gt;,&lt;/code&gt;&lt;/bdi&gt; and un-selecting &quot;&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;Live Autocompletion&lt;/bdi&gt;&quot;.
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] Tool-maintainers who use the Graphite system for tracking metrics, need to migrate to the newer Prometheus system. They can check [https://grafana.wikimedia.org/d/K6DEOo5Ik/grafana-graphite-datasource-utilization?orgId=1 this dashboard] and the list in the Description of the [[phab:T350592|task T350592]] to see if their tools are listed, and they should claim metrics and dashboards connected to their tools. They can then disable or migrate all existing metrics by following the instructions in the task. The Graphite service will become read-only in April. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/KLUV4IOLRYXPQFWD6WKKJUHMWE77BMSZ/]
* [[File:Octicons-tools.svg|12px|link=|class=skin-invert|Advanced item]] The [[mw:Special:MyLanguage/NewPP parser report|New PreProcessor parser performance report]] has been fixed to give an accurate count for the number of Wikibase entities accessed. It had previously been resetting after 400 entities. [https://phabricator.wikimedia.org/T279069]

'''Meetings and events'''
* A [[mw:Special:MyLanguage/Wikimedia_Language_and_Product_Localization/Community meetings#29 November 2024|Language community meeting]] will take place November 29 at [https://zonestamp.toolforge.org/1732896000 16:00 UTC]. There will be presentations on topics like developing language keyboards, the creation of the Mooré Wikipedia, the language support track at [[m:Wiki Indaba|Wiki Indaba]], and a report from the Wayuunaiki community on their experiences with the Incubator and as a new community over the last 3 years. This meeting will be in English and will also have Spanish interpretation.

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/48|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W48&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:42, 25 November 2024 (UTC)
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== Tech News: 2024-49 ==

&lt;section begin=&quot;technews-2024-W49&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/49|Translations]] are available.

'''Updates for editors'''
* Two new parser functions were added this week. The &lt;code dir=&quot;ltr&quot;&gt;&lt;nowiki&gt;{{&lt;/nowiki&gt;[[mw:Special:MyLanguage/Help:Magic words#interwikilink|#interwikilink]]&lt;nowiki&gt;}}&lt;/nowiki&gt;&lt;/code&gt; function adds an [[mw:Special:MyLanguage/Help:Links#Interwiki links|interwiki link]] and the &lt;code dir=&quot;ltr&quot;&gt;&lt;nowiki&gt;{{&lt;/nowiki&gt;[[mw:Special:MyLanguage/Help:Magic words#interlanguagelink|#interlanguagelink]]&lt;nowiki&gt;}}&lt;/nowiki&gt;&lt;/code&gt; function adds an [[mw:Special:MyLanguage/Help:Links#Interlanguage links|interlanguage link]]. These parser functions are useful on wikis where namespaces conflict with interwiki prefixes. For example, links beginning with &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;&lt;code&gt;MOS:&lt;/code&gt;&lt;/bdi&gt; on English Wikipedia [[phab:T363538|conflict with the &lt;code&gt;mos&lt;/code&gt; language code prefix of Mooré Wikipedia]].
* Starting this week, Wikimedia wikis no longer support connections using old RSA-based HTTPS certificates, specifically rsa-2048. This change is to improve security for all users. Some older, unsupported browser or smartphone devices will be unable to connect; Instead, they will display a connectivity error. See the [[wikitech:HTTPS/Browser_Recommendations|HTTPS Browser Recommendations page]] for more-detailed information. All modern operating systems and browsers are always able to reach Wikimedia projects. [https://lists.wikimedia.org/hyperkitty/list/wikitech-l@lists.wikimedia.org/thread/CTYEHVNSXUD3NFAAMG3BLZVTVQWJXJAH/]
* Starting December 16, Flow/Structured Discussions pages will be automatically archived and set to read-only at the following wikis: &lt;bdi&gt;arwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;cawiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;frwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;mediawikiwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;orwiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;wawiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;wawiktionary&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;wikidatawiki&lt;/bdi&gt;{{int:comma-separator/en}}&lt;bdi&gt;zhwiki&lt;/bdi&gt;. This is done as part of [[mw:Special:MyLanguage/Structured_Discussions/Deprecation|StructuredDiscussions deprecation work]]. If you need any assistance to archive your page in advance, please contact [[m:User:Trizek (WMF)|Trizek (WMF)]]. [https://phabricator.wikimedia.org/T380910]
* This month the Chart extension was deployed to production and is now available on Commons and Testwiki. With the security review complete, pilot wiki deployment is expected to start in the first week of December. You can see a working version [[testwiki:Charts|on Testwiki]] and read [[mw:Special:MyLanguage/Extension:Chart/Project/Updates|the November project update]] for more details.
* View all {{formatnum:23}} community-submitted {{PLURAL:23|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, a bug with the &quot;Download as PDF&quot; system was fixed. [https://phabricator.wikimedia.org/T376438]

'''Updates for technical contributors'''
* In late February, temporary accounts will be rolled out on at least 10 large wikis. This deployment will have a significant effect on the community-maintained code. This is about Toolforge tools, bots, gadgets, and user scripts that use IP address data or that are available for logged-out users. The Trust and Safety Product team wants to identify this code, monitor it, and assist in updating it ahead of the deployment to minimize disruption to workflows. The team asks technical editors and volunteer developers to help identify such tools by adding them to [[mw:Trust and Safety Product/Temporary Accounts/For developers/Impacted tools|this list]]. In addition, review the [[mw:Special:MyLanguage/Trust and Safety Product/Temporary Accounts/For developers|updated documentation]] to learn how to adjust the tools. Join the discussions on the [[mw:Talk:Trust and Safety Product/Temporary Accounts|project talk page]] or in the [[discord:channels/221049808784326656/1227616742340034722|dedicated thread]] on the [[w:Wikipedia:Discord|Wikimedia Community Discord server (in English)]] for support and to share feedback. 

'''''[[m:Special:MyLanguage/Tech/News|Tech news]]''' prepared by [[m:Special:MyLanguage/Tech/News/Writers|Tech News writers]] and posted by [[m:Special:MyLanguage/User:MediaWiki message delivery|bot]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News#contribute|Contribute]]&amp;nbsp;• [[m:Special:MyLanguage/Tech/News/2024/49|Translate]]&amp;nbsp;• [[m:Tech|Get help]]&amp;nbsp;• [[m:Talk:Tech/News|Give feedback]]&amp;nbsp;• [[m:Global message delivery/Targets/Tech ambassadors|Subscribe or unsubscribe]].''
&lt;/div&gt;&lt;section end=&quot;technews-2024-W49&quot;/&gt;

&lt;bdi lang=&quot;en&quot; dir=&quot;ltr&quot;&gt;[[User:MediaWiki message delivery|MediaWiki message delivery]]&lt;/bdi&gt; 22:22, 2 December 2024 (UTC)
&lt;!-- Message sent by User:Quiddity (WMF)@metawiki using the list at https://meta.wikimedia.org/w/index.php?title=Global_message_delivery/Targets/Tech_ambassadors&amp;oldid=27873992 --&gt;

== Tech News: 2024-50 ==

&lt;section begin=&quot;technews-2024-W50&quot;/&gt;&lt;div class=&quot;plainlinks&quot;&gt;
Latest '''[[m:Special:MyLanguage/Tech/News|tech news]]''' from the Wikimedia technical community. Please tell other users about these changes. Not all changes will affect you. [[m:Special:MyLanguage/Tech/News/2024/50|Translations]] are available.

'''Weekly highlight'''
* Technical documentation contributors can find updated resources, and new ways to connect with each other and the Wikimedia Technical Documentation Team, at the [[mw:Special:MyLanguage/Documentation|Documentation hub]] on MediaWiki.org. This page links to: resources for writing and improving documentation, a new &lt;bdi lang=&quot;zxx&quot; dir=&quot;ltr&quot;&gt;#wikimedia-techdocs&lt;/bdi&gt; IRC channel on libera.chat, a listing of past and upcoming documentation events, and ways to request a documentation consultation or review. If you have any feedback or ideas for improvements to the documentation ecosystem, please [[mw:Wikimedia Technical Documentation Team#Contact us|contact the Technical Documentation Team]].

'''Updates for editors'''
[[File:Edit Check on Desktop.png|thumb|Layout change for the Edit Check feature]]
* Later this week, [[mw:Special:MyLanguage/Edit check|Edit Check]] will be relocated to a sidebar on desktop. Edit check is the feature for new editors to help them follow policies and guidelines. This layout change creates space to present people with [[mw:Edit check#1 November 2024|new Checks]] that appear ''while'' they are typing. The [[mw:Special:MyLanguage/Edit check#Reference Check A/B Test|initial results]] show newcomers encountering Edit Check are 2.2 times more likely to publish a new content edit that includes a reference and is not reverted.
* The Chart extension, which enables editors to create data visualizations, was successfully made available on MediaWiki.org and three pilot wikis (Italian, Swedish, and Hebrew Wikipedias). You can see a working examples [[testwiki:Charts|on Testwiki]] and read [[mw:Special:MyLanguage/Extension:Chart/Project/Updates|the November project update]] for more details.
* Translators in wikis where the [[mw:Special:MyLanguage/Content translation/Section translation#Try the tool|mobile experience of Content Translation is available]], can now discover articles in Wikiproject campaigns of their interest from the &quot;[https://test.wikipedia.org/w/index.php?title=Special:ContentTranslation&amp;campaign=specialcx&amp;filter-type=automatic&amp;filter-id=collections&amp;active-list=suggestions&amp;from=es&amp;to=en All collection]&quot; category in the articles suggestion feature. Wikiproject Campaign organizers can use this feature, to help translators to discover articles of interest, by adding the &lt;code dir=ltr&gt;&lt;nowiki&gt;&lt;page-collection&gt; &lt;/page-collection&gt;&lt;/nowiki&gt;&lt;/code&gt; tag to their campaign article list page on Meta-wiki. This will make those articles discoverable in the Content Translation tool. For more detailed information on how to use the tool and tag, please refer to [[mw:Special:MyLanguage/Translation suggestions: Topic-based &amp; Community-defined lists/How to use the features|the step-by-step guide]]. [https://phabricator.wikimedia.org/T378958]
* The [[mw:Special:MyLanguage/Extension:Nuke|Nuke]] feature, which enables administrators to mass delete pages, now has a [[phab:T376379#10310998|multiselect filter for namespace selection]]. This enables users to select multiple specific namespaces, instead of only one or all, when fetching pages for deletion.
* The Nuke feature also now [[phab:T364225#10371365|provides links]] to the userpage of the user whose pages were deleted, and to the pages which were not selected for deletion, after page deletions are queued. This enables easier follow-up admin-actions. Thanks to Chlod and the Moderator Tools team for both of these improvements. [https://phabricator.wikimedia.org/T364225#10371365]
* The Editing Team is working on making it easier to populate citations from archive.org using the [[mw:Special:MyLanguage/Citoid/Enabling Citoid on your wiki|Citoid]] tool, the auto-filled citation generator. They are asking communities to add two parameters preemptively, &lt;code dir=ltr&gt;archiveUrl&lt;/code&gt; and &lt;code dir=ltr&gt;archiveDate&lt;/code&gt;, within the TemplateData for each citation template using Citoid. You can see an [https://en.wikipedia.org/w/index.php?title=Template%3ACite_web%2Fdoc&amp;diff=1261320172&amp;oldid=1260788022 example of a change in a template], and a [https://global-search.toolforge.org/?namespaces=10&amp;q=%5C%22citoid%5C%22%3A%20%5C%7B&amp;regex=1&amp;title= list of all relevant templates]. [https://phabricator.wikimedia.org/T374831]
* One new wiki has been created: a {{int:project-localized-name-group-wikivoyage}} in [[d:Q9240|Indonesian]] ([[voy:id:|&lt;code&gt;voy:id:&lt;/code&gt;]]) [https://phabricator.wikimedia.org/T380726]
* Last week, all wikis had problems serving pages to logged-in users and some logged-out users for 30–45 minutes. This was caused by a database problem, and investigation is ongoing. [https://www.wikimediastatus.net/incidents/3g2ckc7bp6l9]
* [[File:Octicons-sync.svg|12px|link=|class=skin-invert|Recurrent item]] View all {{formatnum:19}} community-submitted {{PLURAL:19|task|tasks}} that were [[m:Special:MyLanguage/Tech/News/Recently resolved community tasks|resolved last week]]. For example, a bug in the [[mw:Special:MyLanguage/Help:Growth/Tools/Add a link|Add Link]] feature has been fixed. Previously, the list of sections which are excluded from Add Link was partially ignored in certain cases. [https://phabricator.wikimedia.org/T380455][https://phabricator.wikimedia.org/T380329]

'''Updates for technical contributors'''
* [[mw:Special:MyLanguage/Codex|Codex]], the design system for Wikimedia, now has an early-stage [[git:design/codex-php|implementation in PHP]]. It is available for general use in MediaWiki extensions and Toolforge apps through [https://packagist.org/packages/wikimedia/codex Composer], with use in MediaWiki core coming soon. More information is available in [[wmdoc:design-codex-php/main/index.html|the documentation]]. Thanks to Doğu for the inspiration and many contributions to the library. [https://phabricator.wikimedia.org/T379662]
* [https://en.wikipedia.org/api/rest_v1/ Wikimedia REST API] users, such as bot operators and tool maintainers, may be affected by ongoing upgrades. On December 4, the MediaWiki Interfaces team began rerouting page/revision metadata and rendered HTML content endpoints on [[testwiki:|testwiki]] from RESTbase to comparable MediaWiki REST API endpoints. The team encourages active users of these endpoints to verify their tool's behavior on testwiki and raise any concerns on the related [[phab:T374683|Phabricator ticket]] before the end of the year, as they intend to roll out the same change across all Wikimedia projects in early January. These changes are part of the work to replace the outdated [[mw:RESTBase/deprecation|RESTBase]] system.
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      <sha1>39hqwvn3ff7602mm0a8onoxzlch1y3c</sha1>
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    <title>Social Victorians/Timeline/1896</title>
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      <text bytes="166553" sha1="2k8igw1qzkifqx81njde7a955wrdj0w" xml:space="preserve">[[Social Victorians/Timeline/1850s | 1850s]] [[Social Victorians/Timeline/1860s | 1860s]] [[Social Victorians/Timeline/1870s | 1870s]] [[Social Victorians/Timeline/1880s | 1880s Headlines]] [[Social Victorians/Timeline/1890s | 1890s Headlines]] [[Social Victorians/Timeline/1890 | 1890]] [[Social Victorians/Timeline/1891 | 1891]] [[Social Victorians/Timeline/1892 | 1892]] [[Social Victorians/Timeline/1893 | 1893]] [[Social Victorians/Timeline/1894 | 1894]] [[Social Victorians/Timeline/1895 | 1895]] 1896 [[Social Victorians/Timeline/1897 | 1897]] [[Social Victorians/Timeline/1898 | 1898]] [[Social Victorians/Timeline/1899 | 1899]]  [[Social Victorians/Timeline/1900s|1900s]] [[Social Victorians/Timeline/1910s|1910s]] [[Social Victorians/Timeline/1920s-30s|1920s-30s]]

==Sometime in 1896==
Sometime in the first quarter of 1896 [[Social Victorians/People/William Butler Yeats|William Butler Yeats]] moved to No. 18 Woburn Buildings, London, possibly January, but for sure by March (Harper 80 76, n. 12, 3-4)
==January 1896==
===1 January 1896, Wednesday, New Year's Day===
===13 January 1896, Monday===
On the 24th ''The Literary World'' reports the following: &quot;The Memorial Institute to Mrs. Elizabeth Barrett Browning was opened at Ledbury last week by Mr. Rider Haggard. The institute is a charming building in the half-timbered perpendicular style of architecture, and it occupies one of the most commanding positions in the town. It is bult of Lebury limestone and Etonfield sandstone, with oak timbering, with a clock-tower at the corner. The total cost was £2,330, and it is satisfactory to know that the whole of that sum was obtained from more than 1,000 subscribers, with the exception of about £300. Already several important gifts have been made to the library, including a complete set of the works of Robert and Elizabeth Barrett Browning, presented by Mr. George Malton Barrett, the brother of the poetess; and about one hundred volumes of books, presented by Dr. Furnival, the late president of the Browning Society. Mr. Haggard, who spoke for a considerable time to an appreciative concourse, sketched the life of the poetess in enthusiastic terms, and paid a generous tribute to the memory of 'the greatest poetess the English-speaking people have yet produced.'&quot; &quot;Table Talk,&quot; The Literary World, 24 January 1896, vol. 53, p. 77, col. 1. (Accessed 9 October 2009 in Google Books.)

===18 January 1896, Saturday===
On the 24th ''The Literary World'' reports the following: &quot;A pleasant gathering took place in Edinburgh on Saturday last to do honour to Mr. Andrew Stewart, who for a quarter of a century has been connected with The People's Friend. Mr. Stewart began his career on The Friend as sub-editor, under Mr. David Pae, and among his contributors had such men as the late George Gilfillan and Professor Blackie. Such novelists as Annie S. Swan and Adeline Sergeant have written much of their best work for The Friend, and through its pages their stories were read weekly in a quarter of a million homes. Mr. W. C. Leng, one of the proprietors, took the chair, in the absence of Sir John Leng, who is abroad, and speeches were delivered by Mr. Anderson, Mr. Robert Ford, Mr. J. C. Hadden, Mrs. Lawson, Miss A. S. Falconer, and others.&quot; &quot;Table Talk,&quot; The Literary World, 24 January 1896, vol. 53, p. 77, col. 2. (Accessed 9 October 2009 in Google Books.)

===22 January 1896, Wednesday===
On the 24th ''The Literary World'' reports the following (but really it occurred on the 15th?): &quot;A meeting of the Society of Public Librarians was held at the Whitechapel Public Library on Wednesday evenng last. (Mr. Frowde in the chair), when two excellent papers were delivered -- 'Subject Index to English Literature,' by Mr. Bagguley, and 'Lady Assistants in Public Libraries,&quot; by Mr. Snowsill. The whole of the members present were practically, if ungallantly, strongly opposed to the introduction of females as attendants in public libraries.&quot; &quot;Table Talk,&quot; The Literary World, 24 January 1896, vol. 53, p. 78, col. 1. (Accessed 9 October 2009 in Google Books.)

===24 January 1896, Friday===
The 31 January 1896 ''Literary World'' reports the following: &quot;The recently-formed Publishers' Associaton had a meeting of its members on Friday last, most of the leading publishers attending. No defnite appointments were made, but it is pretty generally understood that the office of President lies between Mr. Charles Longman and Mr. John Murray.&quot; &quot;Table Talk,&quot; The Literary World, 31 January 1896, vol. 53, p. 103, col. 2. (Accessed 9 October 2009 in Google Books.)

===27 January 1896, Monday===
Authors' Society meeting, talk by Mr. Hall Caine on international copyright. &quot;Table Talk,&quot; The Literary World, 31 January 1896, vol. 53, p. 101, col. 3. (Accessed 9 October 2009 in Google Books.)

==February 1896==
Lecture at the Westminster Town Hall reported by ''The Literary World'' on 14 February 1896: &quot;'The transmission of personality is the creed of literature as it is of religion,' said Mr. Birrell in the course of a lecture on Dr. Johnson, at Westminster Town Hall, and the ober dictum is worthy of all acceptation. Mr. Asquith presided, and the audience including 'all the talents,' Lord Roseberry, Mr. Arthur Balfour, Mr. Thomas Hardy, Mr. Henry James, and Mr. Herbert Paul occupying chairs in the front row. / Mr. Asquith uttered the usual orthodoxies concerning the author 'who lived so little by his writings and so much by his personality.' That is a view which we confess we do not share. ...&quot; &quot;Table Talk,&quot; The Literary World, 14 February 1896, vol. 53, p. 149, col. 1. (Accessed 9 October 2009 in Google Books.)

The annual meeting of the Authors' Society, reported on in the 21 February 1896 ''Literary World'': &quot;The annual meeting of the Authors' Society passed off pleasantly, in spite of the minatory motion that stood in the name of Mr. W. H. Wilkins regarding the unfortunate 'Address' to the authors of America, a motion that was gracefully withdrawn in view of the committee's resoluton that the 'Address' had no official character. We congratulate the Society on the access of 14 new members during the year and on the evidence of practical work afforded by the fact that two-thirds of the members had applied for advice and assistance, to say nothing of the MSS. submitted for the same purpose. The printed report, of which a copy has reached us, is full of exceedingly sound advice, of especial value to young or inexperienced authors.&quot; &quot;Table Talk,&quot; The Literary World, 14 February 1896, vol. 53, p. 172, col. 3. (Accessed 9 October 2009 in Google Books.)

===3 February 1896, Monday===
Sometime this week, probably, was a meeting of the Society of Public Librarians, reported on in the 14 February 1896 ''Literary World'': &quot;A meeting of the Society of Public Librarians was held at the Canning Town branch of the West Ham Public Libraries last week, when Mr. Foskett, of the Camberwell Public Libraries, delivered 'A Contribution to Occult Literature,' and Mr. Whitwell, of West Ham, read a paper entitled, 'Some Critical Remarks on the Works of Thomas Love Peacock.' Both papers were very well received, and gave rise to interesting discussions.&quot; &quot;Table Talk,&quot; The Literary World, 14 February 1896, vol. 53, p. 150, col. 1. (Accessed 9 October 2009 in Google Books.)
===5 February 1896, Wednesday===
Dinner: [[Social Victorians/People/George Bernard Shaw|G. B. Shaw]], Richard Burton Haldane, [[Social Victorians/People/Asquith|H. H. Asquith]], [[Social Victorians/People/Balfour|Arthur Balfour]] (Gibbs 124).
===22 February 1896, Saturday===
According to the 28 February 1896 ''Literary World'', &quot;On Saturday last, at Hampstead, the ceremony was witnessed of unveiling the memorial tablet in the house in John-street in which John Keats resided. It was expected that Sir Walter Besant would take part n the ceremony; gout, unhappily, prevented his doing so, but he sent a letter in his place, which was read in due course. Sir Charles Dilke, Mr. S. Colvin, Mr. Edmund Gosse, Dr. Robertson Nicol, and Prof Hall Griffin were among those present. The proceedings were simple in the extreme. Prof. Griffin, in a brief speech, dwelt on the historical nature of the surroundings from a literary point of view and the ceremony terminated, leaving Lawn-Bank, John-street, with the addition of a tablet bearing the following inscription: Erected by the Society of Arts. / JOHN KEATS, / Poet, / Lived in this House. / B. 1795. / D. 1821.&quot; &quot;Table Talk,&quot; The Literary World, 28 February 1896, vol. 53, p. 196, col. 2. (Accessed 9 October 2009 in Google Books.)

==March 1896==
Sometime in March 1896, the Inner Order of the Golden Dawn moved its headquarters to 62 Oakley Square, where it stayed until September 1897 (Howe 126).

Sometime in the first quarter of 1896 W. B. Yeats moved to No. 18 Woburn Buildings, London, possibly January, but for sure by March (Harper 80 76, n. 12, 3-4)
===5 March 1896, Thursday===
&quot;The wedding of Miss Lily Caine, sister of the novelist, with Mr. George Day will take place on March 5 at St. George's, Hanover-square.&quot; &quot;Table Talk,&quot; The Literary World, 28 February 1896, vol. 53, p. 196, col. 2. (Accessed 9 October 2009 in Google Books.)
===7 March 1896, Saturday===
Gilbert and Sullivan's ''The Grand Duke, Or the Statutory Duel'' opens at the Savoy.

=== 11 March 1896, Wednesday ===
Queen's Drawing Room hosted by Alexandra, Princess of Wales, as reported in the London Evening ''Standard'' on Thursday, 12 March 1896. The long list of names is rendered as an ordered or numbered list here to save space and make referring to people easier; the original newspaper story puts each one on a new line as a new paragraph.&lt;blockquote&gt;THE DRAWING ROOM.

The Princess of Wales held the first Drawing Room of the season at Buckingham Palace yesterday afternoon, on behalf of the Queen. Carriages conveying ''débutantes'' commenced to arrive shortly after noon, and by one o'clock the line of vehicles reached right away to Marlborough-yard. The weather was mild though somewhat gloomy, and a large crowd collected in the Mall. Tho number of presentations was about the same as usual; but, from an outsider's point of view, there was an unusual absence of colour. The Princess of Wales was accompanied by the Princesses Victoria and Maud and Prince Charles of Denmark, and the Duke and Duchess of Saxe-Coburg and Gotha and Princess Alexandra, the Duke and Duchess of Connaught, and the Duke and Duchess of York, were present. Escorted by a troop of Life Guards, the State carriage, conveying the Princess of Wales, her two daughters, and Prince Charles of Denmark, arrived at Buckingham Palace from Marlborough House almost precisely at three o'clock. The National Anthem was played as their Royal Highnesses passed into the Palace, and there was general uncovering and cheering among the crowd in front of the Palace gates. The Princess was received by the Officers of State, and conducted to the Throne Room, when the presentations commenced.

The Drawing Room was to a large extent a mourning function as regards dress. All the Royal personages were in black, even the two brides-elect, Princess Maud of Wales and Princess Alexandra of Coburg. The Princesses Victoria and Maud of Wales were dressed alike in black satin, prettily arranged with hart's-tongue fern leaves of lisse outlined in jet on the bodices and skirt foot, and rich black satin ribbon at the waist. The material chosen by the Duchess of Coburg was rich black moiré, and the Princess Alexandra's black satin gown was veiled in gauze brocaded in a small floral design. The Duchess of York was dressed in black silk of English manufacture. The Duchess of Buccleuch, like all the other ladies belonging to the Royal Households, wore black plumes and veil. Her gown was of richest poult de soie, trimmed on the corsage with folds of crape and jet ornaments. The Duchess of Buccleuch presented her niece, Lady Victoria Kerr, daughter of the Marquess of Lothian and goddaughter of the Queen, who wore a charming white satin gown, the bodice veiled in lisse held with bands of silver embroidery. The neck was softened by a drapery of lisse, on which was laid, with very natural effect, a spray of apple-blossom. From the silver waistband fell a scarf of silvered lisse to the bottom of the skirt, fastened there by a bunch of apple-blossoms. The train of striped white brocade was bordered with lisse, knotted at intervals with clusters of apple-blossom. Lady Helen Kerr was also in white satin, with an exceedingly pretty corsage arranged with mousseline de soie, and graceful trails of mauve and white convolvuli. There were folds of mousseline de soie carried down the front of the skirt, widening towards the foot, and enframed by the convolvuli. The pale mauve brocade train had a lace-like pattern in cream silk, and was bordered with the flowers and mousseline de soie. Lady Tweedmouth's black velvet toilet was ornamented with fine jet on the corsage, and had full tulle sleeves. The train was fastened to the shoulder by a large knot and lined with a new material, moiré mouillée. Lady Howard Vincent chose a chène silk gown with design of roses and violets, trimmed on the bodice with a fringe of violets, and shoulder-straps of roses. There was a softening of pink lisse about the neck, and the heliotrope and white train came from under the arms, and was fastened with a coquille bow at the back. Susan, Countess of Malmesbury — presented on her marriage — wore a gown of pearl grey satin, draped with exquisite old needlepoint lace, forming a fichu on the bodice. The train was of black brocade. Lady Eva Cotterell — also presented on her marriage — wore white satin, embroidered in silver, and trimmed on the train with lovely lace and knots of silver ribbon. Lady Emma Crichton was in black satin, embroidered in sapphires and silver swallows, and draped with creamy lace. The black velvet train was lined with white satin. Lady Codrington's heliotrope satin gown was made with pointed Court bodice and stomacher of fine embroidery wrought in gilt thread, and pale rubies and diamonds. The shoulder-pieces, of wine-toned velvet, were ornamented to match, and a large poppy, with diamond heart, was fastened at the side. The train was of velvet. Miss James, niece of Lord James of Hereford, wore a black satin gown, richly worked on overskirt and bodice with jet and brilliants in design of knots and floral sprays. The black velvet train was lined with white satin.

Lady Algernon Gordon Lennox was in black and satin, the train being trimmed with tulle ruches, wide at the hem and narrowing towards the waist. The bodice was softened by folds of tulle caught with diamonds, and a long chain of pearls passed over one shoulder and encircled the figure. Lady Feo Start chose a gown of pinkish mauve satin, embroidered half way down the front seams with bunches of wheat, the leaves and stems being wrought in fine silver and the wheat ears in diamonds. At the foot a larger cluster appeared gracefully tapering to the side. The corsage was embroidered to match, and filled in round the neck by silver tulle, and the embroidered sleeves fell off the arm, held by shoulder-straps of silver. The train was in the same shade of velvet, with bold groups of the wheat at each corner, and diminishing to a point midway up the sides. This train was lined with white satin, and fastened to each shoulder by splendid diamond ornaments. The Countess of Lathom's black satin dress was ornamented with fine jet embroidery, and worn with a black brocade train draped with Chantilly lace. Lady Bertha Wilbraham accompanied her mother, wearing a satin dress in a delicate shade of French grey, very prettily trimmed with chiffon, old lace, and clusters of lilies of the valley. The Countess of Clanwilliam had a rich black broché train with a black satin gown richly pailletted. Lady Elizabeth Meade's white satin gown had a very smart bodice arranged with kilted chiffon forming scollop-shaped frills on the shoulders, and groups of Eucharist lilies were fastened at the bust and waist. Lady Beatrice Meade was in white moiré deftly arranged with lisse, embroidered lace, and white narcissi, which also trimmed the white satin train. Viscountess Cross was attired in a black moiré bengaline satin, bordered with Brussels Point caught with fine jet ornaments, and a black satin gown. The Hon. Mary Cross wore black satin, relieved by a vest and sleeves of silver embroidered white satin and a pearl grey satin train. The ivory satin gown selected by the Hon. Margaret Cross was embroidered in a charming design executed in silver, gold, and steel, and had billowy chiffon sleeves, and a train of striped white satin. Lady Arthur Hill was in black satin, draped with costly old lace, and wore pearl and diamond ornaments. Lady Arthur presented Miss Nina Hill in a sweetly pretty white satin toilet, veiled in Brussels net. The corsage was finished by an ostrich feather ruche in front, and frills of net round the shoulders and back. The train fell from both shoulders, like white wings, showing the figure between. Lady Aline Wentworth Beaumont wore white satin, the corsage softened with chiffon, and the waist encircled by a deep silver band. The handsome train was of gold and cream brocade, with a design of shaded tulips, and was turned back at the corner with bunches of tulips. The Countess of Lytton's black peau de chine dress was trimmed up the side with bows of satin ribbon, and worn with a brocade train. The bodice was arranged with jetted lace. Lady O'Conor wore a black velvet gown, the bodice draped across rich jet embroidery and finished by jet butterflies and roses on the shoulders. The train was of black satin.

The Lady Mayoress of London was beautifully dressed in ivory satin, embroidered in frosted silver, forming a festooned floral design round the skirt foot. There were touches of turquoise blue velvet on the corsage, matching the train, which was lined with primrose satin, and ornamented with bunches of large white ostrich plumes. Lady Wilkin presented her daughter in a charming ''débutante's'' gown of white satin under net. From each side of the waist fell clusters of lilies of the valley and mimosa, stray blossoms of the flowers being scattered in a shower to the skirt foot. The train was trimmed with silver cord and bunches of flowers. Mrs. H. M. Stanley was becomingly attired in grey satin, embroidered in steel paillettes, forming irregular lines about the hips. The bodice was trimmed with grey chiffon and steel embroidered guipure, and the train was of grey and gold brocade. Lady Mary Lygon, in attendance upon the Duchess of York, had a black velvet train, and a black satin gown trimmed with chiffon and jet. Viscountess Chelsea's white satin dress was very beautifully embroidered in diamonds. Lady Playfair was in black satin. The Dowager Lady Westbury wore a black and white brocade [Col. 2/3] gown, trimmed with rare old Spanish lace, and a black velvet train. Viscountess Trafalgar's becoming toilet was carried out in delicate tones of green and pink. Viscountess Dalrymple wore a superb white brocade gown. Lady Rivers Wilson was presented, on her marriage, wearing an oyster-toned satin gown, made in Louis XV. period, with long corsage, trimmed with rare Point de Gaze, caught up with bouquets of white poppies enveloped in tulle. The train of silver tissue formed a Venetian mantle falling under a hood of the lace, and was lined with mauve satin, matching a large straggling branch of orchids which were laid on at the side. Lady Mount-Stephen wore a gown of rich black brocade, with a large design of roses and little trailing blossoms. The bodice was filled in, back and front, with cream satin under filmy lace, and was embroidered in jet. The sleeves were of white chiffon and lace, and the train of rich black velvet. Viscountess Knutsford's black brocade gown was enriched with fine jet embroidery, and her black satin train was trimmed with lace and jet.

By command of the Queen, a Drawing Room was held yesterday afternoon, at Buckingham Palace, by her Royal Highness the Princess of Wales, on behalf of her Majesty.

Presentations to her Royal Highness at this Court are, by the Queen's pleasure, considered as equivalent to presentations to her Majesty.

Their Royal Highnesses the Princess of Wales, Princess Victoria, and Princess Maud of Wales, accompanied by his Royal Highness Prince Charles of Denmark, attended by Lady Suffield (Lady in Waiting), Miss Knollys (Bedchamber Woman in Waiting), Lord Colville of Culross, K.T. (Chamberlain to the Princess of Wales), General Sir D. M. Probyn (Comptroller and Treasurer to the Prince of Wales), Sir Francis Knollys (Private Secretary to the Prince of Wales), and Major General Stanley Clarke (Private Secretary to the Princess of Wales), escorted by a detachment of the Ist Life Guards, arrived at the garden entrance of the Palace from Marlborough House.

Their Royal Highnesses the Duke and Duchess of Saxe-Coburg and Gotha, and Princess Alexandra of Saxe-Coburg and Gotha, arrived from Clarence House, attended by Miss Colville and Captain the Hon. D. Monson.

Their Royal Highnesses the Duke and Duchess of Connaught and Strathearne, attended by Lady Elphinstone and Colonel Alfred Egerton, were present at the Drawing Room.

Their Royal Highnesses the Duke and Duchess of York arrived from York House, attended by Lady Mary Lygon, Major General Sir F. De Winton, and Sir Charles Cust.

His Royal Highness Prince Christian of Schleswig-Holstein and his Highness Prince Christian Victor of Schleswig-Holstein arrived from Cumberland Lodge, attended by the Hon. C. Eliot. His Highness the Duke of Teck was present at the Drawing Room. 

Her Majesty's Body Guard of the Honourable Corps of Gentlemen-at-Arms was on duty in the State Saloons, under the command of Lord Belper (the Captain). The Royal Body Guard of the Yeomen of the Guard were on duty in the interior of the Palace, under the command of Lieut. Colonel H. P. Vance, the Lieutenant (in the unavoidable absence of the Captain, the Eari of Limerick). A Guard of Honour of the 1st Battalion of Grenadier Guards, with the Band of the Regiment, was mounted in the Quadrangle of the Palace, and a Guard of Honour of the Ist Life Guards, with their Band, was stationed in the Courtyard of the Palace; and the Park party was furnished by the Royal Horse Guards. 

The Princess of Wales, accompanied by the other members ol the Royal family, entered the Throne Room at three o'clock, and the Princess of Wales took her station in front of the Throne. 

Her Royai Highness the Princess of Wales wore a gown of black silk embroidered in jet, corsage and train to correspond. Headdress — Tiara of diamonds, black feathers, and veil. Ornaments — Pearls and diamonds. Orders — Victoria and Albert, Crown of India, St. Catherine of Russia, St. John of Jerusalem, the Saxe-Coburg and Gotha, and the Danish Family and Golden Wedding Orders. 

Their Royal Highnesses the Princesses Victoria and Maud of Wales wore gowns of black satin, corsages embroidered with jet applique in the shape of leaves, sleeves of vandyke chiffon with straps of fine jet, the same kind of jet forming the waistbelt; trains of black satin to correspond. Ornaments — Pearls and diamonds. Orders — Victoria and Albert, Crown of India, Danish Golden Wedding, Saxe-Coburg and Gotha, and Jubilee Commemoration Medal. 

Her Royal Highness the Duchess of York wore a dress of rich black English watered silk, embroidered and trimmed with jet and feathers; corsage and train to correspond. Headdress — Tiara, feathers, and veil. Ornaments — Pearls and diamonds. Orders — Victoria aud Albert, Crown of India, St. John of Jerusalem, Saxe-Coburg and Gotha, and Jubilee Commemoration Medal. 

The Foreign Ambassadors and Ministers having been introduced in the order of precedence, the following presentations were made in the Diplomatic Circle: — 

By Countess Deym, Princess Alex Thurn Taxis (''née'' Princess Hohenlohe), Countess Clary Aidringen (''née'' Countess Kinsky), Madame Geoffray, and Mdlle. Demidoff. 

By Mrs. Bayard, Mrs. William Sheffield Cowles. 

By Madame de Bille, Madame de Salis. 

By the Marchioness of Salisbury, Madame Kato and Countess Lewenhaupt. 

The following presentations to the Princess of Wales, on behalf of the Queen, were made, the names having been previously left at the Lord Chamberlain's office, and submitted for her Majesty's approval: — [The long list of names is rendered as an ordered or numbered list here to save space and make referring to people easier; the original newspaper story puts each one on a new line as a new paragraph.] 

# Arnold, Lady, by Lady Suffield. 
# Adair, Mrs. Charles H., by Lady Salmon. 
# Anstruther, Miss Rosamond, by the Hon. Mrs. Anstruther. 
# Ardagh, Lady (Dowager Countess of Malmesbury), by Viscountess Knutsford. 
# Bedford, Lady, by Mrs. Goschen. 
# Bainbridge, Miss Gwendolen, by her mother, Mrs. Hugh Bainbridge. 
# Bell, of Scatwell, Lady, by the Hon. Mrs. Rennel [?] 
# Bannerman, Miss, by the Countess of Ellesmere. 
# Birney, Miss Kerrow, by Lady Hart. 
# Bellew, the Hon. Mrs. Richard, on her marriage, by the Lady Bellew. 
# Bums, Mrs. James C., by the Lady Gertrude Cochrane. 
# Butler, Miss Blanche, by her mother, Hon. Mrs. Robert Butler. 
# Baylis, Mrs. Philip, by Mrs. Wharton Hood. 
# Brown, Miss Hargreaves, by her mother, Mrs. A. Hargreaves Brown. 
# Boodle, Miss Marion Florence, by her mother, Mrs. H. Trelawny Boodle. 
# Baker, Miss Katharine, by her mother, Mrs. George Barrington Baker. 
# Buxton, Mrs. Edward, on her marriage, by her mother, Mrs. Gurney. 
# Brabazon, Lady Mary, by the Countess of Lathom. 
# Buxton, Miss Hilda, by her mother, Hon. Mrs. Francis Buxton. 
# Beilew, The Lady, by the Lady Alexandrina Beaumont. 
# Boulton, Mrs. Oscar, by Mrs. S. B. Boulton. 
# Barclay, Mrs. George, by the Hon. Mrs. Francis Buxton. 
# Bostock, Mrs. Ashton, by Lady Russell Reynolds. 
# Bairstow, Mrs. Walter, by Mrs. Ingilby. 
# Bucknall, Mrs. Sydney, by her mother, Lady Sidgreaves. 
# Bevan, Miss Mary Pauline, by her mother, Mrs. Thomas Bevan. 
# Bruce, The Hon. Mary, by Lady Balfour of Burleigh. 
# Brassey, Lady Violet, by Lady Evelyn Cotterell. 
# Bankes, Mrs. Ralph Vincent, on her marriage, by Mrs. Mount. 
# Beach, Miss Susan Hicks, by her mother, Lady Lucy Hicks Beach. 
# Cotterell. Lady Evelyn, by Hon. Lady Cotterell. 
# Curtis, Miss (of the United States), by Mrs. Bayard. 
# Curtis, Miss Clara (of the United States), by Mrs. Bayard. 
# Campbell, Mrs. Alexander, by the Hon. Mrs. Townley Mitford. 
# Cooper. Mrs. J. R., by the Hon. Lady Ridley. 
# Clay, Miss Sybil, by her mother, Mrs. Walter Holbech. 
# Craven, Miss, by Lady King. 
# Cockerell, Miss Patience, by her mother, Mrs. William Cockerell. 
# Cunningham, the Hon. Lady, by Lady George Hamilton. 
# Cooper, Mrs. Harry, by Lady Comrnerell. 
# Craig, Miss Gibson, by Lady Gibson Craig. 
# Craig, Miss Alice Gibson, by Lady Gibson Craig. 
# Crossley, Miss, by the Hon. Mrs. Montagu Forbes. 
# Cole, Lady Florence, by Countess of Enniskillen. 
# Chaplin, Miss Bertha, by her mother, Mrs. Cecil Chaplin. 
# Colomb, Miss Gwenda, by her mother, Lady Colomb. 
# Clifford, Miss Alice, by Lady Pollock. 
# Callaghan, Mrs. George, by the Hon. Lady Fremantle. 
# Callaghan, Miss Dorothy, by Mrs. George Callaghan. 
# Coddington, Lady, by Viscount Cranborne. 
# Clarkson, Miss, by Mrs. Laurenco Edye. 
# Crossman, Mrs. Douglas, by Lady Grant Duff. 
# Da Costa, Mrs. Oscar, on her marriage, by Mrs. Bertram Ward. 
# De la Rue, Miss Sybil, by Mrs. T. Andros de la Rue. 
# Dale, Mrs., by Lady Dale. 
# Dale, Lady, by the Marchioness of Ripon. 
# Dawnay, Miss Helen, by her mother, Lady Adelaide Dawnay. 
# Digby, Miss Lettice, by her mother, the Hon Mrs. Kenelm Digby. 
# Dalgety, Miss Gladys, by her sister, Viscountess Trafalgar. 
# Dunphie, Mrs. Alfred, on her marriage, by Mrs. Anderson Critchett. 
# Douglass, the Hon. Mrs., on her marriage, by the Hon. Mrs. Paton. 
# Dalison, Miss Joan, by her mother, Mrs. Maximilian Dalison. 
# Dunlop, Mrs. William H., by Mrs. Frank Addison Brace 
# Evans, Miss Gwladys, by Lady Evans. 
# Edge, Miss Kathleen, by Lady Barnes. 
# Egerton, Lady Katharine, by her mother, the Countess of Ellesmere. 
# Earle, Miss Caroline, by Lady Earle. 
# Earle, Miss Evelyn, by Lady Earle. 
# Eustace, Miss Adelaide, by her mother, Lady Katharine Eustace. 
# Eustace, Miss Violet, by her mother, Lady Katharine Eustace. 
# Frere, Mrs. Arthur, on her marriage, by the Countess of Lathom. 
# Fremantle, Honble. Lady, by Mrs. Goschen. 
# Foley, Lady Mary, on her marriage, by Lady Feodorowna Sturt. 
# Fletcher, Mrs. H. Morley, by the Hon. Mrs. Walter R. D. Forbes. 
# Fenwick, Miss Elfreda Gabriel, by her mother, Mrs. Fenwick Fenwick. 
# Forwood, Lady, by the Marchioness of Salisbury. 
# Forwood, Miss Ida, by her mother, Lady Forwood. 
# Fielden, Miss Lorna, by her mother, Mrs. Thomas Fielden. 
# Fowler, Miss, by Mrs. Forrest. 
# Finlay, Lady, by the Marchioness of Salisbury. 
# Fortescue, Hon. Mrs. Lionel, on her marriage, by Lady Lucy Hicks Beach. 
# Floyd, Mrs. Henry, by the Countess of Clanwilliam. 
# Fordham, Mrs. R. Oswald (Lady O'Malley), on her marriage, by Lady Flower. 
# Fowler, Miss Anna, by Mrs. Christie-Miller. 
# Fielden, Miss Gertrude, by her sister-in-law, Mrs. Thos. Fielden.  
# Firebrace, Mrs. Frederick, on her marriage, by the Lady Reay. 
# Finch, Miss Essex, by Mrs. Finch. 
# Gillford, the Lady, by the Countess of Clanwilliam. 
# Greenly, Miss Lucy, by Lady Florence King King. 
# Gambier, Miss Gore, by Mrs. Murdoch. 
# Horsfall. Miss Eva, by Lady Charles Scott. 
# Heygate, Lady, by the Countess Waldegrave. 
# Hill, Miss, by her mother, Lady Arthur Hill. 
# Hutchinson, Mrs. Edward, by Lady Dale. 
# Howard, Miss Gertrude, by her mother, Mrs. John Howard. 
# Hogg, Miss Ethel, by her aunt, Mrs. Horner. 
# Hall, Mrs. Thomas, by Mrs. Chamberlain. 
# Hely-Hutchinson, Lady Evelyn, by Countess of Donoughmore. 
# Hutton, Mrs. Stamford, on her marriage, by her mother, Mrs. Fenwick Fenwick. 
# Hanbury, Mrs. Everard, by her mother, Mrs. Murdoch. 
# Herbert, Miss Gwladys, by Mrs. Edmund M'Clure. 
# Hoskyns, Mrs. [P?]aget, by the Dowager Lady Westbury. 
# Hawke, Hon. Catharine I., by the Lady Hawke. 
# Jowers, Miss Ethel, by Lady George Campbell. 
# Jenkins, Mrs. Lawrence, by Lady George Hamilton. 
# Jervis. Hon. Mrs. Bonald, by Lady Harris. [Col. 3/4] 
# Kennard, Miss Winifred Hegan, by her mother, Mrs. Hegan Kennard.
#Knutsford, the Viscountess, by the Marchioness of Salisbury. 
#King, Miss Alice King, by her mother, Lady Florence King King. 
#Kemble, Miss Dorothea, by her mother, Mrs. Horace Kemble. 
#Kerr, Lady Victoria, by her aunt, the Duchess of Buccleuch. 
#Low, Miss Olive, by Lady Low. 
#Low, Lady, by the Lady Ida Low. 
#Low, Miss Helen, by Lady Low. 
#Loch, Lady, by the Marchioness of Ripon. 
#Leverson, Mrs. George B. C., on her marriage, by the Hon. Mrs. Mostyn. 
#Mount, Miss Evelyn, by her mother, Mrs. Mount. 
#Morris, Miss Lilian, by Mrs. Malcolm Morris. 
#Mackay, Mrs. Alexander Dunlop, by her mother, Hon. Mrs. Townley Mitford. 
#Maunsell, Mrs. Mark, by the Countess of Lauderdale. 
#Mitford, Miss Constance, by her mother, Mrs. Robert Sidney Mitford. 
#MacLeod, Miss Flora, by her aunt, the Hon. Lady Northcote. 
#The Lady Mayoress, by the Marchioness of Salisbury. 
#Maitland, Lady Nora, by the Countess of Lauderdale. 
#Micklethwaite, Mrs., on her marriage, by the Hon. Mrs. Baillie of Dochfour. 
#Mackenzie, Mrs. G. Mackay, on her marriage, by Lady Charley. 
#Maguire, Hon. Mrs., by her aunt, Lady Peel. 
#McDonald, Mrs. Archibald, by Mrs. Edmund McClure. 
#Marshall, Miss, by Mrs. Victor Marshall. 
#Markham, Miss June, by Mrs. Edwin Markham
#Morris, Mrs. Malcolm, by the Countess of Lytton. 
#Noel, Miss Charlotte, by her mother, Mrs. Gerard Noel. 
#[[Social Victorians/People/Oppenheim|Oppenheim]], Miss Linda, by Mrs. Henry [[Social Victorians/People/Oppenheim|Oppenheim]]. 
#Pery, The Lady Florence, by her mother, the Countess of Limerick. 
#Paynter, Mrs. Hugh, by Viscountess Cross. 
#Peckover, Miss Alexandrina, by the Hon. Mrs. Arthur Brand. 
#Pound, Mrs. John, by the Hon. Lady Ridley. 
#Palmer, Mrs. Norman Craig, by the Hon. Mrs. Hanbury Lennox. 
#Phillips, Miss Faudel, by Mrs. Faudel Phillips. 
#Phillips, Miss Norah Faudel, by Mrs. Faudel Phillips. 
#Playfair, Miss, by the Hon. Mrs. Playfair. 
#Parr, Miss Katharine, by Mrs. Charlton Parr. 
#Pakington, the Hon. Mary, by her mother, Lady Hampton. 
#Page, Mrs. Ernest, by Mrs. William Court Gully. 
#Pilcher, Miss Margaret, by Mrs. Henry Drayson Pilcher. 
#Ritchie, Mrs. (of the United States), by Mrs. Bayard. 
#Reid, Lady, by Lady Harcourt. 
#Riddel, Mrs. D. McN., on her marriage, by Lady M'Clintock. 
#Royds, Miss Kathleen, by Mrs. Clement Molyneux Royds. 
#Reynardson, Miss Alice Birch, by her mother, Mrs. Birch Reynardson. 
#Ravenhill, Mrs. Frederick, by Mrs. Richard B. Martin.
#Russell, Miss Edith, by Mrs. Joseph Chamberlain. 
#Stern, Miss Violet, by her mother, Mrs. James Stern. 
#Savile, Miss Beatrice Mary, by the Viscountess Pollington. 
#Scott, the Lady Constance, by the Duchess of Buccleuch. 
#Sterling, Miss Margaret, by her mother, Mrs. Sterling. 
#[[Social Victorians/People/Schreiber|Schrelber]], Miss, by her mother, Mrs. Ernest [[Social Victorians/People/Schreiber|Schrelber]]. 
#[[Social Victorians/People/Schreiber|Schrelber]], Miss Evelyn, by her mother, Mrs. Ernest [[Social Victorians/People/Schreiber|Schrelber]]. 
#Smith, Mrs. Alwyn Dudley, by Mrs. Dudley Smith. 
#Townsend, Mrs. George, by the Lady Rayleigh. 
#Tarbutt, Miss Dorothy Percy, by Mrs. Percy Tarbutt. 
#Thornycroft, Mrs., by Mrs. Gerard Noel. 
#Thornycroft, Miss Ruth, by her mother, Mrs. Thornycroft. 
#Tennant, Mrs. Coombe, on her marriage, by Mrs. Henry Morton Stanley. 
#Tufton, the Hon. Rosamond, by her mother, Lady Hothfield. 
#Tritton, Mrs. Joseph Herbert, by the Viscountess Torrington. 
#Tritton, Miss Elizabeth Mary, by her mother, Mrs. Joseph Herbert Tritton. 
#Tritton, Mrs. Herbert Leslie Melville, by her mother-in-law, Mrs. Joseph Herbert Tritton. 
#Troughton, Miss Lilian Adeline, by Mrs. Gubbins. 
#Vincent, Lady, by the Hon. Lady Ridley. 
#Vandeleur, Miss Evelyn Norah, by her mother, Mrs. Vandeleur. 
#Verney, Hon. Mrs., by Mrs. Oswald. 
#Wright, Miss, by Mrs. George Townsend. 
#Wright, Miss Ettie, by Mrs. George Townsend. 
#Wilkin, Miss, by her mother, the Lady Mayoress. 
#Worcester, the Marchioness of, on her marriage, by the Duchess of Abercorn. 
#Whiteley, Mrs. George, by Mrs. Robert Yerburgh. 
#Warrington, Mrs. Thomas Rolls, by Mrs. Matthew Ingle Joyoe. 
#Walker, Mrs. Frowd, on her marriage, by Mrs. Chamberlain. 
#Wyld, Miss Beatrice, by her mother, Mrs. Wyndham Bewes. 
#Wyld, Miss Violet, by her mother, Mrs. Wyndham Bewes. 
#Wilson, the Hon. Lady Rivers, on her marriage, by the Hon. Mrs. Mostyn. 
#Wood, Mrs. Henry James Theodore, by Lady Powell. 
#Worrall, Miss Katharine, by her mother, Mrs. James Worrall. 
#Walsh, Mrs. William Hussey, on her marriage, by her mother-in-law, Mrs. Hussey Walsh.&lt;ref&gt;&quot;The Drawing Room.&quot; London Evening Standard 12 March 1896, Thursday: 3 [of 10], Col. 2a–4b [of 7]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0000183/18960312/014/0003.&lt;/ref&gt; 
&lt;/blockquote&gt;

===12 March 1896, Thursday===
[? Date is a guess.] The 20 March 1896 ''Literary World'' reports the following: &quot;Last Thursday week, at Stationers' Hall, the first meeting of the newly-formed Publishers' Union was held, about ninety members, representing nearly fifty of the leading publishing-houses, being present. Mr. C. J. Longman was elected president, Mr. John Murray vice-president, and Mr. Frederick Macmillan treasurer, with ten members of council.&quot; &quot;Table Talk,&quot; The Literary World, 20 March 1896, vol. 53, p. 270, col. 1. (Accessed 10 October 2009 in Google Books.)

=== 25 March 1896, Wednesday ===
On 11 April 1896 the ''Morning Post'' reported on a committee to raise funds for a memorial to George Augustus Sala and for aid to his widow:&lt;blockquote&gt;A Committee, of which the Duke of Abercorn is President and hon. treasurer, was formed at a meeting he[l]d on Wednesday, March 25, at Hampden House, Mayfair, with a view to raise a fund for a memorial to the late George Augustus Sala, in recognition of his long and distinguished journalistic career. It has been decided to devote the proceeds of the fund to the erection of a monument over Mr. Sala's grave and to aiding his widow. The following have consented to act on the Committee:— The Duke of Abercorn, President and hon. treasurer; the Duke of Fife, the Earl of Rosebery, the Marquis of Dufferin and Ava, Lord Glenesk, Lord Ronald Gower, Sir Arthur Otway, Sir Charles Dilke, Sir Henry De Bathe, Sir Douglas Straight, Sir David Salomons, Sir Hugh Gilzean Reid, Sir George Arthur, Sir Walter Besant, Sir George Newnes, Sir Benjamin Ward Richardson, Sir Richard Quain, Sir Harry Bodkin Poland, Sir Eyre Massey Shaw, Sir Augustas Harris, Sir Somers Vine, the Hon. W. F. D. Smith, M.P., the Hon. [[Social Victorians/People/Bourke|Algernon Bourke]], Mr. T. P. O'Connor, M.P., Messrs. Collingridge, Messrs. Lloyd, Messrs. J. S. Wood, A. J. Warden, H. Tiedeman (President Foreign Press Association), W. G. Thistle, M.D., Frederick Gordon, J. L. Toole, George Alexander, George Edwardes, John Hollingshead, E. Routledge, William Black, Wellesley Hammond, Horne Payne, Q.C., Charles Morton, E. A. Goodchild, Tom Bird, M.D., John Leighton, A. P. Watt, Alfred Beyfus, Walter Weblyn, Elwin Drew, John Whitley, Alfred Harmsworth, G. M. Kelson, James R. Parday, Caton Woodville, and M. Hall. The Executive Committee consists of Sir Somers Vine, Sir Thomas Straight, and Messrs. H. Tiedeman, Wellesley Hammond, John Hollingshead (Chairman), and W. G. Thistle (hon. secretary). Subscriptions are invited, and cheques, &amp;c., may be made payable to the Duke of Abercorn, and will be acknowledged by the hon. secretary. Lists of the subscriptions received will be from time to time published in the columns of the Morning Post. The bankers of the fund are Messrs. Coutts and Co.&lt;ref&gt;&quot;The Sala Memorial Fund.&quot; ''Morning Post'' 11 April 1896, Saturday: 5 [of 10], Col. 6c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18960411/074/0005. Print p. 5.&lt;/ref&gt;&lt;/blockquote&gt;

===27 March 1896, Friday===
The 20 March 1896 ''Literary World'' reports the following: &quot;Mr. J. M. Barrie, Mr. Anthony Hope, Sir Douglas Straight, Mr. Henry James, and Mr. James Bryce will be amongst the guests at the quarterly dinner of the Omar-Khayyäm Club next Friday.&quot; &quot;Table Talk,&quot; The Literary World, 20 March 1896, vol. 53, p. 271, col. 2. (Accessed 10 October 2009 in Google Books.)

==April 1896==
===3 April 1896, Friday===
Good Friday
===5 April 1896, Sunday===
Easter Sunday
===6 April 1896, Monday===
The 17 April 1896 ''Literary World'' reports the following: &quot;The Vagabonds flocked to the Holborn Restaurant last week to do honour to Mr. Linley Sambourne — and to be photographed. Mr. à Becket introduced his colleague on Punch in a witty and charming little speech, and Mr. Sambourne replied with a short but eloquent description of the changes in 'black and white' art since he began his career.&quot; &quot;Table Talk,&quot; The Literary World, 17 April 1896, vol. 53, p. 364, col. 2. (Accessed 13 October 2009 in Google Books.) The 17 April 1896 ''Literary World'' also reports the following: &quot;Another function held during the past week was the dinner given by Sir Stuart Knill at the Mansion House to 'The Sette of Odde Vlumes,' of which coteries he has been elected president.&quot; &quot;Table Talk,&quot; The Literary World, 17 April 1896, vol. 53, p. 365, col. 1. (Accessed 13 October 2009 in Google Books.)
===11 April 1896, Saturday===
The 17 April 1896 ''Literary World'' reports the following: &quot;The London Press Society held their annual gathering on Saturday last at Anderton's Hotel, with Mr. L. W. Lason presiding. The chairman, in proposing the chief toast, drew an interesting parallel between the Press of our empire and that of foreign nations. The Continental and American Press were too often coarse and vituperative in their attacks on rivals and political opponents, he remarked; but, taking our Press all round, it could not be denied that it shone to advantage in honest, purity, and quiet courage.&quot; &quot;Table Talk,&quot; The Literary World, 17 April 1896, vol. 53, p. 366, col. 2. (Accessed 13 October 2009 in Google Books.)
===14 April 1896, Tuesday===
The 17 April 1896 ''Literary World'' reports the following: &quot;The twentieth anniversary meeting of the supporters of the Bethnal-green Free Library was held on Tuesday last at Grosvenor House, the Rev. C. J. Ridgeway presiding in the unavoidable absence of the Duke of Westminster.&quot; &quot;Table Talk,&quot; The Literary World, 17 April 1896, vol. 53, p. 365, col. 1. (Accessed 13 October 2009 in Google Books.)
===19 April 1896, Sunday===
&quot;The celebration on Sunday of the anniversary which members of the Primrose League deem suitable for a gentle demonstraiton of Conservative political sentiment, as well as of regard for the interesting personality of the late Lord Beaconsfield, was observed with the customary floral rites and tributes, especially displaed around the pedestal of his statue outside Westminster Abbey. At Hughenden Manor, his country house, and at his tomb in the churchyard there, some pilgrims of this memorial vocation assembled. Other places associated with some incidents of his life — the houses in London where he resided at different periods, and his reputed birthplace, which as been a matter of doubt and discussion — were spoken of, though not formally visited, upon / the same occasion. It now appears to be the most probable opinion that Benjamin Disraeli was born, not in the house at the corner of Bloomsbury Square, or in the house in the Adelphi. where some years of his childhood were passed, but in a house situated in Theobald's Road, overlooking Gray's Inn Gardens, which was certainly occupied by his father, Mr. Isaac Disraeli, at that date.&quot; (&quot;Primrose Day at Westminster.&quot; Illustrated London News (London, England), Saturday, April 25, 1896; pg. 515; Issue 2975, Cols. B-C)
===20 April 1896, Monday===
Not sure of date: the 1 May 1896 ''Literary World'' reports the following: &quot;The new Publishers' Association held their first meeting at Stationer's Hall last week, when the President, Mr. C. J. Longman, delivered a lengthy address, in the course of which he touched on many points of contention in the relations between authors and publishers, and other topics of interest and importance to the book-trade. Amongst those present were Mr. John Murray, Mr. Frederick Macmillan, Mr. R. B. Marston, Mr. Oswald Crawford, Mr. William Heinemann, Mr. T. Fisher Unwin, Mr. Edwin Arnold, Colonel / Routledge, Mr. Richard Bentley, Mr. Edward Bell, and Mr. R. J. Smith.&quot; &quot;Table Talk,&quot; The Literary World, 1 May 1896, vol. 53, p. 412, cols. 1-2. (Accessed 13 October 2009 in Google Books.)
===25 April 1896, Saturday===
The 1 May 1896 ''Literary World'' reports the following: &quot;The fifteenth annual dinner of the Press Club, which took place on Saturday last at the Freemasons' Tavern, was a great and unqualified success. Mr. John Morley, who was enthusiastically received, criticised modern journalism in a speech of some length, reminding his hearers in the course of it that he had been called to his present course from the desk where he was writing his leading article. Sir Frank Lockwood also spoke, as did Mr. Spencer Hughes. Lord Wolseley and Lord Charles Beresford were present, and the chair was taken by Mr. Charles Williams.&quot; &quot;Table Talk,&quot; The Literary World, 1 May 1896, vol. 53, p. 412, col. 2. (Accessed 13 October 2009 in Google Books.)
===27 April 1896, Monday===
The 1 May 1896 ''Literary World'' reports the following: &quot;Sir Walter Besant was prevented, by an attack of incipient influenza, from presiding at last Monday's dinner at the Authors' Club. There was a larger attendance than usual in expectation of seeing him in the chair.&quot; &quot;Table Talk,&quot; The Literary World, 1 May 1896, vol. 53, p. 415, col. 2. (Accessed 13 October 2009 in Google Books.) Column 1 in the same &quot;Table Talk&quot; narrates a story told apparently at this same Authors' Club: &quot;Now that the May Meetings are upon us, a story of Exeter Hall in the old days may be quoted from the recollection of a gentleman who told it at the Authors' Club. The occasion was a meeting for advancing the cause of Foreign Missions, and several speakers had deplored the fact that so many converts had recanted. A young midshipman, who was present, felt moved to get on his feet, and say that he knew of at least one case where a convert had not recanted. Being urged to give details he told how he had once been in a boat at sea with a Kaffir chief. Pushing the chief overboard he had asked him if he would be a Christian. The chief declined as energetically as he could with his mouth half full of water, and the midshipman holding on to his scalp. The latter soused him under again, and in a few seconds pulled him to the surface to ask the same question. The chief still refusing, he was dipped again, and then, on regaining the surface, he loudly declared himself a believer. 'I thereupon,' said the midshipman, 'put him under for ten minutes, and I can assure you that convert never recanted.'&quot; &quot;Table Talk,&quot; The Literary World, 1 May 1896, vol. 53, p. 415, col. 1. (Accessed 13 October 2009 in Google Books.) 

Monday, 1896 April 27, [[Social Victorians/People/Muriel Wilson|Muriel Wilson]] was a bridesmaid the wedding of Lady Angela St. Clair Erskine and James Stewart Forbes (1896-04-28 Aberdeen Journal).

Here is the report of the wedding from the ''Inverness Courier'', with the gift list set as an unordered list to save space and simplify finding people:&lt;blockquote&gt;MARRIAGE OF LADY ANGELA ST. CLAIR ERSKINE.

Yesterday afternoon, at the increasingly fashionable church of St Paul’s, Knightsbridge, S.W., and in the presence of very large and fashionable assembly, the marriage took place of Mr James Stewart Forbes, and Lady Angela Selina Blanche St Clair Erskine. The bridegroom, Mr James Stewart Forbes, of the 9th Lancers, is the only son of the late Mr George Stewart Forbes (who was senior partner in the well-known Indian mercantile firm of Forbes, Forbes, &amp; Co., in the city of London), nephew of Helen Lady Forbes of Newe, Aberdeenshire, and cousin of the present baronet. The bride, Lady Angela Selina Blanche St Clair Erskine, is the charming and accomplished youngest daughter of the late Earl of Rosslyn, and of Blanche, Countess of Rosslyn, of Rosebank, Mid-Lothian, and 20 Charles Street, Berkeley Square, London. She is a sister of the present Peer and also of the Duchess of Sutherland and Countess of Westmoreland, and half-sister of the Countess of Warwick and Lady Algernon Gordon-Lennox.

The service was fully choral, and the Church handsomely decorated with tall palms banked with white flowers, while the altar vases had been specially refilled with white blooms for the ceremony. The Rev. James Fleming, Canon of York and Vicar of St Michael Square. S.W., officiated, assisted by the Rev. Montagu Villiers, M.A., of St Paul’s; the Rev. J. Thompson, domestic Chaplain to the Earl of Rosslyn. The bride arrived with her brother the Earl of Rosslyn, who during the singing of the nuptial hymn, Lead us, heavenly father, lead us,” conducted her to the chancel entrance and gave her away. The bridegroom was supported by his brother officer, Mr F. Allhusen of the 9th Lancers as “best man.” There were eight bridesmaids in attendance upon the bride. These young ladies were — Lady Marjorie Blanche Eva Greville, the daughter of the Earl and Countess of Warwick; Miss Ivy Gordon-Lennox, the daughter of Lord and Lady Algernon Gordon- Lennox, nieces of the bride; Miss Keith Fraser (daughter of General James Keith Fraser, C.M.G., and Mrs Keith Fraser), cousin of the bridegroom; the Hon. Ethel Gerard (daughter of Lord and Lady Gerard), Miss Diana Isabel Sturt (daughter of the Hon. Humphrey and Lady Feadovouno Sturt; Miss Edith Chaplin (daughter of the Right Hon. Henry Chaplin, M.P.); the Hon. Muriel Agnes Stewart Erskine (daughter of Lord and Lady Cardross), and [[Social Victorians/People/Muriel Wilson|Miss Muriel Wilson]] (daughter of Mr and Mrs Arthur Wilson). The bridesmaids were charmingly gowned in white muslin dresses, Louis LVI. style, over satin with frilled fichu, and ruched sleeves to wrist finished with frills and broad white satin ribbon sash. They also wore very handsome white and bright scarlet velvet cloaks, slung from one shoulder, lined with white satin, and large felt white picture hats with white ostrich feathers, and knots of scarlet velvet. The bridegroom’s presents to them were enamel chain bangles with enamel heart in centre, each of different design, and carrying nosegays of lilies of the valley in foliage.

Two smart pages (nephews of the bride), the Marquis of Stafford and Lord Alistair Clair Leveson-Gower (sons of the Duke and Duchess of Sutherland), followed the bride as trainbearer, picturesquely attired in white satin Court costume, with full blouse of gold Indian muslin, and point de Alencon lace chabot and sleeves, ruffles, white shoes and silk stockings, the breeches being fastened at the knee with diamond buckles, and scarlet velvet cloaks from shoulders, “Cavalier&quot; style, to match bridesmaids; white felt “Cavalier&quot; hats, fastened on one side with strap of red velvet, clasped with a diamond ornament, and white ostrich feathers falling over the brim, the bride’s present them being diamond fox-head pins. Lady Angela St Clair Erskine selected a “wedding gown” consisting of white satin Duchesse petticoat “Josephine,” over dress of Brussels lace, with entredeux of fine Indian muslin, the bodice being of satin, with inforcement of Brussels lace and Indian muslin, with bands of Brussels lace, “Mount de cour&quot; of the richest white satin, with very delicate embroidery of sprays of lilies of the valley, wrought in diamonds and silver. Her fine tulle veil covered coronet of real orange blossoms. Her ornaments were pearls, and she carried a bridal bouquet of lilies of the valley, tied with white satin streamers.

The scene inside the church was a most brilliant one. Quite an hour before the time fixed for the ceremony, the large edifice was nearly filled, and at the hour even standing room could not be had. The carriages outside had completely blocked Wilton Place, where the church is situated. The first to arrive was the Dowager-Countess Lovelace, wearing a gown of grey brocaded satin, with black velvet cape. Soon after came Isabella Countess of Wilton, wearing dark purple velvet, Lady Blythswood in black, Lord and Lady Newton Butler, Lord Algernon Gordon Lennox, Viscountess Hood, the Countess of Rosslyn, Lady Esher, Lady Algernon Gordon Lennox, and the Countess of Warwick (the former in pale heliotrope, the latter in white silk, with lovely cape of turquoise blue velvet trimmed with silver). The young Duke and Duchess of Marlborough next arrived. This is the Duchess's first appearance at a society wedding since her marriage. She looked very well in black satin, and wore some magnificent diamonds. The Duke and Duchess of Sutherland followed, the latter in white muslin, arranged with pale yellow silk, and large white hat, ornamented with white plumes and yellow bows. There were also the Duchess of Westminster, Earl and Countess of Westmoreland [sic], Blanch, Countess of Rosslyn, Lord Thorpe, Lady Alwyn Compton, Lady Clementine Walsh, Lady Hothfield, Hon. Rosamond Tufton, Earl of Crewe, Earl of Dunraven, Lady Mabel Kenyon, Lady Slaney, Countess Cairns, Sir Allan and Lady Mackenzie (wearing black and white striped silk), Marchioness of Downshire, Major and Lady Kathleen Pilkington, Mr Wm. Gilett, Marchioness of Tweeddale, Sir Charles and Lady Forbes of Newe, Mrs George Forbes, Miss Forbes, Lady Maud Keppell, Lady Evelyn Dawnay, Lord and Lady William Nevill, Countess of Essex, Sir W. H. Wilkins, Lady St Oswald, Lady Ducane [sic], Lady Lilian Wemyss, Helen Lady Forbes of Newe, Mrs Menzies, Col. Baillie, Mrs Farquharson, Mr Hugh Fraser, Mr Dudley Ward, Mr and Mrs Grenfell, Lady Gerard, Mrs Charles Wilson, Mrs Arthur Wilson of Tranby Croft, Captain Foley, Hon. George and Mrs Curzon, Lady Sarah Wilson, Lady Georgina Curzon, Lord Rowton, Sir George Chetwynd, Mr and Mrs Clayton Glyn, Sir Charles and Lady Hartopp, Countess Deym, Lady Vivian, Lord Vivian, Mr Percy Wyndham, Hon. Mrs Keith Falconer, Mrs Alfred Somerset, Mr Dundas, Miss G. Harvey, Mrs Ernest Chaplin, Colonel and Mrs Gore, Sir Arthur Holkett, Lady Meysey Thompson, Mr and Mrs Alfred Loder, Sir William and Lady Russell, Mr and Lady Mary Jenkins, Hon. Mrs Eliot, Hon. Mrs Percy Mitford, Mrs Balfour, Captain Leigh, and many others.

The procession up the aisle looked very pretty, the unique design of the bridesmaids' gowns and cloaks causing great admiration. Diamonds were the principal ornaments worn, and most of the ladies present wore bright colours, heliotrope and green shades appearing to be the favourites, and it is seldom that London sees such a brilliant gathering. The Prince of Wales would have attended the church, but was unable to do so owing to the levee. He, however, attended the reception, and heartily congratulated the happy pair.

During the service the hymn &quot;O perfect love, all human thoughts transcending,&quot; was sung with great effect, and after the signing of the register, the bridal party adjourned to Stafford House, where Blanche Countess of Rosslyn, gave a large reception. Early in the afternoon Mr James and Lady Angela Forbes left for Easton Lodge, Dunmow, Essex, a seat of the Earl and Countess of Warwick, where the early days of the honeymoon will be spent. The going-away dress was of pale grey canvas, with large white satin collar and revers, and green sash, and large black picture hat, with green feather and shaded yellow roses.

The presents, which numbered over 600, were exhibited in the drawing-room of Stafford House. They included the following:—

* His Royal Highness the Prince of Wales — Sapphire and diamond curb bracelet
*H.S.H. Princess Adolphus of Teck — Ruby and sapphlre safety pin
*The Duke and Duchess of Sutherland and the Earl and Countess of Warwick — A magnificent diamond tiara [Col. 2c / 3a]
*Bridegroom to Bride — Ruby and diamond ring, emerald and diamond bracelet, large diamond bow, enamel and gold muff chain, diamond heart, emerald and diamond necklace, large leather fan with &quot;Angela&quot; in diamonds
* Blanche, Countess of Rosslyn — Old Brussels lace, three rows of pearls, and a long rope of pearls
* Mrs George Forbes — Complete set of silver plate
* The Earl of Crewe — Opal and diamond pendant
* Adelaide, Countess of Westmoreland, and Lady M. Spicer — Umbrella handle
* Lady Sarah Wllson — Shagreen card case
* Mrs Wilfred Marshall — Heart-shaped links
* Mr J. Oswald — Silver-topped toilet bottle
* Mrs Oswald — Silver-mounted memorandum book
* Mrs Scarisbrick — Photo. frame
* Mr Kennard — Silver candles
* Lady Keith Ashley — Silver tea knives
* Rev. Mr and Mrs Pigott — Small silver tray
* Mr and Mrs H. Cherrington — Gold-topped salts-bottle
* Colonel and Lady Mabel Slaney — Picture of Warwick Castle
* Lady Bettine Taylor — Cushion
* Mr and Mrs Alfrel Loder — Card case
* Mr and Mrs de Winton — Fan
* Hon. A. Macdonell — Stationery case
* Lady Edmonstone — Brooch
* Sir John Willoughby — Ruby and diamond bracelet
* The Ladies Cecilie and Mary Willoughby — Photo. frame
* The Earl of Rosslyn — Turquoise bangle and Victoria
* Miss Balfour — Silver box
* The Countess of Ancaster — Fan
* Lord and Lady Burton — Fox and fan
* Mrs Macdonald — Paper knife
* Colonel and Mrs Baillie — Tortoiseshell and silver box
* Mrs Dowdall —Book
* Lady Alwyn Compton — Tortoiseshell and turquoise-handled umbrella
* Hon. John Ward — Small gold and enamel photo. frame
* Lord Herbert Vane Tempest — Turquoise bangle
* Viscountess Hood — Book
* Misses L. and D. de Bremner — Parasol
* Mrs Farquharson — Parasol
* Colonel Poynter — Silver candlesticks
* Count Larisch — Enamel and pearl muff chain
* Mrs Woodhouse — Book
* Mrs Finch — Silver tray
* The Austrian Ambassador — Feather fan
* The Countess of Cork — Diamond and black pearl brooch
* Miss Fleetwood Wilson — Silver-mounted clock
* Comte and Comtesse A Munster — Clock
* The Ladies F. and L. Cecil — Silver tray
* Mrs Baird — Sugar castor
* Mrs L. de Rothschild — Ruby and diamond bangle
* Mrs A. Sassoon — &quot;Duck&quot; brooch
* Mr and Mrs Hufa [sic] Williams — Old gilt candlesticks and shade
* Lord Kenyon — Diamond crescent
* Lord and Lady Raincliffe — Turquolse and diamond bangle
* Mr and Lady Eva Dugdale — Cabinet for miniatures
* [[Social Victorians/People/Holden|Mr Henry Holden]] — Silver-mounted salts bottle
* Countess Cairns — Fan
* Tenants on Lord Rosslyn's Estate — Silver candlesticks
* Mr and Mrs Stuart Menzies — Silver pot
* Lord Cardross — Old tortoiseshell box
* Lady Evelyn Bertie — Smelling bottle
* Lord Hy. Grosvenor — Silver toast racks
* Earl and Countess of Essex — Lamp shade
* Hon. Baillie of Dochfour — Miniature case
* Mrs Gore — Small tray
* Countess Howe — Silver ornaments
* Lady Southampton — Silver box
* Miss Keith Falconer — Photo frame
* Lord and Lady Rothschild — Antique silver tea and coffee service in case
* Mrs Gerard Leigh — Silver-mounted note book
* Mr and Mrs A. Bourke [Rourke?] — Box for miniature
* Major-General Sir Henry Ewart — Two gold candlesticks
* [[Social Victorians/People/Muriel Wilson|Miss Muriel Wilson]] — Diamond and pearl bangle
* Major Davidson — Links
* Mr and Mrs D. Cooper — Old tortoiseshell tray
* Lady de Trafford — Large green travelling cushion
* Mr Bristow — Tortoiseshell umbrella
* Mr Sykes — Whip
* Mr Dowell — Book
* Mr and Mm F. Hartmann — Old box
* The Countess of Westmorland — Old three-fold gilt screen
* Mrs Forbes — Diamond swallow
* Lady Cotterell — Silver photo. frame
* Lord and Lady Lindsey — Silver paper knife
* Mr and Mrs R. Vigner — Turquoise and diamond ring
* Lady Blanche Conyngham — Silver hand bell
* Mrs Mitford — Silver ornament
* Mrs Asquith — Butterfly brooch
* Lady Dorchester — Silver dish and spoon
* Mr and Mrs Frewer — Louis XVI. candlesticks
* Canon and Mrs Fleming — Silver handled paper knife
* Colonel and Mrs Oldham — Tortoiseshell box
* Mr and Mrs Alwyn Greville — Two old gilt looking-glasses
* Lord Ronald Gower — Old print
* Mr and Mrs J. Lowther — Large gold-topped salts bottle
* Mr and Miss Tufnell — Large box
*Mrs Wall and the Servants at Rosslyn Rest — Silver inkstand
*Hon. Sydney Greville — Silver photo frame
*Mr and Mrs Adrian Hope and Mrs Farnham — Case for writing paper
*Lord Rosebery — Sapphire and diamond bracelet
*Lord Rowton — Silver cup
*Mr R. Charters — Driving whip
*Mrs Lawrence Currie — Amethyst heart brooch
*Captain and Mrs Drummond — Book-case
*The Duchess of Wellington — Enamel clock
*Lady Cardross — Dryfons frame [sic: Dryfus? type seems clear enough...]
*Helen Lady Forbes — Silver teapot, sugar basin, and cream jug
*Mrs R. Brett — Diamond and ruby pin
*Lord W. and Lord R. Nevill — Two gold cups
*Isabella, Countess of Wilton — Silver box
*Duke of Grafton — Coral necklace
*Mr Cough Craven — Turquoise and diamond ring
*The Duke and Duchess of Marlborough — Diamond ring
*Sir Charles and Lady Hartopp — Green travelling bag
*Lord Willoughby de Broke — Fox head safety pin
*Lady Caroline Gordon Lennox — Frame
*Sir William Russell — Book
*Sir George Chetwynd — Saphire [sic] and diamond bangle
*Mrs Bischoffsheim — Parasol
*Mr and Mrs Watson Taylor — Hand-painted fan
*Mr Barclay — Turquoise and diamond ring
*Mr and Madam Von Andre — Gold-mounted travelling bag
*Mr Corbet — Whip
*Viscount Royston — Writing table
*Mrs Marshall — Tortoiseshell salts bottle
*Mrs Somerset — China handled stick
*Countess of Chesterfield — Writing set
*Lord Hy. Bentinck — Photo. frame
*Sir Allan and Lady Mackenzie — Two old silver bowls 
*Mr F. Murray Honey — Menu holders 
*Lady L. Wemyss — Safety pin 
*Mr Tynedale — White candlesticks 
*Mr Cecil Foley — Fox head pin 
*Lord and Lady Curzon — “En tout cas,&quot; with China handle 
*Lord Stafford, Lord Alastair Leveson Gower, Ladv Rose Mary Leveson Gower, Miss K. [R.?] Chaplin, and Miss F. Chaplin — Small watch set with diamonds 
*Mr and Mrs Harry Lawson — Silver mirror 
*Mrs Glyn — Cushions 
*Lord Blythswood — Old Worcester teapot 
*Mrs Hartmann — Louis XVI. settee 
*Mrs George Curzon — Frame 
*Dowager Countess of Warwick—Writing table 
*Lady Wolvarton—Two small silver coffee pots 
*Miss Blanche Forbes — Antique mustard pot 
*Miss Forbes — Silver tea and coffee set in case 
*Mr F. Allhuson — Tortoiseshell and gold box 
*Baron and Baroness de Hirsch de Gererk — Gold coffee set on tray 
*Mr Powell — Gilt basket 
*Viscount Brackley — Six &quot;Initial&quot; menu holders 
*[[Social Victorians/People/Keppel|Mrs George Keppel]] — Box 
*Lord and Lady St Oswald — Three small silver cruets 
*Hon. R. Ward — Luncheon basket 
*Sir Samuel Scott — Luncheon basket 
*Madame de Falbe — Gilt tea set 
*Madame Offenheim — Gilt coffee set 
*Lady Filmer [? Fihaer? Fihner? ] — Ebony and silver paper cutter 
*Lady Du Cane — Silver seal 
*Lady Sandhurst — Gun metal and gold pocket knife 
*Lady M. Jenkins — Two silver boot lifters in case 
*Lady Esher — Silver paper clip 
*Lord and Lady William Nevill — Two silver trays 
*Countess of Romney — Silver cigarette case 
*Mr and Mrs Arthur Sasson [sic] — Silver box 
*Household Servants of Mrs Forbes (Burleigh) — Silver cigarette case 
*Mary Lady Edmonstone — Silver holder 
*Household Servants of Mrs George Forbes — Silver salver 
*Lady C. Walsh — Small silver salver 
*Hon. R. Brett — Silver candlesticks 
*Baron Ferdinand de Rothschild — Silver vase 
*The Marquis of Camden —Two silver candlesticks 
*Lady Evelyn Dawny — Two silver candlesticks &amp;c.&lt;ref&gt;&quot;Marriage of Lady Angela St. Clair Erskine.&quot; ''Inverness Courier'' 28 April 1896 Tuesday: 5 [of 8], Cols. 2a–3c [of 6]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000446/18960428/037/0005.&lt;/ref&gt; 
&lt;/blockquote&gt;

==May 1896==
===3 May 1896, Monday===
The 8 May 1896 ''Literary World'' reports the following: &quot;Speaking at the Booksellers' dinner in the week, Dr. Welldon remarked that there was a time in history when the dissatisfied author could complain to the Archbishop of Canterbury, who had in certain cases legal authority to redress his grievances. He was sorry this cusom had died out in the profession. It would have been instructive to hear what price the Archbishop would have put on 'Robert Elsmere,' &quot;The Heavenly Twins,' 'The Sorrows of Satan,' or 'Barabbas.'&quot; The next item is also related to the Booksellers' dinner: &quot;Mr. Crockett was also on hand with one or two good stories. One of the best of these concerned himself. Mr. Crockett told how he recently was introduced to a lady, to whom his profession was mentioned. 'Mr. Crockett,' she said during the evening, 'I hear you are an author. Have you published any of your works yet?'&quot; &quot;Table Talk,&quot; The Literary World, 8 May 1896, vol. 53, p. 436, col. 2. (Accessed 13 October 2009 in Google Books.)

===4 May 1896, Tuesday===
&quot;MARLBOROUGH HOUSE, May 6.&quot; &lt;quote&gt;Sir Horace Farquhar, M.P., and Lady Farquhar entertained at dinner last evening at their resident in Grosvenor-square the Duchess of Devonshire, the Duke of Leeds, the Marchioness of Salisbury and Lady Gwendolen Cecil, the Marquis and Marchioness of Londonderry, the Lord Lieutenant of Ireland and Countess Cadogan and Lady Sophie Cadogan, the Countess of Derby, the Earl and Countess of Onslow, the Earl of Dudley, Viscount Royston, Lord James of Hereford, Lord Stanley, M.P., Lady George Hamilton, the Right Hon. George Curzon, M.P., and Mrs. Curzon, the Hon. St. John Brodrick, M.P., and Lady Hilda Brodrick, Sir Samuel Scott, and Mr. Victor Cavendish, M.P. Subsequently Lady Farquhar gave a reception. Those present included the Astro-Hungarian Ambassador and Countess Deym and Countess Isabella Deym, the Brazilian Minister, the Duke of Norfolk, and the Duke of Devonshire.&lt;/quote&gt;&lt;cite&gt;(&quot;Court Circular.&quot; Times [London, England] 7 May 1896: 9. The Times Digital Archive. Web. 2 May 2013.)&lt;/cite&gt;.

===5 May 1896, Wednesday===
&lt;quote&gt;MARLBOROUGH HOUSE, May 6. [/] His Royal Highness the Prince of Wales gave a dinner party this evening, at which the following were present:— His Royal Highness the Duke of Connaught; the German Ambassador, Count Hatzfeldt; the Austro-Hungarian Ambassador, Count Deym; the United States Ambassador, the Hon. T. F. Bayard; the French Ambassador, Baron de Courcel; the Italian Ambassador, Lieutenant-General A. Ferrero; the Spanish Ambassador, Count de Casa Valencia; the Turkish Ambassador, Costaki Anthopoulo Pasha; the Count de Ficalho, Grand Maître de la Cour to the King of Portugal; the Archbishop of Canterbury; the Lord Chancellor, Lord Halsbury; the Lord President of the Council, the Duke of Devonshire; the Marquis of Lansdowne, the Marquis of Salisbury, the Earl of Lathom, the Earl of Rosebery, the Earl of Kimberley, Lord George Hamilton; Field-Marshal Viscount Wolseley, Lord Herschell; the Right Hon G. J. Goschen, the Chancellor of the Exchequer; the Right Hon. J. Chamberlain, the Right Hon. Sir William Vernon Harcourt, the Right Hon. A. J. Balfour, the Right Hon. Sir Henry Fowler, the Right Hon. John Morley, General the Right Hon. Sir Redvers Buller, the Right Hon. Sir Matthew White Ridley, the Right Hon. H. Asquith; the President of the Royal Society, Sir Joseph Lister; General Sir Evelyn Wood, Admiral Sir Frederick Richards; the President of the Society of Antiquaries, Sir Augustus Wollaston Franks; the Director of the Natural History Museum, Sir William Flower; Rear-Admiral Sir John Fisher, Rear-Admiral Sir Frederick Bedford; the Principal Librarian and Secretary of the British Museum, Sir Edward Maunde Thompson; the President of the Royal Geographical Society, Mr. Clements R. Markham; the President of the Royal College of Surgeons, Mr. Christopher Heath; the President of the Royal College of Physicians, Dr. Samuel Wilks; Colonel Alfred Egerton, in attendance on His Royal Highness the Duke of Connaught; and General Sir Dighton Probyn and Major-General A. Ellis, in attendance on His Royal Highness the Prince of Wales. [/] The following were unavoidably prevented from obeying His Royal Highness's command:— The Russian Ambassador, M. de Staal; the Speaker, the Right Hon. W. C. Gully; the President of the Royal Academy, Sir John E. Millais. [/] During dinner the band of the Grenadier Guards, under the direction of Lieutenant Dan Godfrey, played the following selection of music:— [/]&lt;blockquote&gt;March, &quot;Hepp, Hepp, Hurrah!&quot; -- Kràl.&lt;br /&gt;Overture, &quot;Le Singe de Brésil&quot; -- Lindpaintner.&lt;br /&gt;Waltzer, &quot;Gartenlaube&quot; -- Johann Strauss.&lt;br /&gt;Selections of Melodies -- Greig.&lt;br /&gt;March, &quot;Mit Hörnerklang durch Wald und Flur&quot; -- Kohout.&lt;br /&gt;Fantasia, &quot;Hänsel und Gretel&quot; -- Humperdinck.&lt;br /&gt;Polish Dances -- Franz Morgan.&lt;br /&gt;Selection, &quot;Donna Juanita&quot; -- Suppé.&lt;br /&gt;Waltzer, &quot;Mondnacht auf der Alster&quot; -- Fétras.&lt;br /&gt;&lt;/blockquote&gt;

===10 May 1896, Monday===
The 15 May 1896 ''Literary World'' reports the following: &quot;A brilliant gathering took place on Monday last at the Galleries of the Royal Society of British Artists, where, on the invitation of Mr. and Mrs. W. T. Madge, nearly a thousand authors and pressmen, peers and members of Parliament came together to meet the proprietors and editors of the newspapers of the United Kingdom. An excellent musical programme was given under the direction of Mr. William Ganz, and the reception was altogether a thorough success.&quot; &quot;Table Talk,&quot; The Literary World, 15 May 1896, vol. 53, p. 462, col. 3. (Accessed 13 October 2009 in Google Books.)

===13 May 1896, Thursday===
The 22 May 1896 ''Literary World'' reports the following: &quot;Mr. Frankfort Moore, in a racy speech, introduced Mr. Harold Frederic, the London representative of The New York Times, and the author of 'Illumination,' and other well-known novels, to the members of the New Vagabond Club on Thursday, the 14th, and Mr. Frederic responded to the toast of his health in a speech full of point and humour. He touched upon the international question, and quietly hinted that it was only here that the fuss was made, not in America. We daresay he is right as regards the people, but the New York newspapers occasionally give one a different impression. Perhaps what Mr. Frederic meant to convey, but was too courteous to say in an assembly of Englishmen, was that all the American talk about the Venezuela busines from the beginning to end was only a way of pulling the British lion's tail so as to enjoy hearing him roar and to make capital out of the incident for election purposes. Mr. Frederic passed a compliment upon Englishmen as regards their 'splended cosmopolitanism,' as shown in the capacity of Englishmen to live up to everything that is demanded of an Imperial race. He alluded to the number of American authors, from Bret Harte downwards, who had made their homes here — 'not that they loved America less, but that they loved London more.' Among those who attended to do honour to Mr. Harold Frederic were Mr. Grant Allen, Mr. William Le Queux, Mr. C. J. Tibbits, Mr. G. B. Burgin, Mr. Morris / Colles, Mr. Coulson Kernahan, Mr. Bertram Mitford, Mr. Walter Jerrold, and Mr. Douglas Sladen.&quot; &quot;Table Talk,&quot; The Literary World, 22 May 1896, vol. 53, p. 484, cols. 1-2. (Accessed 13 October 2009 in Google Books.)

===14 May 1896, Friday===
The 8 May 1896 ''Literary World'' reports the following: &quot;Mr. Hermann Vezin will assist at the Tenth Annual Public Reading of the Shakespeare Reading Society, to be given at the Steinway Hall on Friday evening, May 15. The play Julius Caesar is arranged and rehearsed under the direction of Mr Wm. Poel; the harp will be plaed by Miss Mary Chatterton. The Reading will be repeated on the following evening to students who are preparing the play for the Oxford and Cambridge local examination.&quot; &quot;Table Talk,&quot; The Literary World, 8 May 1896, vol. 53, p. 436, col. 2. (Accessed 13 October 2009 in Google Books.) The 15 May 1896 Literary World confirms the date: &quot;The Shakespearian Reading Society will meet at the Steinway Hall, Lower Seymour-street, W., to-night at 8.30 p.m., when Julius Caesar will be read by its members, assisted by Mr. Hermann Vezin and Mr. Wm. Poel.&quot; &quot;Table Talk,&quot; The Literary World, 15 May 1896, vol. 53, p. 461, col. 2. (Accessed 13 October 2009 in Google Books.)

===23 May 1896, Sunday===
Whit Sunday

1896 May 23 (or the weekend before, so Saturday May 16?), [[Social Victorians/People/Muriel Wilson|Muriel Wilson]] is at a weekend country-house party at [[Social Victorians/People/Warwick|Warwick Castle]]: &lt;quote&gt;Among the guests entertained by the Earl and Countess of Warwick at Warwick Castle for the weekend were Sir John Willoughby, the Countess of Rosslyn, Lord and Lady Algernon Gordon Lennox, Miss Muriel Wilson, and Miss Tufnell.&lt;/quote&gt; (1896-05-23 Leamington Spa Courier).

===26 May 1896, Wednesday===
The 10 April 1896 ''Literary World'' reports the following: &quot;Mr. George A. Macmillan will preside at the booksellers' Dinner to be held at the Holborn Restaurant on the 27th of next month. He will be assisted by Mr. Joseph W. Darton, and several leading authors and publishers will be there.&quot; &quot;Table Talk,&quot; The Literary World, 10 April 1896, vol. 53, p. 341, col. 2. (Accessed 13 October 2009 in Google Books.)

==June 1896==
The Women Journalist Club's &quot;midsummer party to which all literary, artistic and social London is bidden&quot; (Krout, Mary H., &quot;Women's Clubs,&quot; Chapter 9, A Looker-On in London. Rpt in Victorian London: Publications: Social Investigation/Journalism. Online: www.victorianlondon.org [August 2005].). Here, from victorianlondon.org, is Krout's description of that event:
&lt;blockquote&gt;In June, 1896, this great function was held at Stafford House - the town residence of the Duke and Duchess of Sutherland, and there was such a demand for invitations that the committee was forced to announce through the columns of the morning newspapers that no more cards would be issued, those which had been sent having been inexorably marked &quot;strictly non-transferable.&quot; The invitations included every artist, man or woman, every journalist, author, musician and actor of note in London, with scientists, members of Parliament, cabinet ministers, diplomats and those who lived simply to enliven and adorn the social world. Long before ten o'clock there was a line of carriages stretched down Pall Mall, each awaiting its turn at the entrance in the shadow of the great porte cochère around which was stationed an array of footmen in black and gold livery. The guests were received by the President, Mrs. Craigie, a woman of striking beauty and dignity, who was assisted by Mrs. Johnson, the editor of ''The Gentlewoman'', and other women journalists.

A remarkably varied programme had been arranged, literally suited to all tastes, and the names of the artists who had contributed their services included Mine. [Mme] Albani and Cissy Loftus, Arthur Roberts, the comedian and Johannes Wolff the violinist, Alice Gomez, the contralto of the St. James concerts, and Letty Lind of the Empire Music Hall. Mme. Albani did not appear, but the beautiful and fascinating Cissy Loftus did not disappoint the company, and she gave an extremely clever imitation of a popular actress whose mannerisms were then the delight [-85-] of the Music Hall artists, and a source of pecuniary profit as well. [The page break in the original print copy is marked in the text as &quot;[-85-].&quot;] (A Looker-On in London, by Mary H. Krout, 1899 - Chapter 9 -Women's Clubs)&lt;/blockquote&gt;
===1 June 1896, Monday===
Not sure of date. The 12 June 1896 ''Literary World'' reports the following: &quot;It is authoritatively understood that the offer of one of the most important literary positions in London has been made to Mr. Edward W. Bok, editor of The Ladies' Home Journal, of Philadelphia, who is at present in England. Not alone is the position offered Mr. Bok of the most desirable character, but the honorarium attached to it is reported to be several times larger than the salary received by any editor in England. In addition to this, a ten-year lease of a Grosvenor-quare mansion is included in the offer. The position would require Mr. Bok's permanent residence in London. ... / An offer of the magnitude which the negotiations with Mr. Bok are reported to assume is particularly significant from the fact of the recipient's youth. Mr. Bok, if we err not, has just passed the thirty line in point of age, and is the youngest of all the American magazine editors. He was born in Holland, and comes of excellent Dutch / [col. 2] stock. He came to America at the age of six, and his rise there has been phenomenal. ... / Mr. Bok has been a much-dined and fèted man during his present visit to London. Last week, Lady Morell Mackenzie gave a dinner in his honour, and this week will entertain him with a country house-party at her place at Wargrave.&quot; &quot;Table Talk,&quot; The Literary World, 12 June 1896, vol. 53, p. 556, cols. 1-2. (Accessed 13 October 2009 in Google Books.)

===3 June 1896, Wednesday===
Derby Day at Epsom Downs, so the [[Social Victorians/People/Louisa Montagu Cavendish|Luise Friederike Auguste Montagu]], Duchess of [[Social Victorians/People/Devonshire|Devonshire]], hosted a ball at Devonshire House that night?

Georgiana, Lady Dudley: &lt;quote&gt;After all that umbrella holding she [Georgiana, Lady Dudley] deserved to be the one whom the Prince chose to sup with on the happiest day of his life. This was June 3, 1896, when H.R.H. won the Derby with Persimmon to tumultuous applause. After the usual dinner at the Jockey Club, Albert Edward, so his engagement diary records, went on to 'midnight supper with Lady Dudley'&lt;/quote&gt;&lt;cite&gt;(Leslie 74)&lt;/cite&gt;.

===7 June 1896, Sunday===
Of Mr. Edward W. Bok, &quot;Last Sunday Mr. Bok was the special guest of Madame Adelina Patti at a luncheon of thirty.&quot; &quot;Table Talk,&quot; The Literary World, 12 June 1896, vol. 53, p. 556, col. 2. (Accessed 13 October 2009 in Google Books.)

===8 June 1896, Monday===
Of Mr. Edward W. Bok, Lady Morell Mackenzie &quot;this week will entertain him with a country house-party at her place at Wargrave.&quot; &quot;Table Talk,&quot; The Literary World, 12 June 1896, vol. 53, p. 556, col. 2. (Accessed 13 October 2009 in Google Books.) Also, &quot;Sir Douglas also entertained Mr. Bok at dinner a few evenings ago. Mrs. C. D. Gibson gave him a luncheon; he led the Portland House cotilion with the young Duchess of Marlborough, while Anthony Hope, Jerome K. Jerome, Sir Arthur Sullivan, and Beerbohm Tree have all entertained him.&quot; &quot;Table Talk,&quot; The Literary World, 12 June 1896, vol. 53, p. 556, col. 2. (Accessed 13 October 2009 in Google Books.)

=== 10 June 1896, Wednesday ===
[[Social Victorians/People/Schreiber|Mr. Schreiber]] was present at the fashionable wedding of the Hon. John Tufton and Lady Ierne Hastings, which was reported in the &quot;Court Circular&quot; section of the ''Morning Post'' for 11 June 1896:&lt;blockquote&gt;The marriage of the Hon. John Tufton, eldest son of Lord Hothfield, and Lady Ierne Hastings, third daughter of the late Earl of Huntingdon, was solemnised yesterday at St. Anselm's Church, Davies-street, at half-past two o'clock. The ceremony was performed by the Rev. Herbert Moore, vicar of St. Anselm's, assisted by the Rev. W. F. B. Ward, private Chaplain to the Duke of Newcastle. The bride was given away by her brother, the Earl of Huntingdon. The bridesmaids were Lady Rowena and Lady Noreen Hastings, sisters of the bride; Lady Kathleen Hastings and Miss Pasley, nieces of the bride; Lady Muriel Parsons and Miss Campbell. The bridegroom was attended by the Hon. Charles Wyndham, lst Life Guards. Amongst the immediate relatives and friends in the church and afterwards at Grosvenor-square were the Duke of Newcastle, the Countess of Huntingdon, Mr. and Lady Irene Campbell, Sir Thomas and Lady Constance Pasley, Major and the Hon. Mrs. Candy, Major and the Hon. Mrs. Stirling, Lady Dora Yeoman, Lady Sarah Wilson, Lady Ventry and the Hon. Maud de Moleyns, the Hon. Lady Acland Hood, Lady and the Misses Wilson, General Stracey, Colonel Stracey, Scots Guards; Mr. W. Campbell, Mr. Herbert Wilson, the Countess of Cottenham and Lady M. Pepys, the Countess of Ranfurly, Marchesa Santurce, the Viscountess Galway, Lady Churston, the Countess of Rosse, Viscount and Viscountess Wolseley and the Hon. F. Wolseley, Mrs. Adrian Hope and Miss Hope, Mr. W. Gillett, Mr. Hastings Parker, Sir Hubert Miller, Captain Milner, lst Life Guards; [[Social Victorians/People/Schreiber|Mr. Schreiber, 1st Life Guards]]; Lord Lovat, 1st Life Guards; Captain Boyce, and many others. The Duchess of Newcastle was prevented by illness from being present. Mr. and Lady lerne Tufton left London for the Isle of Wight in the afternoon.&lt;ref&gt;&quot;Court Circular.&quot; ''Morning Post'' 11 June 1896, Thursday: 7 [of 12], Col. 6c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18960611/072/0007.&lt;/ref&gt;&lt;/blockquote&gt;

===12 June 1896, Friday===
[[Social Victorians/People/Muriel Wilson|Muriel Wilson]] is at Epsom for the races in Mr, and Mrs.’s D’Arcy’s private stand, which they had lent to Lord and Lady William Nevill, who then “entertained a large party on the Derby and Oak days.” &lt;quote&gt;Mr and Mrs D’Arcy owing to their absence on the Continent, lent their private stand at Epsom to Lord and Lady William Nevill, who entertained a large party on the Derby and Oaks days. The Company comprised the Duke and Duchess of Marlborough, the Duke of Manchester, the Marquess of Abergavenny, the Marchioness of Worcester, Marquis Camdea [?], the Marquis of Waterford, the Marchesa di Serramezzena [?], Donna Flori, Count Palfly, the Earl and Countess of Yarborough, the Earl and Countess Delawarr, Countess Cowley, Lord Sudeley, Lord Suffield, Lady Sandhurst, Lord and Lady Henry Nevill and Miss Nevill, Lord and / Lady George Nevill, Lady Alice Morland, Mr and Lady Violet Brassey, Lady Clementine Walsh, Lady Cicely Gathorne Hardy, and Miss Gathorne Hardy, Hon. Sidney Greville, Hon. F. Stanley, Hon. H. Henniker, Hon. Miriam Thellusson, Hon. R. Molyneux, Hon. Jas. Mansfield, Hon. W. Edwards, Hon. Mrs Oliphant, Sir Edward and Lady Colebrooke, Sir Frederick and Lady Milner, Sir George and Lady Lewis and Miss Lewis, Mr Arnold Morley, Hr and Mrs Henry Labouchere, Mr and Mrs Beerbohm Tree, Mr E. Hatch, [[Social Victorians/People/Arthur Stanley Wilson|Mr and Mrs Wilson]], and Miss Muriel Wilson, Mr Chas. Wyndham, Mr and Mrs Beresford Melville, and Miss Clay, Madame Van André, Mrs Leslie, Mr and Mrs Adrian Hope, Miss Mary Moore, Mr and Mrs George Alexander, Captain Ellison, Captain Peel, Hr H. Spender Clay, Mr George Ellison, Miss Rollit, Mr and Mrs C. Van Raalte, Mr and Mrs Arthur James, Mrs H. V. Higgens, Mr J. B. Leigh, Mr Walter Leslie, Mr and Mrs B. Crawshay, Mr Brinton, Captain Oswald Ames, and many others.&lt;/quote&gt; (1896-06-12 The Courier)

===14 June 1896, Sunday===
Of Mr. Edward W. Bok, &quot;For Sunday next Sir Douglas Straight has invited a party of friends to take the young editor on his private steam-launch for a cruise on the Thames.&quot; &quot;Table Talk,&quot; The Literary World, 12 June 1896, vol. 53, p. 556, col. 2. (Accessed 13 October 2009 in Google Books.)

===15 June 1896, Monday===
The 5 June 1896 ''Literary World'' reports the following: &quot;We understand that Dr. Conan Doyle will preside at the Ladies' dinner of the New Vagabond Club, on the 15th inst., as Mr. Jerome will be absent from London on that date. Eighteen literary ladies have been invited as guests.&quot; (&quot;Table Talk,&quot; The Literary World, 5 June 1896, vol. 53, p. 532, col. 1. [Accessed 13 October 2009 in Google Books].) The 19 June 1896 Literary World goes on at length about the dinner: &quot;The Ladies' Dinner of the New Vagabond Club, held on Monday in the King's Hall, Holborn, was a great success. The most interesting feature was the really able speeches given by the two ladies, Mrs. Burnett Smith (Annie S. Swan) and Mrs. Fenwick Miller, who responded for themselves and their fellow-guests to the toast of their health proposed by Dr. Conan Doyle. If this sort of thing grows, male speakers will soon be at a discount, and no public function will be complete without an oration or two from members of the fair sex. When Mrs. Miller rose to follow Mrs. Burnett Smith in thanking her hosts for their entertainment, the happy thought struck her that it would be as well to observe the strict rule on such occasions; so she desired the other lady guests to stand up while she spoke. This request was complied with, and afforded the audience a better opportunity of distnguishing the special guests of the evening from the larger number who occupied seats at the high table. Their names were, in addition to the two speakers, Mrs. Burton Harrison, Mrs. Flora A. Steel, Mrs. Gertrude Atherton, Mrs. Edith E. Cutbell, Mrs. Andrew Dean, 'George Egerton,' 'Helen Mathers,' Miss Mathilde Blind, and Miss Nora Vynne. / The Vagabonds and their guests could not have been less than 500 in number, and overflowed from the floor of the hall into the galleries. But there was no crowding, and the principal speeches were heard better than is customary at such dinners, probably owing to the eagerness of all to hear, thus preventing the usual under-current of chat. It is impossible to enumerate here all the many literary and otherwise distinguished persons who made up the audience, but it included Sir James Linton, Henniker Heaton, M.P., 'Max O' Rell,' Edward W. Bok, Dr. Moncure D. Conway, Silas K. Hocking, Frankfort Moore, A. W. / [col. 2] a' Beckett, Walter Crane, Oswald Crawford, C.M.G., G. Manville Fenn, Robert Barr, Couldon Kernshan, Dr. Todhunter, Walter Jerrold, W. Morris Colles, D. Havelock Fisher, G. Thompson Hutchinson, and, of course, the vice-chairman, Douglas Sladen and G. B. Burgin, to whom the success of the club is largely due. / ... Dr. Doyle concluded by saying that it would be strange if the New Vagabond Club did not make these ladies welcome, for women had always been noted for being charitable to beggars. / Mrs. Burnett Smth began by remarking that twenty years ago such a meeting as that would have caused a flutter in the breast of Mrs. Grundy. But she was delighted to think that women could thus meet their brothers on equal terms of kindliness and goodwill. Whatever might be said of the 'new woman' movement, it made for good in one direction. A number of old-fashioned ideas about women who write had disappeared. She felt the truth of the quotation: / Woman's cause is man's, / They rise or sink together. / She made an eloquent protest against the old theory that the husband of the literary woman lived in a chronic state of buttonless shirts, undarned socks, and ill-cooked dinners. Her wide experience taught her that wmen writers were conspicuous for their excellent housekeeping. / [Col. 3] Mrs. Fenwick Miller, in her splendidly enunciated little speech, took up the same strain of protest against the prejudiced view of women who write. For herself she was glad to live in this age, as in no previous one had comradeship been so strong. It was possible for a woman to believe that her greatest happiness consisted in the love of one man and the pleasures of one home, and yet to learn to extend her sympathies and so gain more happiness. The world was made for both sexes, and not, as some seemed to imagine, for one.&quot; &quot;Table Talk,&quot; The Literary World, 19 June 1896, vol. 53, p. 581, cols. 1-3. (Accessed 13 October 2009 in Google Books.)

===17 June 1896, Wednesday===
Of Mr. Edward W. Bok: &quot;On Wednesday next he will sail home.&quot; &quot;Table Talk,&quot; The Literary World, 12 June 1896, vol. 53, p. 556, col. 2. (Accessed 13 October 2009 in Google Books.)

===20 June 1896, Saturday===
The 26 June 1896 ''Literary World'' reports the following: &quot;Mr. Clement Shorter, by the way, presided at the gathering of the Omar Khayyám Club at Marlow last Saturday evening. Several prominent writers were present, including 'Maarten Maartens,' Mr. Grant Allen, Mr. J. M. Barrie, Mr. Harold Frederic, Mr. Edmund Gosse, and Mr. George Gissing.&quot; &quot;Table Talk,&quot; The Literary World, 26 June 1896, vol. 53, p. 605, col. 3. (Accessed 14 October 2009 in Google Books.)

===22 June 1896, Monday===
The 26 June 1896 ''Literary World'' reports the following: &quot;Speaking at the Women Writers' Annual Dinner, on Monday last, at the Criterion Restaurant, Mrs. Sydney Webb pleaded for consideration for writers of books which she classed apart from literature, though precisely why dd not transpire. Speaking from her own experience, Mrs. Webb declared that the occupation was a hard one, and that the women who took it up needed all the encouragement presumably that the authors of successful 'Pioneer' and 'Pseudonym' novels could give them. Amongst the speakers were Miss Mary Kingsley, who described the doubtful pleasures of exploring, Miss Clementina Black, and Miss Ella Curtis, who had some serious problems concerning reviewers and reviewing to place before her audience. Others present at the dinner were Mrs. Flora Annie Steel, Mrs. Molesworth, Mrs. Caffyn ('Iota'), Mrs. Sidgwick, 'Helen Mathers,' 'Annie S. Swan,' and Miss Sarah Doudney.&quot; &quot;Table Talk,&quot; The Literary World, 26 June 1896, vol. 53, p. 604, cols. 1-2. (Accessed 14 October 2009 in Google Books.)

===26 June 1896, Friday===
There was apparently a regular celebration of [[Social Victorians/People/Arthur Collins|Arthur Collins]]' birthday, 26 June, by Bret Harte, George Du Maurier, Arthur Sullivan, Alfred Cellier, Arthur Blunt, and John Hare (Nissen, Axel. Brent Harte: Prince and Pauper: 239. [http://books.google.com/books?id=WEDewmUnapcC]). Choosing 1885–1902 as the dates because those apparently are the dates of the close relationship between Harte and Collins, ending in Harte's death in 1902.

===29 June 1896, Monday===
The 19 June 1896 ''Literary World'' reports the following: &quot;The Authors' Club will entertain Dr. Conan Doyle at dinner at the Club-house on June 29, and Sir Walter Besant will take the chair.&quot; &quot;Table Talk,&quot; The Literary World, 19 June 1896, vol. 53, p. 581, col. 3. (Accessed 13 October 2009 in Google Books.)

[[Social Victorians/People/Muriel Wilson|Muriel Wilson]] was present (among those who “accepted invitations to this function”) at the wedding of Lady Sophie Cadogan and Sir Samuel Scott at Holy Trinity Church, Sloane Street., W., London. The [[Social Victorians/People/Albert Edward, Prince of Wales|Prince]] and [[Social Victorians/People/Alexandra, Princess of Wales|Princess of Wales]] were there, as were “hundreds” from “Society.” The list of notable guests, which includes [[Social Victorians/People/Arthur Stanley Wilson|Mrs. Arthur Wilson]] and Muriel Wilson, precedes the groom’s name in the story (1896-06-30 Belfast News-Letter).

Here is the report of the wedding from the ''Morning Post'', with the list of people who attended set as an unordered list to save space and simplify finding people; commas for that list have been silently deleted. The newspaper article made the gift list easy to navigate by using all caps for people's names, so that list is set as it was.&lt;blockquote&gt;MARRIAGE OF SIR SAMUEL SCOTT AND LADY SOPHIE CADOGAN.

Holy Trinity Church, Sloane-street, was filled to its utmost capacity yesterday afternoon, on the occasion of the marriage of Lady Sophie Cadogan, younger daughter of the Lord Lieutenant of Ireland and Countess Cadogan, with Sir Samuel Scott, Bart., Royal Horse Guards. The chancel of the spacious building was beautifully adorned with odontoglossum Alexandrae, white hydrangea, lilium Harrissii, white roses, carnations, azaleas, and ferns, and a variety of palms — Kentias, Seaforthias, and cocos — were arranged on each side and disposed about the choir with excellent taste. The nave was lined by non-commissioned officers and troopers of the bridegroom's regiment in full uniform with cuirasses.

Their Royal Highnesses the Prince and Princess of Wales, accompanied by the Princesses Victoria and Maud, and attended by the Countess of Macclesfield and Major-General Stanley Clarke, arrived shortly before half-past two o'clock, and were shown to seats reserved for them facing the chancel on the bride's side. Their Royal Highnesses the Duke and Duchess of York, who had previously arrived, with Lady Mary Lygon and the Hon. Derek Keppel in waiting, occupied seats on the bridegroom's side, near Sir Horace and Lady Farquhar. His Royal Highness the Duke of Cambridge was also present.

Punctually at half-past two the choir advanced to the west door to receive the bride, while the organist played the Bridal March from &quot;Lohengrin.&quot; Lady Sophie entered the church a few minutes afterwards, accompanied by the Lord Lieutenant. The bridal procession was then formed, and passed up the nave between the ranks of Guardsmen, singing, **0 Jesus, I have promised.&quot; Sir Samuel, who awaited his bride at the chancel steps, was attended by Marquis Camden as best man. The bride wore a trained gown of the richest ivory satin duchesse, handsomely embroidered round the hem in silver and small pearls in a floral design, and finished at the edge with three tiny ruches of chiffon. The bodice had a yoke of narrowly tucked chiffon, bordered with exquisite Brussels lace, from which soft draperies of chiffon inserted with silver and pearl embroidery was drawn into a high satin sash. The transparent sleeves, of drawn chiffon, were finished at the top with bow epaulettes of embroidered satin. Lady Sophie wore a coronet of orange blossoms and a tulle veil. Her lovely bouquet was of rare white orchids intermixed with lilies of the valley. Master Green, nephew of the bridegroom, acted as page, wearing a pale blue satin costume trimmed with lace, the cape lined with white satin. He carried a large white felt hat with blue feathers. Nine bridesmaids followed, Lady Anne Coventry, Lady Helen Craven, and Miss Margaret Van de Weyer, cousins of the bride; Lady Helen Stewart, Lady Isobel Stanley, Lady Kathleen Cole, Miss Bridget Bulkeley, and two little girls, the Hon. Sybil Cadogan, niece of the bride, and the Hon. Victoria Stanley. The elder bridesmaids wore gowns of ivory mousseline-de-soie over kilted glacé silk, the full bodices veiled with deep cream lace lightly embroidered with diamonds, with long lace sleeves and Pompadour sashes fastened on one side with diamond buttons. Their hats were of white fancy straw, draped round the crowns with folds of white tulle, glacé ribbon to match their sashes, and black tulle, with a plume of feathers at the side. The children were in quaint frocks to match the elder ladies, but their picturesque hats were of kilted chiné ribbon, with loops of bébé ribbon and clusters of white feathers. The bridegroom presented each with a gold chain bracelet with turquoise acorn pendant, the cup of which was of diamonds, and a bouquet of pink roses tied with white satin, the children carrying small baskets of pink rosebuds.

The Lord Primate of Ireland performed the nuptial rite, assisted by the Rev. Canon Eyton (late rector of the parish), rector of St. Margaret's, Westminster; the Rev. Henry Bevan, rector of Holy Trinity; the Rev. Gerald Blunt, rector of Chelsea; the Rev. J. J. Roumieu, rector of Culford, Bury St. Edmunds; and the Rev. Edward Symonds, domestic chaplain to Earl Cadogan. The bride was given away by her father. The Service was fully choral, and before the address the hymn, &quot;O perfect life of love,&quot; was sung as an anthem, the solo being taken by a boy soprano. After the Benediction, given by the Archbishop, the choir and congregation sang, &quot;Lead us, Heavenly Father, lead us.&quot; The bride and bridegroom, preceded by the Archbishop of Armagh and clergy, then passed to the vestry to sign the register, Earl Cadogan escorting the Princess of Wales, and the Prince of Wales accompanying Countess Cadogan. Sir Horace Farquhar gave his arm to the Duchess of York, and the Duke of York offered his to Lady Farquhar. A wedding favour of shamrock, white heather, and orange blossom was placed on the seat of each guest. While the registers were being attested a member of the choir sang, &quot;Be thou faithful unto death,&quot; from Mendelssohn's &quot; St. Paul.&quot;

Their Royal Highnesses the Prince and Princess of Wales, the Princesses Victoria and Maud, and the Duke and Duchess of York, with their ladies and gentlemen in waiting, attended the reception afterwards held by Earl and Countess Cadogan at Chelsea House. Those present at the ceremony and reception included

# the Austro-Hungarian Ambassador and Countess Deym and Countess Isabella Deym
# the Brazilian Minister
# Count Koziebrodski
# Princess Pless
# the Duchess of Devonshire
# the Duchess of Manchester
# the Duke and Duchess of Abercorn and Lady Alexandra Hamilton
# the Duke and Duchess of Marlborough
# the Duchess of Buccleuch and the Ladies Scott
# the Duke of Grafton
# the Duke of Manchester
# Sir Horace aud Lady Farquhar
# the Marchioness of Ormonde and Lady Beatrice Butler
# the Marquis and Marchioness of Salisbury
# the Marchioness of Londonderry and Lady Helen Stewart
# the Marchioness of Headfort aud Lady Beatrice Taylour
# the Marchioness of Lansdowne and Lady Beatrice Fitzmaurice
# the Marchioness of Hastings and Miss Olive Chetwynd
# the Archbishop of Armagh and Miss Alexander
# the Earl and Countess of Coventry and Lady Barbara Coventry
# Elizabeth Countess of Wilton and Mr. Pryor
# the Earl of Crewe
# the Earl and Countess of Romney
# the Earl of March
# the Earl and Countess of Kilmorey
# the Earl of Rosse
# Countess Howe and Lady Evelyn Eyre
# the Earl of Clarendon
# the Earl and Countess of Craven
# Evelyn Countess Craven
# the Countess of Lathom and the Ladies Wilbraham
# Countess Grosvenor and Lady Constance Grosvenor
# the Countess of Derby and Lady Isobel Stanley
# Victoria Countess of Yarborough and Mr. Richardson
# the Countess of Ancaster
# Lady Alice Willoughby and Lady Cecelie Goff
# the Countess of Powis
# the Earl of Hardwicke
# the Earl of Listowel and Lady Beatrice Hare
# the Countess of Erne
# the Countess of Enniskillen and Lady Florence Cole
# Georgiana Countess of Dudley
# the Earl of Kimberley
# the Countess of Caledon
# the Countess of Gosford
# the Countess of Huntingdon
# Viscountess Marsham
# Viscount and Viscountess Chelsea
# Viscountess Newport and the Hon. Miss Bridgeman
# Viscountess Helmsley
# Viscount Castlereagh
# Viscountess Coke
# Viscount and Viscountess Deerhurst and Lady Dorothy Coventry
# Viscount and Viscountess Curzon
# Viscount and Viscountess Cross
# the Lord Chancellor of Ireland
# Lord and Lady Lurgan
# Lady Ashbourne and the Hon. Violet Gibson
# Lady Halsbury and the Hon. Evelyn Giffard
# Lord and Lady Glenesk
# Lord and Lady Hastings
# Lord and Lady William Nevill
# Lord and Lady Castletown
# Lady Norreys
# Lady Alice Stanley
# Lady Stratheden
# Lady Arthur Wellesley and Miss Wellesley
# Lady Lucy Hicks-Beach and Miss Hicks-Beach
# Lord and Lady Iveagh
# Lady Barbara Smith
# Lord and Lady Burton and Miss Thorn
# Lord Rowton
# Lord H. Vane-Tempest
# Lady Julia Wombwell and Miss Wombwell
# Lady Aline Beaumont, Lord Charles Montagu
# Lord and Lady Algernon Gordon Lennox
# the Earl of March
# Lady Angela Forbes
# Lord Inchiquin and the Hon. Miss O'Brien
# Lord Berkeley Paget and Miss Paget
# the Hon. Lady and Miss Ridley
# Emily Lady Ampthill
# Lady St. Oswald
# Lady Wolverton
# Lady Musgrave and the Hon. Miss Harbord
# Lady Caroline Gordon Lennox
# Lady De Trafford
# Lady Bulkeley
# Lady Hindlip
# Lady and Miss Forbes
# Lord and Lady Arthur Hill
# Lady Tweedmouth
# Lord and Lady Balfour of Burleigh
# the Dowager Lady Lurgan
# Lady Clementine Walsh
# Lady Jeune and Miss Stanley
# the Hon. Mrs. Maguire
# the Hon. Mrs. Corbett
# the Hon. Mrs. Arthur Cadogan
# the Hon. Mr. and Mrs. Marsham Townshend and Miss Eva Hoare
# the Hon. Mrs. Charles Hay and Miss Hay
# the Hon. Mrs. Charles Cadogan
# the Hon. Mrs. George Campbell
# the Hon. Humphry and Lady Feodore Sturt
# the Hon. Sidney Greville
# Admiral the Hon. John Yorke
# the Hon. Mrs. Algernon Bourke
# the Hon. W. Coventry
# the Hon. C. Brownlow
# the Hon. Otway and Mrs. Cuffe
# the Lord Chief Justice of Ireland and Lady and Miss O'Brien
# the Lord Mayor of Belfast and Mrs. Pirrie
# the Right Hon. G. J. and Mrs. Goschen
# Mr. and Lady Emily Van de Weyer
# Captain and Lady Jane Van Koughnet
# Mr. and Lady Victoria Hamilton
# Captain and Lady Sarah Wilson
# Mr. and Lady Margaret Loder
# Captain the Hon. A. and Mrs. Somerset
# Sir Archibald and Lady Edmonstone
# Sir Albert Rollit, M.P.
# Sir Henry Edwardes
# Mr. Algernon Peel
# Mr. Seymour Corkran
# Sir Charles and Lady Hartopp
# Captain and Mrs. Philip Green
# Baroness and Miss de Brienen
# Mr. and Mrs. Henry White
# Mr. and Mrs. Douglas Gordon
# Colonel and Mrs. Fludyer
# Captain and Mrs. Anstruther Thomson
# Mrs. A. Paget
# Mr. and Mrs. A. Sassoon
# Mr. and Mrs. Arthur Hay
# Mr. and Mrs. W. H. Grenfell
# Mr. Hatch, M.P.
# Mr. and Mrs. Sassoon
# Mr. Horace Cadogan
# Mr. and Mrs. Claud Hay
# Colonel and Miss Crichton
# Mrs. Charrington
# Mrs. James
# Mrs. Cecil Reid and Miss Reid
# Mrs. Arthur Wilson and Miss Wilson
# Mrs. Charles Wilson and Miss Wilson
# Mrs. Molyneux and Miss Dawnay
# Mr. and Mrs. Cornwallis West and Miss West
# Mr. and Mrs. Coles Child
# Mrs. Owen Williams
# Mrs. B. Martin
# Mrs. Prothero
# the Misses Caldwell
# Mrs. Hatford Harter
# Mrs. Henry Villiers
# Mrs. Pease and Miss Pereira
# Mrs. Hwfa Williams
# Mr. and Mrs. F. Sassoon
# Mr. Paley
# Mr. Graham Vivian
# the Misses Montgomery
# Mr. W. Clay
# Mrs. and Miss Ritchie
# Mr. and Mrs. E. Walter Greene and the Misses Greene
# Mr. and Mrs. Helicar and Miss Helicar
# Mrs. Hungerford
# Mrs. Chute. Major and Mrs. de Freville
# Mr. and Mrs. David T. Arnott, Rev. Mr. and the Hon. Mrs. Bevan
# and many others.

The house was beautifully decorated with palms and white flowers, and the Band of the Royal Horse Guards, stationed on the Terrace, played some spirited music during the afternoon. The Royal guests took their departure at four o'clock.

The newly-wedded pair left soon afterwards for Castle Rising Hall, King's Lynn, the seat of Sir Horace Farquhar. [Col. 2c / 3a] Lady Sophie Scott went away in a gown of ivory crêpe de chine, the bodice draped with point d'Alençon, caught with pale malmaisons, the tight-fitting sleeves were of rucked I mousseline-de-soie, and the softly hanging skirt was edged with ruches of mousseline and lace insertion. The bride wore a large white hat covered with white feathers.

The wedding presents were exhibited in the ball-room, and included a large number of costly jewels.

The Prince and Princess of W&lt;small&gt;ALES&lt;/small&gt; presented the bride with a diamond aigrette set with two large turquoises.

The Princesses V&lt;small&gt;ICTORIA&lt;/small&gt; and M&lt;small&gt;AUD&lt;/small&gt; sent her a gold bonnet-pin encrusted with diamonds and a large turquoise.

The Duke and Duchess of Y&lt;small&gt;ORK&lt;/small&gt;'&lt;small&gt;S&lt;/small&gt; gift was a gold parasol handle set round with turquoises and diamonds. 

The Duke and Duchess of F&lt;small&gt;IFE&lt;/small&gt; sent four silver dessert baskets.

The gems given by the B&lt;small&gt;RIDEGROOM&lt;/small&gt; to his bride comprised a superb diamond tiara, a broad diamond collar formed of seven rows of stone, another collar of diamonds and sapphires, a magnificent diamond bracelet, a set of half hoop rings— diamond, ruby, emerald, and sapphire— and a diamond bow brooch.

Sir H&lt;small&gt;ORACE&lt;/small&gt; and Lady F&lt;small&gt;ARQUHAR&lt;/small&gt; gave a valuable parure of sapphires and diamonds, including a coronet, necklace, bracelet, star, &amp;c.

Earl C&lt;small&gt;ADOGAN'S&lt;/small&gt; presents to his daughter were a lovely diamond necklace, formed of five bows, with clustered centres and tassels, terminating with large pear-shaped stones, the bows connected by festoons of smaller diamonds; and a magnificent bracelet of emeralds and diamonds  Countess C&lt;small&gt;ADOGAN&lt;/small&gt; gave her a sapphire and diamond bracelet. Lady L&lt;small&gt;URGAN'S&lt;/small&gt; gifts were a diamond butterfly and gold curb bracelet; Viscount C&lt;small&gt;HELSEA&lt;/small&gt;, gold-mounted dressing case; the Hons. G&lt;small&gt;ERALD&lt;/small&gt;, L&lt;small&gt;EWIN&lt;/small&gt;, W&lt;small&gt;ILLIAM&lt;/small&gt;, E&lt;small&gt;DWARD&lt;/small&gt;, and A&lt;small&gt;LEXANDER&lt;/small&gt; C&lt;small&gt;ADOGAN&lt;/small&gt;, diamond locket with crystal centre; Viscountess C&lt;small&gt;HELSEA&lt;/small&gt;, a chain bracelet with turquoise and diamond drop; Lord L&lt;small&gt;URGAN&lt;/small&gt;, gold-mounted dressing bag.   

The other gifts to the bride included:— From Princess A&lt;small&gt;DOLPHUS&lt;/small&gt; of T&lt;small&gt;ECK&lt;/small&gt;, stick with tortoiseshell and gold handle; Prince and Princess E&lt;small&gt;DWARD&lt;/small&gt; of S&lt;small&gt;AXE&lt;/small&gt;-W&lt;small&gt;EIMAR&lt;/small&gt;, a hand-painted photograph screen; Prince and Princess H&lt;small&gt;ENRY&lt;/small&gt; of P&lt;small&gt;LESS&lt;/small&gt;, large white ostrich feather fan; the Marquis and Marchioness of S&lt;small&gt;ALISBURY&lt;/small&gt;, pink enamel pendant watch, set round with pearls, and attached to an enamel bow; the Duke of M&lt;small&gt;ANCHESTER&lt;/small&gt;, diamond and sapphire combs; the Marquis and Marchioness of L&lt;small&gt;ONDONDERRY&lt;/small&gt;, diamond and turquoise bracelet; the Marquis and Marchioness of Z&lt;small&gt;ETLAND&lt;/small&gt;, turquoise and diamond shamrock brooch; the Earl and Countess of C&lt;small&gt;OVENTRY&lt;/small&gt;, green enamel and pearl bracelet; the Earl and Countess of D&lt;small&gt;ERBY&lt;/small&gt;, bracelet composed of large pearls and diamonds alternately, with pendant heart encrusted with diamonds; the Lord Lieutenant's A&lt;small&gt;IDES&lt;/small&gt;-&lt;small&gt;DE&lt;/small&gt;-C&lt;small&gt;AMP&lt;/small&gt;, inkstand, blotter, and envelope case in tortoiseshell and silver; the Earl of D&lt;small&gt;URHAM&lt;/small&gt;, ruby, diamond, and pearl shamrock brooch; the Earl of R&lt;small&gt;OSEBERY&lt;/small&gt;, massive silver-mounted mirror; the Right Hon. A. J. B&lt;small&gt;ALFOUR&lt;/small&gt;, pair of silver-mounted toilet bottles; G&lt;small&gt;EORGINA&lt;/small&gt; Countess of D&lt;small&gt;UDLEY&lt;/small&gt;, a pearl and diamond bracelet; Lord and Lady R&lt;small&gt;OTHSCHILD&lt;/small&gt;, a lace fan set with monogram in diamonds; the Earl of C&lt;small&gt;HESTERFIELD&lt;/small&gt;, a table; the Earl and Countess B&lt;small&gt;ATHURST&lt;/small&gt;, an enamel watch; the Earl and Countess of L&lt;small&gt;ATHOM&lt;/small&gt;, jewelled hat pin; the Duchess of S&lt;small&gt;UTHERLAND&lt;/small&gt;, a silver hunting flask; the Earl and Countess of C&lt;small&gt;RAVEN&lt;/small&gt;, a silver looking-glass; the Earl of S&lt;small&gt;UFFOLK&lt;/small&gt;, an armchair; the Earl and Countess of A&lt;small&gt;NCASTER&lt;/small&gt;, a silver bowl; the Countess of M&lt;small&gt;ACCLESFIELD&lt;/small&gt;, a Dresden china coffee set; the Duke and Duchess of W&lt;small&gt;ESTMINSTER&lt;/small&gt;, a screen; the Earl of C&lt;small&gt;LARENDON&lt;/small&gt;, a pair of cut glass and silver perfume bottles; the Earl and Countess of C&lt;small&gt;ALEDON&lt;/small&gt;, a silver bowl; the Earl and Countess of E&lt;small&gt;RNE&lt;/small&gt;, a clock; the Earl and Countess of D&lt;small&gt;ALKEITH&lt;/small&gt;, a pair of silver candlesticks; the Dowager Countess of C&lt;small&gt;RAVEN&lt;/small&gt;, a silver tea set; Sir W&lt;small&gt;ILLIAM&lt;/small&gt; and Lady K&lt;small&gt;AYE&lt;/small&gt;, a marble-topped writing table; Mr. A&lt;small&gt;LFORD&lt;/small&gt;, a table; Sir P&lt;small&gt;ETER&lt;/small&gt; and Lady O'B&lt;small&gt;RIEN&lt;/small&gt;, hunting crop; Lord H&lt;small&gt;ERBERT&lt;/small&gt; V&lt;small&gt;ANE&lt;/small&gt;-T&lt;small&gt;EMPEST&lt;/small&gt;, tortoiseshell boxes; Mr. and Mrs. A. S&lt;small&gt;ASSOON&lt;/small&gt;, a diamond bow brooch; Lady M&lt;small&gt;USGRAVE&lt;/small&gt;, a diamond wing brooch; Lord and Lady A&lt;small&gt;RTHUR&lt;/small&gt; H&lt;small&gt;ILL&lt;/small&gt;, a bezique table; the L&lt;small&gt;ORD&lt;/small&gt; C&lt;small&gt;HANCELLOR&lt;/small&gt; of I&lt;small&gt;RELAND&lt;/small&gt; and Lady A&lt;small&gt;SHBOURNE&lt;/small&gt;, a silver cup; Baroness H&lt;small&gt;IRSCH&lt;/small&gt;, jewelled fly; Viscount and Viscountess C&lt;small&gt;ASTLEROSSE&lt;/small&gt;, a jewelled brooch; Lord and Lady L&lt;small&gt;ANGFORD&lt;/small&gt;, a gold curb bracelet set with sapphires and diamonds; the Earl and Countess of A&lt;small&gt;RRAN&lt;/small&gt;, a pair of links; the Countess of G&lt;small&gt;OSFORD&lt;/small&gt;, a muff chain; the Earl and Countess of E&lt;small&gt;SSEX&lt;/small&gt;, a handsome screen; Lord and Lady T&lt;small&gt;WEEDMOUTH&lt;/small&gt;, a gold and diamond heart-shaped smelling-bottle; the Right Hon. G&lt;small&gt;ERALD&lt;/small&gt; and Lady B&lt;small&gt;ETTY&lt;/small&gt; B&lt;small&gt;ALFOUR&lt;/small&gt;, set of enamelled buttons; the Ladies D&lt;small&gt;OROTHY&lt;/small&gt; and A&lt;small&gt;NNE&lt;/small&gt; C&lt;small&gt;OVENTRY&lt;/small&gt;, a diamond buckle; Lady C&lt;small&gt;ONSTANCE&lt;/small&gt; G&lt;small&gt;ROSVENOR&lt;/small&gt;, two jewelled pins; Mr. G&lt;small&gt;EORGE&lt;/small&gt; C&lt;small&gt;ORNWALLIS&lt;/small&gt; W&lt;small&gt;EST&lt;/small&gt;, a jewelled handle for parasol; Mr. P&lt;small&gt;RYOR&lt;/small&gt; and the Countess of W&lt;small&gt;ILTON&lt;/small&gt;, a miniature snuff-box; Mr. and Mrs. B&lt;small&gt;RADLEY&lt;/small&gt; M&lt;small&gt;ARTIN&lt;/small&gt;, a gold chain purse set with sapphires and diamonds; Lady H&lt;small&gt;ELEN&lt;/small&gt; S&lt;small&gt;TEWART&lt;/small&gt; and Lord C&lt;small&gt;ASTLEREAGH&lt;/small&gt;, a gold and glass scent-bottle set with pearls and diamonds; Mr. C&lt;small&gt;YRIL&lt;/small&gt; F&lt;small&gt;OLEY&lt;/small&gt;, a gold, diamond, and turquoise brooch; Miss B&lt;small&gt;LANCHE&lt;/small&gt; F&lt;small&gt;ORBES&lt;/small&gt;, a gold and diamond pin; Captain D&lt;small&gt;UNDAS&lt;/small&gt;, gold case set with diamonds and turquoises; the Duke and Duchess of D&lt;small&gt;EVONSHIRE&lt;/small&gt;, a diamond and ruby safety-pin; Lady W&lt;small&gt;OLVERTON&lt;/small&gt;, a pearl and diamond brooch; Mr. A&lt;small&gt;LFRED&lt;/small&gt; de R&lt;small&gt;OTHSCHILD&lt;/small&gt;, a diamond and ruby double horse-shoe brooch; Lord and Lady A&lt;small&gt;LICE&lt;/small&gt; S&lt;small&gt;TANLEY&lt;/small&gt;, a diamond and enamelled hat-pin; Colonel and Mrs. A&lt;small&gt;RTHUR&lt;/small&gt; P&lt;small&gt;AGET&lt;/small&gt;, a ruby and diamond bracelet; the Earl of C&lt;small&gt;REWE&lt;/small&gt;, a pearl half-hoop bracelet; the Knight of K&lt;small&gt;ERRY&lt;/small&gt; and Lady F&lt;small&gt;ITZGERALD&lt;/small&gt;, a silver scent-bottle; Viscount and Viscountess D&lt;small&gt;EERHURST&lt;/small&gt;, a fan; Lady Victoria H&lt;small&gt;AMILTON&lt;/small&gt;, a pair of silver baskets; the Hon. C&lt;small&gt;HARLES&lt;/small&gt; C&lt;small&gt;OVENTRY&lt;/small&gt;, a torquoise brooch; Viscountess C&lt;small&gt;URZON&lt;/small&gt;, a silver box; Mr. A&lt;small&gt;RTHUR&lt;/small&gt; V&lt;small&gt;ICARS&lt;/small&gt;, a hunting horn and kettle extinguisher; Mrs. C&lt;small&gt;AVENDISH&lt;/small&gt; B&lt;small&gt;ENTINCK&lt;/small&gt;, a watch; Mr. and Lady E&lt;small&gt;MILY&lt;/small&gt; V&lt;small&gt;AN DE&lt;/small&gt; W&lt;small&gt;EYER&lt;/small&gt;, an inlaid writing-desk and table; Lord and Lady B&lt;small&gt;URTON&lt;/small&gt;, a silver bowl; Mr. and Mrs. L&lt;small&gt;EOPOLD DE&lt;/small&gt; R&lt;small&gt;OTHSCHILD&lt;/small&gt;, a diamond comb for the hair: Mr. E&lt;small&gt;DWARD&lt;/small&gt; P&lt;small&gt;ACKE&lt;/small&gt;, a diamond heart-shaped locket on chain: Mr. and Mrs. F. S&lt;small&gt;ASSOON&lt;/small&gt;, six silver shell-shaped salt cellars; Mr. and Lady B&lt;small&gt;ETTY&lt;/small&gt; B&lt;small&gt;ALFOUR&lt;/small&gt;, a pair of links; Mr. C&lt;small&gt;HARLES&lt;/small&gt; B&lt;small&gt;ALFOUR&lt;/small&gt;, set of enamelled pins; the Hon. L&lt;small&gt;EWIN&lt;/small&gt; C&lt;small&gt;ADOGAN&lt;/small&gt;, a scent bottle and Prayer-book; the Hon. A&lt;small&gt;LEXANDER&lt;/small&gt; P&lt;small&gt;ARKER&lt;/small&gt;, a driving whip; Lady H&lt;small&gt;ELEN&lt;/small&gt; C&lt;small&gt;RAVEN&lt;/small&gt;, a silver pepper castor; Viscountess M&lt;small&gt;ARSHAM&lt;/small&gt; and Mrs. P. G&lt;small&gt;REEN&lt;/small&gt;, a fishing rod; Colonel and Mrs. C&lt;small&gt;ORNWALLIS&lt;/small&gt; W&lt;small&gt;EST&lt;/small&gt;, a parasol; Mr. and Mrs. C&lt;small&gt;HARLES&lt;/small&gt; W&lt;small&gt;ILSON&lt;/small&gt;, a gold necklace with diamond and turquoise drops; Mr. A. S&lt;small&gt;PIERS&lt;/small&gt; and Lady A&lt;small&gt;NNE&lt;/small&gt; S&lt;small&gt;PIERS&lt;/small&gt;, a pair of silver lyre candlesticks; Lord A&lt;small&gt;LEXANDER&lt;/small&gt; P&lt;small&gt;AGET&lt;/small&gt;, a gold-topped toilet bottle; Mrs. P&lt;small&gt;ALEY&lt;/small&gt;, a silver inkstand; the Countess of E&lt;small&gt;NNISKILLEN&lt;/small&gt;, a tortoiseshell and gold toilet bottle; Mr. J&lt;small&gt;AMES&lt;/small&gt; M&lt;small&gt;ANSFIELD&lt;/small&gt;, a silver snuff box; Miss K&lt;small&gt;ATHLEEN&lt;/small&gt; S&lt;small&gt;COTT&lt;/small&gt;, silver bonbon tray; Sir F&lt;small&gt;RANCIS&lt;/small&gt; and Lady J&lt;small&gt;EUNE&lt;/small&gt;, a tortoiseshell pen tray with silver figure; Mr. A&lt;small&gt;RTHUR&lt;/small&gt; P&lt;small&gt;ALEY&lt;/small&gt;, a silver powder puff box; Lord R&lt;small&gt;OWTON&lt;/small&gt;, an old silver flask; E&lt;small&gt;MILY&lt;/small&gt; Lady A&lt;small&gt;MPTHILL&lt;/small&gt;, a tortoiseshell and silver-mounted clock; Viscount B&lt;small&gt;RACKLEY&lt;/small&gt;, a glass specimen case; Mr. and Mrs F. S&lt;small&gt;ANDFORD&lt;/small&gt;, an old French ormolu clock; Mr. O&lt;small&gt;SBERT&lt;/small&gt; C&lt;small&gt;RAVEN&lt;/small&gt;, a carriage clock; the R&lt;small&gt;ECORDER&lt;/small&gt; of D&lt;small&gt;UBLIN&lt;/small&gt;, a fine old Irish lace collar; the S&lt;small&gt;OLICITOR&lt;/small&gt;-G&lt;small&gt;ENERAL&lt;/small&gt; of I&lt;small&gt;RELAND&lt;/small&gt; and Mrs. K&lt;small&gt;ENNY&lt;/small&gt;, an antique silver clock; Miss C&lt;small&gt;LARE&lt;/small&gt; O'B&lt;small&gt;RIEN&lt;/small&gt;, a silver stamp-box; Lord H&lt;small&gt;ENRY&lt;/small&gt; V&lt;small&gt;ANE&lt;/small&gt;-T&lt;small&gt;EMPEST&lt;/small&gt;, a silver-mounted writing set, with pair of silver candlesticks; the Rev. G. B&lt;small&gt;LUNT&lt;/small&gt;, two silver muffineers; Mrs. J&lt;small&gt;OHN&lt;/small&gt; W&lt;small&gt;OODFORD&lt;/small&gt;, Venetian glass toilet bottle; Lord and Lady B&lt;small&gt;URTON&lt;/small&gt;, a silver-gilt bowl with cover; Mr. C&lt;small&gt;LAUDE&lt;/small&gt; Y&lt;small&gt;ORKE&lt;/small&gt;, two silver-mounted photo frames; Mr. and Mrs. J. B. D&lt;small&gt;OUGHETY&lt;/small&gt;, a large silver and glass toilet bottle; the L&lt;small&gt;ORD&lt;/small&gt; M&lt;small&gt;AYOR&lt;/small&gt; of B&lt;small&gt;ELFAST&lt;/small&gt; and Mrs. P&lt;small&gt;IRRIE&lt;/small&gt;, a beautiful Irish embroidered handkerchief; Lady H&lt;small&gt;INDLlP&lt;/small&gt;, worktable requisites in crystal casket; M&lt;small&gt;EMBERS&lt;/small&gt; of the I&lt;small&gt;RISH&lt;/small&gt; I&lt;small&gt;NDUSTRIAL&lt;/small&gt; S&lt;small&gt;OCIETY&lt;/small&gt;, beautiful lace handkerchiefs; Mrs. O&lt;small&gt;PPENHEIM&lt;/small&gt;, a silver-gilt mounted blotting book; L&lt;small&gt;ILY&lt;/small&gt; Duchess of M&lt;small&gt;ARLBOROUGH&lt;/small&gt; and Lord W&lt;small&gt;ILLIAM&lt;/small&gt; B&lt;small&gt;ERESFORD&lt;/small&gt;, four tall silver candlesticks; Colonel and Mrs. F&lt;small&gt;LUDYER&lt;/small&gt;, a silver-gilt and glass toilet bottle; Mr. and Lady B&lt;small&gt;ARBARA&lt;/small&gt; S&lt;small&gt;MITH&lt;/small&gt;, an antique covered silver tankard; Mr. B&lt;small&gt;ATES&lt;/small&gt; V&lt;small&gt;AN DE&lt;/small&gt; W&lt;small&gt;EYER&lt;/small&gt;, four silver bonbon baskets; the Marchioness of H&lt;small&gt;ASTINGS&lt;/small&gt;, a silver-mounted photo frame; Miss C&lt;small&gt;HEYWYND&lt;/small&gt;, a silver heart-shaped box; Colonel F&lt;small&gt;ORESTER&lt;/small&gt;, two silver dessert baskets; Mr. and Mrs. A&lt;small&gt;RTHU&lt;/small&gt;R L&lt;small&gt;IDDELL&lt;/small&gt;, an antique silver box; Lord and Lady A&lt;small&gt;MPTHILL&lt;/small&gt;, sealing-wax holder and seal; Lady C&lt;small&gt;AROLINE&lt;/small&gt; G&lt;small&gt;ORDON&lt;/small&gt;-L&lt;small&gt;ENNOX&lt;/small&gt;, a painted glass flower vase; Miss F&lt;small&gt;ARQUHARSON&lt;/small&gt; and Miss E&lt;small&gt;LO&lt;/small&gt; F&lt;small&gt;ARQUHARSON&lt;/small&gt;, a china parrot; Mr. C&lt;small&gt;ECIL&lt;/small&gt; C&lt;small&gt;ADOGAN&lt;/small&gt;, a carriage clock; E&lt;small&gt;VELYN&lt;/small&gt; Countess of C&lt;small&gt;RAVEN&lt;/small&gt;, an old silver box; Mr. A&lt;small&gt;LFRED&lt;/small&gt; O&lt;small&gt;PPENHEIM&lt;/small&gt;, a French gilt clock; [[Social Victorians/People/Muriel Wilson|Miss M&lt;small&gt;URIEL&lt;/small&gt; W&lt;small&gt;ILSON&lt;/small&gt;]], books (7 vols.); Lady B&lt;small&gt;ULKELEY&lt;/small&gt;, a pair of old china ornaments; Lord and Lady I&lt;small&gt;VEAGH&lt;/small&gt;, a Russian table clock mounted in gold and enamel; Lady E&lt;small&gt;DITH&lt;/small&gt; F&lt;small&gt;RANKLIN&lt;/small&gt;, a silver-mounted toilet bottle; Miss A&lt;small&gt;GAR&lt;/small&gt;, a pocket Prayer-book in case; Mr. F&lt;small&gt;RANK&lt;/small&gt; M&lt;small&gt;ILDMAY&lt;/small&gt;, a silver carriage clock; Mr. and Mrs. E&lt;small&gt;WAN&lt;/small&gt; N&lt;small&gt;EPEAN&lt;/small&gt;, a small silver sauce boat; Miss W&lt;small&gt;ILSON&lt;/small&gt;, book-markers; Mr. J. B. L&lt;small&gt;EIGH&lt;/small&gt;, a basket pincushion; Mr. and the Hon. Mrs. B&lt;small&gt;AILLIE&lt;/small&gt; of Dochfour, a pair of china parrots; General and Mrs. O&lt;small&gt;WEN&lt;/small&gt; W&lt;small&gt;ILLIAMS&lt;/small&gt;, a Chinese figure; Mr. T&lt;small&gt;HOMAS&lt;/small&gt; B&lt;small&gt;ARING&lt;/small&gt;, four china figures; Mr. and Lady V&lt;small&gt;IOLET&lt;/small&gt; B&lt;small&gt;RASSEY&lt;/small&gt;, an inlaid tortoiseshell box; the Hon. Mrs. C&lt;small&gt;ORBETT&lt;/small&gt;, a tortoiseshell purse; Lady E&lt;small&gt;VELYN&lt;/small&gt; C&lt;small&gt;OTTERELL&lt;/small&gt;, a hand-painted glass bowl; the Hon. A&lt;small&gt;RTHUR&lt;/small&gt; and Mrs. C&lt;small&gt;ADOGAN&lt;/small&gt;, a pair of silver candlesticks; Miss M&lt;small&gt;ARGARET&lt;/small&gt; V&lt;small&gt;AN DE&lt;/small&gt; W&lt;small&gt;EYER&lt;/small&gt;, a silver pen and pencil in case; Mr. R&lt;small&gt;AYMOND&lt;/small&gt; G&lt;small&gt;REER&lt;/small&gt;, six turquoise and gold buttons; Lady l&lt;small&gt;SOBEL&lt;/small&gt; S&lt;small&gt;TANLEY&lt;/small&gt;, green and gold etui case; the Dowager Lady L&lt;small&gt;URGAN&lt;/small&gt;, a pair of silver fruit dishes; Chevalier &lt;small&gt;DE&lt;/small&gt; S&lt;small&gt;OUZA&lt;/small&gt; C&lt;small&gt;ORREA&lt;/small&gt;, inlaid box; Mr. W&lt;small&gt;ILLIAM&lt;/small&gt; Y&lt;small&gt;OUNG&lt;/small&gt;, four silver salt-cellars; Lord A&lt;small&gt;NNALY&lt;/small&gt;, a round silver clock; Viscountess H&lt;small&gt;OOD&lt;/small&gt;, an antique painting in oval frame; Mr. and Mrs. L&lt;small&gt;AUNCELOT&lt;/small&gt; L&lt;small&gt;OWTHER&lt;/small&gt;, an old two-handled silver cup; Hon. Mrs. G&lt;small&gt;ERVASE&lt;/small&gt; B&lt;small&gt;ECKETT&lt;/small&gt;, writing table clock in silver-mounted case; V&lt;small&gt;ICTORIA&lt;/small&gt; Countess of Y&lt;small&gt;ARBOROUGH&lt;/small&gt;, a silver-mounted mirror; Miss E&lt;small&gt;NID&lt;/small&gt; W&lt;small&gt;ILSON&lt;/small&gt;, green photo frame; Mr. H&lt;small&gt;ORACE&lt;/small&gt; C&lt;small&gt;ADOGAN&lt;/small&gt;, gold and coral mounted scent-bottle; Countess D&lt;small&gt;EYM&lt;/small&gt;, a small silver coffee pot; Mr. and Mrs. M&lt;small&gt;ARSHAM&lt;/small&gt; T&lt;small&gt;OWNSHEND,&lt;/small&gt; a silver-mounted magnifying glass; Colonel O&lt;small&gt;LIPHANT&lt;/small&gt;, an oval enamelled box; Lady H&lt;small&gt;ELEN&lt;/small&gt; C&lt;small&gt;RAVEN&lt;/small&gt;, a silver saucepan; the Earl and Countess of R&lt;small&gt;OSSE&lt;/small&gt;, a large enamelled casket; the M&lt;small&gt;ANAGER&lt;/small&gt; of the R&lt;small&gt;OYAL&lt;/small&gt; I&lt;small&gt;RISH&lt;/small&gt; S&lt;small&gt;CHOOL&lt;/small&gt; of A&lt;small&gt;RT&lt;/small&gt; N&lt;small&gt;EEDLEWORK&lt;/small&gt;, two beautifully worked covers; Lady M. C&lt;small&gt;RICHTON&lt;/small&gt;-M&lt;small&gt;AITLAND&lt;/small&gt; and Miss C&lt;small&gt;RICHTON&lt;/small&gt;-M&lt;small&gt;AITLAND&lt;/small&gt;, a pair of silver and glass bottles; Major and the Hon. Mrs. S&lt;small&gt;TIRLING&lt;/small&gt;, a pair of silver piano candlesticks; Mr. H&lt;small&gt;ENRY&lt;/small&gt; P&lt;small&gt;ARKER&lt;/small&gt;, a chased silver toilet box; Countess H&lt;small&gt;OWE&lt;/small&gt;, a tortoiseshell letter rack; Miss T&lt;small&gt;HORNHILL&lt;/small&gt;, an enamelled box; Mr. R. C&lt;small&gt;RAVE&lt;/small&gt;N, a coin-handled paper knife; Lady S&lt;small&gt;YKES&lt;/small&gt;, an antique brass and marble inkstand; the Marquis and Marchioness of BATH, a silver card tray; Vicount N&lt;small&gt;EWPORT&lt;/small&gt;, a silver heart-shaped stamp box; Mr. A&lt;small&gt;LGERNON&lt;/small&gt; P&lt;small&gt;EEL&lt;/small&gt;, silver and tortoiseshell paper knife in pink shagreen case; Mr. C&lt;small&gt;ORKRAN&lt;/small&gt;, a pair of tortoise-shell and silver-mounted candlesticks; Sir W&lt;small&gt;ILLIAM&lt;/small&gt; E&lt;small&gt;DEN&lt;/small&gt;, an antique enamelled bracelet; [[Social Victorians/People/Holden|Mr. H&lt;small&gt;ENRY&lt;/small&gt; H&lt;small&gt;OLDEN&lt;/small&gt;]], a shamrock photo frame; the G&lt;small&gt;IRLS'&lt;/small&gt; F&lt;small&gt;RIENDLY&lt;/small&gt; S&lt;small&gt;OCIETY'S&lt;/small&gt; C&lt;small&gt;HELSEA&lt;/small&gt; C&lt;small&gt;LUB&lt;/small&gt;, a pair of silver fruit baskets; the Earl of M&lt;small&gt;ARCH&lt;/small&gt;, a silver-mounted leather casket; Lady Margaret L&lt;small&gt;ODER&lt;/small&gt;, a silver cream jug and sugar basin; the Duke and Duchess of A&lt;small&gt;BERCORN&lt;/small&gt;, Japanese box for counters; Mr. O&lt;small&gt;TWA&lt;/small&gt;Y C&lt;small&gt;UFFE&lt;/small&gt;, an antique silver box; the Countess P&lt;small&gt;OWIS&lt;/small&gt;, a silver-gilt photo frame; Lady R&lt;small&gt;IDLEY&lt;/small&gt;, a set of antique paste buttons: Mrs. N&lt;small&gt;ORTON&lt;/small&gt;, a silver stamp case and porte-monnaie; Mr.C. D&lt;small&gt;ALISON&lt;/small&gt;, a silver pin tray; Earl and Countess C&lt;small&gt;ADOGAN'S&lt;/small&gt; H&lt;small&gt;OUSEHOLD&lt;/small&gt;, silver inkstand and candlestick; Mr. W. C&lt;small&gt;OVENTRY&lt;/small&gt;, a card table; the WOMEN on the C&lt;small&gt;ULFORD&lt;/small&gt; E&lt;small&gt;STATE&lt;/small&gt;, a silver wine cooler; the G&lt;small&gt;AMEKEEPERS&lt;/small&gt; on the C&lt;small&gt;ULFORD&lt;/small&gt; E&lt;small&gt;STATE&lt;/small&gt;, a sporting seat; Mr. and Mrs. W&lt;small&gt;HEELER&lt;/small&gt;, a wine bin; the Earl and Countess of W&lt;small&gt;ESTMORLAND,&lt;/small&gt; a silver-mounted stick; Captain F&lt;small&gt;EILDEN&lt;/small&gt;, umbrella, with pencil; Viscount and Viscountess F&lt;small&gt;OLKESTONE&lt;/small&gt;, a parasol; Lord and Lady A&lt;small&gt;LGERNON&lt;/small&gt; G&lt;small&gt;ORDON&lt;/small&gt;-L&lt;small&gt;ENNOX&lt;/small&gt;, an umbrella; Sir C&lt;small&gt;HARLES&lt;/small&gt; and Lady H&lt;small&gt;ARTOPP&lt;/small&gt;, an umbrella; Viscountess H&lt;small&gt;ELMSLEY&lt;/small&gt;, an umbrella; Colonel and Mrs. C&lt;small&gt;ORNWALLIS&lt;/small&gt; W&lt;small&gt;EST&lt;/small&gt;, an umbrella; the Marchioness of H&lt;small&gt;EADFORT,&lt;/small&gt; a gold-handled umbrella; Lord and Lady G&lt;small&gt;ERARD&lt;/small&gt;, a pale blue silk and gold-handled sunshade; Mr. and Mrs P&lt;small&gt;ERCY&lt;/small&gt; W&lt;small&gt;YNDHAM&lt;/small&gt;, old white and gold china tea set; the P&lt;small&gt;EOPLE&lt;/small&gt; of N&lt;small&gt;ORHT&lt;/small&gt; [sic] H&lt;small&gt;ARRIS&lt;/small&gt;, a length of Harris tweed; Sir R and Lady M. B&lt;small&gt;ULKELEY&lt;/small&gt;, a bookstand; Mrs. W. L&lt;small&gt;AWSON&lt;/small&gt;, a hand-painted fan; Viscount and Viscountess D&lt;small&gt;EERHURST&lt;/small&gt;, a fan; Captain and Lady S&lt;small&gt;ARAH&lt;/small&gt; W&lt;small&gt;ILSON&lt;/small&gt;, a natural ostrich feather fan; Lord and Lady G&lt;small&gt;LENESK&lt;/small&gt;, a fan; the Earl of S&lt;small&gt;EFTON&lt;/small&gt;, a marqueterie and ormolu [Col. 3c / 4a] table; Miss N&lt;small&gt;AYLOR&lt;/small&gt;, an old hand-painted fan; Miss &lt;small&gt;DE&lt;/small&gt; B&lt;small&gt;RIENEN&lt;/small&gt; and Miss Daisy &lt;small&gt;DE&lt;/small&gt; B&lt;small&gt;RIENEN&lt;/small&gt;, an old jewelled fan; Mr. H&lt;small&gt;ORACE&lt;/small&gt; P&lt;small&gt;LUNKETT&lt;/small&gt;, a fan; Mr. O&lt;small&gt;LIPHANT&lt;/small&gt;, an oval glass-topped table; Mr. G&lt;small&gt;RIMSTON&lt;/small&gt;, four silver bonbon dishes in case; the Earl and Countess of L&lt;small&gt;ISTOWEL&lt;/small&gt;, a small table; the Earl and Countess of K&lt;small&gt;ILMOREY&lt;/small&gt;, a green leather revolving bookcase; Lord and Lady A&lt;small&gt;RTHUR&lt;/small&gt; H&lt;small&gt;ILL&lt;/small&gt;, a card table; the Earl of H&lt;small&gt;AREWOOD&lt;/small&gt;, a green blotter and letter case; Lord N&lt;small&gt;ORREYS&lt;/small&gt;, two silver ice pails; [[Social Victorians/People/Arthur Stanley Wilson|Mrs. A&lt;small&gt;RTHUR&lt;/small&gt; W&lt;small&gt;ILSON&lt;/small&gt;]], an embroidered footstool; the C&lt;small&gt;HELSEA&lt;/small&gt; G&lt;small&gt;IRLS'&lt;/small&gt; C&lt;small&gt;LUB&lt;/small&gt;, an embroidered afternoon teacloth; Lady A&lt;small&gt;SHBURTON&lt;/small&gt; and Miss B&lt;small&gt;RASSEY&lt;/small&gt;, an ormulu and marble- topped table; Viscountess D&lt;small&gt;OWNE&lt;/small&gt;, an inlaid Indian stool; the Countess of P&lt;small&gt;ORTARLINGTON&lt;/small&gt;, a green leather bag; Mrs. C&lt;small&gt;HAINE&lt;/small&gt;, a set of pale blue enamelled and gold spoons; Mr. and Mrs. D. C&lt;small&gt;OOPER&lt;/small&gt;, four silver bottle stands; the Rev. E. S&lt;small&gt;YMONDS&lt;/small&gt;, books; Lady H&lt;small&gt;ONORIA&lt;/small&gt; C&lt;small&gt;ADOGAN&lt;/small&gt;, a Prayer-book and salts-bottle; Mr. and Mrs. W&lt;small&gt;ALTER&lt;/small&gt; G&lt;small&gt;REEN&lt;/small&gt;, silver basin; Mr. and Mrs. B&lt;small&gt;ONYNGE&lt;/small&gt;, an embroidered blotter; the M&lt;small&gt;OTHERS'&lt;/small&gt; U&lt;small&gt;NION&lt;/small&gt;, a pair of silver-gilt midget photo frames; the Earl and Countess of H&lt;small&gt;UNTINGDON&lt;/small&gt;, china and silver tea set; Miss V&lt;small&gt;AN DE&lt;/small&gt; W&lt;small&gt;EYER&lt;/small&gt;, silver photo screen; Lady M&lt;small&gt;URIEL&lt;/small&gt; P&lt;small&gt;ARSONS&lt;/small&gt;, antique silver ornament; Mrs. C&lt;small&gt;HARLES&lt;/small&gt; C&lt;small&gt;ADOGAN&lt;/small&gt;, a silver box; Miss K. G&lt;small&gt;REENE&lt;/small&gt;, book in worked cover; Sir C&lt;small&gt;HARLES&lt;/small&gt; H&lt;small&gt;ALL&lt;/small&gt;, set of antique silver ornaments; Lady C&lt;small&gt;OLEBROOKE&lt;/small&gt;, six diamond and enamel buttons; Lord and Lady S&lt;small&gt;ETTRINGTON&lt;/small&gt;, silver pincushion; Mr. J&lt;small&gt;AMES&lt;/small&gt; L&lt;small&gt;OWTHER&lt;/small&gt;, silver bell; Miss E&lt;small&gt;VA&lt;/small&gt; H&lt;small&gt;OARE,&lt;/small&gt; silver bird; Captain and Lady J&lt;small&gt;ANE&lt;/small&gt; V&lt;small&gt;AN&lt;/small&gt; K&lt;small&gt;AUGHNET&lt;/small&gt;, two silver baskets; Miss T&lt;small&gt;HORNEWILL&lt;/small&gt; and Miss J&lt;small&gt;ANE&lt;/small&gt; T&lt;small&gt;HORNEWILL&lt;/small&gt;, large silver bowl; Mrs. C&lt;small&gt;ECIL&lt;/small&gt; R&lt;small&gt;EID&lt;/small&gt;, a silver paper knife; Mr. C&lt;small&gt;ARYL&lt;/small&gt; C&lt;small&gt;RAVEN&lt;/small&gt;, a turquoise and gold bangle; the Hon. J. and Mrs. Y&lt;small&gt;ORKE&lt;/small&gt;, pearl and gold bar brooch: Major and Mrs. &lt;small&gt;DE&lt;/small&gt; T&lt;small&gt;RAVILLE&lt;/small&gt;, a gold fox brooch; Sir H&lt;small&gt;ENRY&lt;/small&gt; E&lt;small&gt;DWARDS&lt;/small&gt;, a gold box; Mrs. C&lt;small&gt;HARLES&lt;/small&gt; B&lt;small&gt;ALFOUR&lt;/small&gt;, two jewelled pins; the Hon. Mrs. M&lt;small&gt;AGUIRE&lt;/small&gt;, a diamond and ruby brooch; Captain J. O&lt;small&gt;RR&lt;/small&gt;-E&lt;small&gt;WING&lt;/small&gt;, enamel and pearl chain; Mr. and Mrs. G&lt;small&gt;EORGE&lt;/small&gt; M&lt;small&gt;ORRIS&lt;/small&gt;, an antique gold and pearl brooch; Lord and Lady A&lt;small&gt;LINGTON&lt;/small&gt;, a gold box; Lady Mary C&lt;small&gt;URRIE&lt;/small&gt;, a gold and jewelled chain; Captain H&lt;small&gt;EDWORTH&lt;/small&gt; L&lt;small&gt;AMBTON&lt;/small&gt;, a pair of links. 

The Bridegroom's presents included:— From the B&lt;small&gt;RIDE&lt;/small&gt;, a pearl and diamond pin; Earl C&lt;small&gt;ADOGAN&lt;/small&gt;, a gold snuff box of exquisite workmanship; Countess C&lt;small&gt;ADOGAN&lt;/small&gt;, an ormolu reading lamp; Lady F&lt;small&gt;ARQUHAR&lt;/small&gt;, silver tantalus on table; Viscount C&lt;small&gt;RICHTON&lt;/small&gt;, four silver pierced dessert dishes; the Marquis of L&lt;small&gt;ONDONDERRY&lt;/small&gt;, a silver cigar box; Lady E&lt;small&gt;DMONSTONE&lt;/small&gt;, set of writing-table requisites; Miss P&lt;small&gt;ACKE&lt;/small&gt;, silver inkstand and calendar; Mr. A&lt;small&gt;LFRED&lt;/small&gt; C&lt;small&gt;OOPER&lt;/small&gt;, a cigarette holder in case; Colonel P&lt;small&gt;ERCY&lt;/small&gt;, a walking-stick; Colonel L&lt;small&gt;ARKING&lt;/small&gt;, a bookcase; Mr. and Mrs. C&lt;small&gt;AMPBELL&lt;/small&gt;, a silver inkpot; Sir R&lt;small&gt;OBERT&lt;/small&gt; M&lt;small&gt;ONCREIFFE&lt;/small&gt;, a silver-crest letter-weight; the Marquis C&lt;small&gt;AMDEN&lt;/small&gt;, a pair of links; the Duchess of S&lt;small&gt;UTHERLAND&lt;/small&gt;, a knife on chain; Mr T. B&lt;small&gt;ROWN&lt;/small&gt;, a hunting crop; Mr. W. H. G&lt;small&gt;REEN&lt;/small&gt;, a silver butterfly letter-clip; Mr. and Mrs. L&lt;small&gt;ENNARD&lt;/small&gt;, an antique silver box; Mr. G&lt;small&gt;ODFREY&lt;/small&gt; H&lt;small&gt;ESELTINE&lt;/small&gt;, a silver shaving-pot; Lady &lt;small&gt;DE&lt;/small&gt; T&lt;small&gt;RAFFORD&lt;/small&gt;, a gold and diamond match-box; Mr. K&lt;small&gt;ENNET&lt;/small&gt; [sic] W&lt;small&gt;ILSON&lt;/small&gt;, a gun-metal match-box; Mr. F. B. M&lt;small&gt;ILLER&lt;/small&gt;, four silver-mounted decanters; Miss P&lt;small&gt;EREIRA&lt;/small&gt;, a walking-stick; Mrs. G&lt;small&gt;EORGE&lt;/small&gt; F&lt;small&gt;ORBES&lt;/small&gt;, a snake seal; Mr. A. R. H&lt;small&gt;AY&lt;/small&gt;, a blotter; Mr. and Mrs. W. H&lt;small&gt;OARE&lt;/small&gt;, a writing set and case; Mr. S&lt;small&gt;TEPHEN&lt;/small&gt; W&lt;small&gt;OMBWELL&lt;/small&gt;, a carriage clock; Lieutenant- General H&lt;small&gt;ANKEY&lt;/small&gt;, a gold pencil case; Lord H&lt;small&gt;ERBERT&lt;/small&gt; V&lt;small&gt;ANE&lt;/small&gt;-T&lt;small&gt;EMPEST&lt;/small&gt;, a silver cigar box; Lord D&lt;small&gt;UNSANDLE&lt;/small&gt;, two silver candlesticks; Mr. J. B. L&lt;small&gt;EIGH&lt;/small&gt;, a silver cigar case; Viscountess M&lt;small&gt;ARSHAM&lt;/small&gt; and Mrs. P&lt;small&gt;HILIP&lt;/small&gt; G&lt;small&gt;REEN&lt;/small&gt;, a pair of silver candlesticks; Mr. E. L&lt;small&gt;EVESON&lt;/small&gt;-G&lt;small&gt;OWER&lt;/small&gt;, a marble clock; Mr. P&lt;small&gt;REDDY&lt;/small&gt; M&lt;small&gt;ENZIES&lt;/small&gt;, a silver and enamel cigarette case and match-box; Mr. R&lt;small&gt;EUBEN&lt;/small&gt; S&lt;small&gt;ASSOON&lt;/small&gt;, a gold-topped cane; Mr. and Mrs. W. J&lt;small&gt;AMES&lt;/small&gt;, four gold and enamel spoons; Sir R&lt;small&gt;ALPH&lt;/small&gt; and Lady F&lt;small&gt;LORENCE&lt;/small&gt; H&lt;small&gt;ARE&lt;/small&gt;, four heart- shaped ash trays; Mr. and Lady A&lt;small&gt;LINE&lt;/small&gt; B&lt;small&gt;EAUMONT&lt;/small&gt;, walking-stick with tortoiseshell handle; Mr. J. P. M&lt;small&gt;ILBANKE&lt;/small&gt;, case containing &quot;Bradshaw,&quot; &quot;A. B. C,&quot; &amp;c.; Viscount M&lt;small&gt;ARSHAM&lt;/small&gt;, a gold-mounted cane with pencil in handle; Mr. and Mrs. C&lt;small&gt;LEMENT&lt;/small&gt; S&lt;small&gt;ATTERTHWAITE&lt;/small&gt;, a silver cigarette case; Mr. A. E. B&lt;small&gt;URNABY&lt;/small&gt;, a silver inkstand with clock on lid; Mr. H&lt;small&gt;UNGERFORD&lt;/small&gt;, a silver shaving-pot; Lord and Lady S&lt;small&gt;TRATHEDEN&lt;/small&gt; and C&lt;small&gt;AMPBELL&lt;/small&gt;, a silver cigarette case; Mr. H. S&lt;small&gt;T&lt;/small&gt;. D'A&lt;small&gt;ETH&lt;/small&gt;, a silver cigar case; Mr. and Mrs. A&lt;small&gt;RTHUR&lt;/small&gt; H&lt;small&gt;AY&lt;/small&gt;, a double silver inkstand; Messrs. J&lt;small&gt;ONES&lt;/small&gt;, walking-stick; Captain V&lt;small&gt;ILLERS [sic]&lt;/small&gt;, a silver cigar cutter and lighter; Captain R&lt;small&gt;ICARDO&lt;/small&gt;, a silver box; Captain and Mrs. G&lt;small&gt;ERALD&lt;/small&gt; F&lt;small&gt;ITZ&lt;/small&gt;G&lt;small&gt;ERALD&lt;/small&gt;, a silver match- box; the Countess of R&lt;small&gt;OMNEY&lt;/small&gt;, a silver cigarette case; Captain and Mrs. A&lt;small&gt;LFRED&lt;/small&gt; J&lt;small&gt;OHNES&lt;/small&gt;, a silver inkstand; Captain and Mrs. A&lt;small&gt;RTHUR&lt;/small&gt; S&lt;small&gt;OMERSET&lt;/small&gt;, a set of antique spoons; Mr. H. F. S&lt;small&gt;COTT&lt;/small&gt;, a pair of diamond links; Baron M&lt;small&gt;AX DE&lt;/small&gt; T&lt;small&gt;UYLL&lt;/small&gt;, an antique china box; Mrs. W. H. G&lt;small&gt;REENFELL&lt;/small&gt;, a pencil case: Mr. E&lt;small&gt;RNEST&lt;/small&gt; H&lt;small&gt;ATCH&lt;/small&gt;, a silver tea-caddy; Mr. E. P&lt;small&gt;ACKE&lt;/small&gt;, a double silver inkstand; the Misses C&lt;small&gt;ALDWELL&lt;/small&gt;, a silver-mounted telegram form book; Captain P&lt;small&gt;HILIP&lt;/small&gt; G&lt;small&gt;REEN&lt;/small&gt;, a weather glass; Mr. R&lt;small&gt;ALPH&lt;/small&gt; L&lt;small&gt;AMBTON&lt;/small&gt;, a luncheon case; Mr. L&lt;small&gt;EONARD&lt;/small&gt; B&lt;small&gt;RASSEY&lt;/small&gt;, a gold match-box; Mr. E. S. J&lt;small&gt;OHNES&lt;/small&gt;, two silver ash trays; Mr. A&lt;small&gt;LGERNON&lt;/small&gt; P&lt;small&gt;EEL&lt;/small&gt; and Mr. V&lt;small&gt;ICTOR&lt;/small&gt; C&lt;small&gt;ORKRAN&lt;/small&gt;, a silver sandwich case; Captain the Hon. E. D&lt;small&gt;AWSON&lt;/small&gt;, weather glass; Mrs. C&lt;small&gt;HARRINGTON&lt;/small&gt;, a cigarette case; Mr. J. S. F&lt;small&gt;ORBES&lt;/small&gt;, a gold seal with pencil; Messrs. C&lt;small&gt;OMYNS&lt;/small&gt; and S&lt;small&gt;ONS&lt;/small&gt;, a blotting book; Mr. C. P. B&lt;small&gt;UCKWORTH&lt;/small&gt;, two silver ash trays; the Duke of M&lt;small&gt;ARLBOROUGH&lt;/small&gt;, a silver inkstand; Lord and Lady W&lt;small&gt;ILLIAM&lt;/small&gt; N&lt;small&gt;EVILL&lt;/small&gt;, a silver-mounted blotting-book; Mr. E. B. C&lt;small&gt;HARTERIS&lt;/small&gt;, a knife; Mr. and Mrs. A&lt;small&gt;LISTAIR&lt;/small&gt; H&lt;small&gt;AY&lt;/small&gt;, four silver pepper castors in the shape of dice; Mr. H&lt;small&gt;ATFIELD&lt;/small&gt; H&lt;small&gt;ARTER&lt;/small&gt;, a paper knife; the T&lt;small&gt;ENANTRY&lt;/small&gt; at North Harris, a dressing bag; the Estate and House S&lt;small&gt;ERVANTS&lt;/small&gt;, G&lt;small&gt;AMEKEEPERS&lt;/small&gt;, and G&lt;small&gt;ILLIES&lt;/small&gt; at North Harris, a deer's-foot inkstand, mounted in silver; the Household and Estate S&lt;small&gt;ERVANTS&lt;/small&gt; at Sundridge Park, a silver kettle; Mr. H. M&lt;small&gt;ELLIDEW&lt;/small&gt;, a knife; the C&lt;small&gt;HAPLAIN&lt;/small&gt; to the Royal Horse Guards, a pair of silver candlesticks; Mr. and Mrs. L&lt;small&gt;EOPOLD&lt;/small&gt; R&lt;small&gt;OTHSCHILD&lt;/small&gt;, a pair of silver candlesticks; Captain the Hon. R and Mrs. G&lt;small&gt;REVILLE&lt;/small&gt;, a pair of silver candlesticks; Mr. A. S&lt;small&gt;ASSOON&lt;/small&gt;, a gold-mounted stick; Mr. and Mrs. A. B&lt;small&gt;OURKE&lt;/small&gt;, a print; Mr. C&lt;small&gt;LAUDE&lt;/small&gt; H&lt;small&gt;AY&lt;/small&gt;, a silver box; the T&lt;small&gt;RADESPEOPLE&lt;/small&gt; at Bromley, a silver teapot, &amp;c.; Mr. E&lt;small&gt;GREMONT&lt;/small&gt; M&lt;small&gt;ILLS&lt;/small&gt;, a book; Colonel R&lt;small&gt;OWNEY&lt;/small&gt;, a pencilcase; Captain the Hon. G. F&lt;small&gt;ORESTER&lt;/small&gt;, a silver-mounted glass jug; Dr. B&lt;small&gt;URNEY&lt;/small&gt; Y&lt;small&gt;EO&lt;/small&gt;, a bezique box; Mrs. H&lt;small&gt;AMMOND&lt;/small&gt;, a knife; the Hon. B. B&lt;small&gt;ATHURST&lt;/small&gt;, a silver box; Mr. B&lt;small&gt;ROWN&lt;/small&gt;, a hunting crop; the Hon. G&lt;small&gt;ERALD&lt;/small&gt; C&lt;small&gt;ADOGAN&lt;/small&gt;, a gold-mounted stick; the Hon. A&lt;small&gt;RTHUR&lt;/small&gt; C&lt;small&gt;OVENTRY&lt;/small&gt;, a gold and tortoiseshell-handled stick; Earl C&lt;small&gt;OWLEY&lt;/small&gt;, a gold-mounted stick; Captain and Mrs. A. S&lt;small&gt;OMERSET&lt;/small&gt;, six antique spoons in case; Major B&lt;small&gt;YNG&lt;/small&gt;, a silver shaving-pot; Mr. T. B. M&lt;small&gt;ILLER&lt;/small&gt;, four silver-mounted claret jugs; Mr. and Mrs. C&lt;small&gt;OLES&lt;/small&gt; C&lt;small&gt;HILD&lt;/small&gt;, a glass and silver match-stand; Mr. H&lt;small&gt;UBERT&lt;/small&gt; E&lt;small&gt;ATON&lt;/small&gt;, a silver-mounted blotter; the Hon. D&lt;small&gt;UDLEY&lt;/small&gt; M&lt;small&gt;ARJORIBANKS&lt;/small&gt;, an antique silver casket; Mr. and Mrs. J. P&lt;small&gt;EASE&lt;/small&gt;, six books; Mr. F. P&lt;small&gt;AYNE&lt;/small&gt; and Mr. J. H. L&lt;small&gt;EPPER&lt;/small&gt;, a silver inkstand.&lt;ref&gt;&quot;Marriage of Sir Samuel Scott and Lady Sophie Cadogan.&quot; ''Morning Post'' 30 June 1896 Tuesday: 4 [of 12], Cols. 2a–4b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18960630/029/0004.&lt;/ref&gt; &lt;/blockquote&gt;

==July 1896==
On 4 July 1896 in ''The Queen'', an article begins, &quot;On Monday last Dr Doyle, who is as much beloved by his friends as by his readers, was entertained at a banquet by his fellow-members of the Authors Club&quot; (Orel 135). Doyle gave a speech at that banquet, which the article reprints.
===2 July 1896, Thursday===
The 5 June 1896 ''Literary World'' reports the following: &quot;The date of the promised revival of Marlow's ''Doctor Faustus'' is fixed for Thursday evening, July 2, when the performance will be given before members of the Elizabethan Stage Society and their friends at St. George's Hall. Marlowe's tragedy differs from Goethe's in this, among other things, that Marlowe wrote, as Goethe could not, in the firm belief in the possibility of the situations he created.&quot;&lt;ref&gt;&quot;Table Talk.&quot; The ''Literary World'', 5 June 1896 (Vol. 53): 533, col. 2. ''Google Books''.&lt;/ref&gt;

=== 3 July 1896, Friday ===
Mrs. Goschen's Dance&lt;blockquote&gt;Mrs. Goschen's dance at the Admiralty last night was a great success. Among the guests were the Duchess of Buccleuch and the Ladies Scott, Captain Jedina, Captain Gulich, [[Social Victorians/People/Hadik|Count Hadik]], Count de Pontavice, M. de la Chaussée, the Lord Chancellor and the Hon. Evelyn Giffard, the Earl of Clanwilliam and the ladies Meade, Earl Granville, the Countess of Belmore and Lady Winifred Corry, the Earl of Donoughmore, the Earl of Eldon and Lady Louisa Scott, the Countess of Dunmore and Lady Victoria Murray, Earl Waldegrave and Lady Mary Waldegrave, and a large contingent of &quot;dancing men.&quot;&lt;ref&gt;&quot;Mrs. Goschen's Dance.&quot; ''St James's Gazette'' 04 July 1896 Saturday: 13 [of 16], Col. 1b [of 2]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001485/18960704/057/0013.&lt;/ref&gt;&lt;/blockquote&gt;

===4 July 1896, Saturday===
The 26 June 1896 ''Literary World'' reports the following: &quot;We understand that one of the principal features of the performance of Marlowe's Doctor Faustus will be the introduction of the Seven Deadly Sins, the designs for which have been taken from engravings belonging to the sixteenth century, and found in the print rom of the British Museum. The first and last parts of the play will reproduce in colour and costume the university life of Marlowe's day. The middle part of the play, the one most difficult for a stage manager to cope with, will consist of tableaux showing Faustus on his travels, and giving living pictures of the Feast of St. Peter, introducing the picturesque incident of the curse with 'bell, book, and candle'; the banquet at the court of Charles V.; and the Flight of Faustus, in his chariot drawn by yoked dragons, 'to scale Olympus top.' In consequence of the heavy expense attending each representation of the play, there will be only one public performance, that on the afternoon of July 4. Mr. Arnold Dolmetsch will supply the music.&quot;&lt;ref&gt;&quot;Table Talk.&quot; The ''Literary World'', 26 June 1896 (Vol. 53): 604, col. 2. Google Books.&lt;/ref&gt;

===13 July 1896, Monday===
[[Social Victorians/People/Muriel Wilson|Muriel Wilson]] attended a Garden Party at Buckingham Palace given by Queen Victoria. Several thousand people were there, it looks like (1896-07-14 Morning Post).

===16 July 1896, Thursday===
The 19 June 1896 ''Literary World'' reports the following: &quot;The idea of Ladies' Dinners seems to have caught on in clubland. A dinner is to be given to Mrs. Hodgson Burnett by the Authors' Club on July 16, and as the accommodation at the Club-house is neither suitable nor adequate, the dinner will be held in the King's Hall of the Holborn Restaurant. Members may take as many guests as they like, either ladies or gentlemen.&quot;&lt;ref&gt;&quot;Table Talk.&quot; The ''Literary World'', 19 June 1896 (Vol. 53): 581, col. 3. ''Google Books''.&lt;/ref&gt;

==August 1896==
Sometime in August 1896 Lady Gregory met William Butler Yeats (I got this from Wade?).

August 1896, the steamer the ''Norse King'' was to take scientists and tourists to the Varanger Fjord to view the solar eclipse. At least in the planning, as reported in January 1896, &quot;The official observers of the joint committee of the Royal Society and the Royal Astronomical Society have arranged to go by the Norse King. Among those on board will be Dr. A. Common, president of the Royal Astronomical Society, and Sir Robert Ball, who has consented to deliver a series of three lecture on the eclipse while the steamer is in the Varanger Fjord.&quot; (From a &quot;special announcement,&quot; quoted in &quot;Table Talk,&quot; The Literary World (3 Januray 1896), vol. 53, p. 13 [accessed 10 October 2009 in Google Books].)

=== 5 August 1896 ===
The ''Gentlewoman'' describes the wedding of the Hon. Terence Bourke and Miss Eveline Haines:&lt;blockquote&gt;The Hon. Terence Bourke to Miss Eveline Haines.

A smart and very pretty ceremony came off in St. Andrew's Church, Westminster, on the 5th inst., on the occasion of the marriage of the Hon. Terence Bourke, son of the late Earl of Mayo, Viceroy of India, with Miss Eveline Haines, daughter of the late Colonel Haines, of Hasketon Manor, Woodbridge. The bride, who was given away by her brother, Captain Haines, York and Lancaster Regiment, was attended by two pages, Master Dicky and Master Valentine Wyndham Quin, in picturesque white and mauve shirts with knee breeches. The five bridesmaids, Miss Mercy Barnes, Miss Robertson, Miss Powell, Miss Constance Mure, and Miss Edith Dods, were prettily gowned in white muslin trimmed with Valenciennes, and wore large white picture hats. Their presents from the bridegroom were gold chains with enamel hearts; the pages' gifts being gold and enamel sleeve links. The rites were solemnised by the Rev. and Hon. George Bourke, Chaplain to the Queen; and Lieut. Cyril Sloane Stanley, 1st Life Guards, supported Mr. Bourke as best man. After the ceremony Mr. and Mrs. Herbert Robertson, uncle and aunt of the bride, held a large reception at Willis's Rooms, amongst those accepting invitations to be present being the Dowager Countess of Mayo, the Earl and Countess of Mayo, Lady Florence Bourke, the Marchioness of Queensberry, Lord Leconfield, Lady Leconfield, Lord and Lady Milton, the Countess of Bective, Major and Lady Eva Wyndham Quin, Lord and Lady Connemara, General the Hon. John Bourke, the Hon. Gerald and Lady Maria Ponsonby, the [[Social Victorians/People/Bourke|Hon. Mrs. Algernon Bourke]], the Hon. Algernon Bourke, the Hon. Percy and Mrs. Wyndham, Mr. Wilfrid and Lady Anne Blunt, Miss Blunt, the Earl and Countess of Roden, Lord and Lady Henry Bentinck, Lord and Lady Douglas, Sir Owen and Lady Agnes Burne, Captain Haines, Master W. Haines, Mrs. and the Misses Dods, Captain and Mrs. Dods, Mr. and Mrs. Powell, Mrs. and Miss Benest, Mr. and Mrs. Ribton, Mr. Erskine and Miss D. Ribton, Captain and Mrs. Inglis, Colonel the Hon. and Mrs. Hubbard, Admiral and Mrs. Saumerez, Mr. and Mrs. B. N. Everard, Mrs. and Miss Drummond, Lady Stawell, &amp;c. In the afternoon the Hon. Terence and Mrs. Bourke took their departure for Paultons, Lieutenant Stanley's place in the New Forest, kindly lent for the honeymoon. The bride left London in a dainty heliotrope canvas costume over white silk, trimmed with white embroidered lisse and [Col. 2c / 3a] chiffon, and a big picture hat arranged with white feathers and purple irises.

The presents, which were very numerous, included: — From bridegroom to bride, diamond tiara, topaz, and diamond necklace, bangle, diamond and ruby ring, ruby ring, fan, stamp album. &amp;c. Mr. Herbert Robertson. M.P., 200 guineas. Mrs. Robertson, silver coffee pot. Miss Robertson and Masters Manning and Nevile Robertson, silver spoons. Captain Haines, silver tea tray, tea pot, sugar basin, and milk jug. The Dowager Countess of Mayo, diamond and turquoise necklace. The Earl and Countess of Mayo, turquoise necklace. Lady Leconfield, pearl, sapphire, and ruby bracelet. Miss E. Dods, pearl bracelet. Lady Eva Wyndham Quin, diamond and sapphire brooch. Lady Florence Bourke, amethyst bangle. The [[Social Victorians/People/Bourke|Hon. and Mrs. Algernon Bourke]], enamel muff chain. Hon. Harry Bourke, cheque. Hon. Edward Bourke, cheque. Hon. J. Bourke, cheque. Lord Connemara, cheque. Mrs. Dods, cheque. Mr. and Mrs. Powell, silver looking-glass. Mrs. Benest, clock. Master Haines, butter dish. Miss M. Haines, cream jug. Mrs. Bischoffsheim, Louis XV. clock. The Hon. Percy and Mrs. Wyndham, pearl and emerald chain and pendant. Marchioness of Queensberry, crown Derby snuff box. Lady Leconfield, silver tea pot. Major and Lady Eva Wyndham Quin, coffee pot and milk pot. Countess of Bective, turquoise and pearl pin. Earl and Countess of Mayo, carriage clock. Lord and Lady Henry Bentinck, gold pencil case. Sir John and Lady Constance Leslie, flower stand. Mr. Wilfrid Blunt, two Arab mares. Sir Raymond and Lady Burrell, Battersea enamel snuff box. Lord Leconfield, cheque £100, &amp;c, &amp;c.&lt;ref&gt;&quot;The Hon. Terence Bourke to Miss Eveline Haines.&quot; ''Gentlewoman'' 15 August 1896, Saturday: 24 [of 54], Col. 2b–3a [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18960815/128/0024. Print p. 232.&lt;/ref&gt;&lt;/blockquote&gt;

=== 19 August 1896, Wednesday ===
Queen Victoria was at Osborne, accompanied by Princess Henry of Battenberg and the Hon. Frances Drummond, and then also Countess Feodore Gleichen. The dinner party Wednesday night also included people who were at Cowes for the yachting:&lt;blockquote&gt;Her Majesty's dinner party last evening included Captain Acland, her Majesty's ship Australia, guardship at Cowes, and the Hon. Mrs. Acland, Mrs. Lawrence Drummond, [[Social Victorians/People/Young|Sir Allen Young, C.B.]], and Major Strong, 2nd Battalion Scottish Rifles.&lt;ref&gt;&quot;Court Circular.&quot; ''Morning Post'' 20 August 1896 Thursday: 5 [of 8], Col. 4a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18960820/072/0005.&lt;/ref&gt;&lt;/blockquote&gt;

===31 August 1896, Monday===
Summer Bank Holiday

==September 1896==
==October 1896==
===31 October 1896, Saturday===
Halloween.
==November 1896==

=== 2 November 1896, Monday ===
The ''Black and White'' hosted a dinner to welcome Special Artist Charles M. Sheldon back from the Soudan:&lt;blockquote&gt;MR. CHARLES M. SHELDON, ''Black and White''&lt;nowiki/&gt;'s Special Artist, received a welcome back from the Soudan at a dinner presided over by the editor and attended by the directors and other officers of the journal, as well as by a large company, including: Messrs. Angus Evan Abbott, A. H. Beaman, A. S. Boyd, F. Whelan Boyle, J. MacLaren Cobban, Oscar Eckhardt, James Greig, Bernard F. Gribble, Paul Hardy, A. S. Hartrick, Lewis Hind, G. Kenealy, A. L. Lazarus, T. J. Lipton, G. G. Manton, Gilbert Marks, G. E. Matheson, F. Frankfort Moore, J. M. Munro, Henry Norman, Barry Pain, R. B. M. Paxton, Ernest Prater, W. Pett Ridge, Charles Robinson, [[Social Victorians/People/Rook|Clarence Rook]], R. Savage, J. A. Shepherd, Alexander Stuart, E. J. Sullivan, Adolf Thirde [?], Enoch Ward, and Edgar Wilson. Letters of apology were intimated from Messrs. J. H. Bacon, Robert Barr, H. Brown, L. Cope Cornford, J. Finnemore, H. W. Massingham, Arthur Morrison, W. Mudford, H. H. S. Pearse, Eden Phillpotts, G. E. Webster, H. G. Wells, H. Seppings-Wright [?], &amp;c., &amp;c. The Chairman proposed the health of Mr. Sheldon, a toast which was honoured with enthusiasm, and the guest ot the evening, in a characteristically modest speech, urged that campaigning was its own reward, though it was enhanced by such a welcome home as had been accorded him. During the night members of the company furnished songs and stories, Mr. Sheldon contributing two notable items to the programme — ''Drill ye tarriers'', an Irish ditty which has, through his instrumentality, become the chief marching song of the Egyptian Army, and ''Black Lulu'', a genuine negro melody.&lt;ref name=&quot;:0&quot;&gt;&quot;Welcome Back from the Soudan.&quot; ''Black and White'' 07 November 1896, Saturday: 9 [of 35], Cols. 1a, 2b. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0004617/18961107/029/0009. Print p. 587.&lt;/ref&gt;{{rp|Col. 1a}}&lt;/blockquote&gt;

The menu for the dinner was printed in the paper because of the puns on North Africa:

* Hors d'oeuvre en Pyramide
* Consomé à la Pacha
* Purée à la Derviche
* Sole à la Noir et Blanc
* Filet de boeuf à la Sirdar Kitchener
* Galatine de Chapon à la Sphinx
* Faisan en Casserole à la Victoria
* Salade Égyptienne
* Souflé à la Khedive
* Bombe à la Dongola
* Petit fours Osman Digna
* Fromage du Nile
* Dessert à la Kartoum&lt;ref name=&quot;:0&quot; /&gt;{{rp|Col. 2b}}

===5 November 1896, Thursday===
Guy Fawkes Day

=== 1896 November 6, Friday ===
The Prince's Club ice-skating rink opened:&lt;blockquote&gt;The building specially constructed at Knightsbridge for the purposes of a skating rink will be formally opened to-morrow. Yesterday the managers invited their friends to a private view of the premises, erected on ground formerly occupied by an oilcloth factory and saw mill. Stoppani's system, which has been in operation for several years at the Palais de Glace and the Pole Nord in Paris, has been adopted, and the sheet of ice, 200ft. long by 50ft. wide, is as perfect as the most fastidious skater could desire. The building is light and airy, and is illuminated at night by arc lamps. M. Olivier Pichat has superintended the decoration of the walls, on which are painted scenes of the Thames, Nile, and Ganges. Among the committee of the club are the [[Social Victorians/People/Lansdowne|Marchioness of Lansdowne]], the [[Social Victorians/People/Londonderry|Marchioness of Londonderry]], the Marchioness of Granby, the [[Social Victorians/People/Minto|Earl and Countess of Minto]], [[Social Victorians/People/Carrington|Countess Carrington]], [[Social Victorians/People/Ribblesdale|Lady Ribblesdale]], Mrs. Cavendish Bentinck, [[Social Victorians/People/Asquith|Mrs. Asquith]], Sir William Hart-Dyke, Bart., M.P., Sir F. Astley Corbett, Bart., Sir [[Social Victorians/People/Arthur Sullivan|Arthur Sullivan]], Admiral Maxse, Messrs. Alfred Lyttelton, M.P., [[Social Victorians/People/Bourke|Algernon Bourke]], W. H. Grenfell, and W. F. Adams. Mr. W. W. Nightingale, who started at Southport in 1878 one of the first artificial ice rinks in the kingdom, is the secretary.&lt;ref&gt;&quot;Prince's Skating Club.&quot; London ''Daily Chronicle'' 06 November 1896, Friday: 9 [of 10], Col. 2c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0005049/18961106/125/0009. Print p. 9.&lt;/ref&gt;&lt;/blockquote&gt;

===23 November 1896, Monday===
&quot;''Little Eyolf'' (in William Archer's translation) opens at the Avenue Theatre, with Janet Achurch, Elizabeth Robins and Mrs Patrick Campbell in the three female roles. [[Social Victorians/People/George Bernard Shaw|[George Bernard] Shaw]]'s review of this production, with a cast including what he described as 'the three best yet undiscovered actresses of their generation', appeared in the ''Saturday Review'' on November 28&quot; (Gibbs 128).

'''1896 November 25, Wednesday''', Lord and Lady Burton hosted a party for Derby Day:&lt;blockquote&gt;Lord and Lady Burton's party at Rangemore for the Derby races included Lady Sarah Wilson, Lord and Lady Hindlip, Lady De Trafford, Sir George Chetwynd, Caytain [sic] Seymour Fortescue, Mr. and Mrs. [[Social Victorians/People/Bourke|Algernon Bourke]], Mr. and Mrs. Baillie, of Dochfour, Mr. and Mrs. Hwfa Williams, Mr. de Murrieta, and Mr. Menzies.&lt;ref&gt;&quot;What the 'World' Says.&quot; ''Sheffield Daily Telegraph'' 25 November 1896, Wednesday: 7 [of 10], Col. 6c [of 8]. ''British Newspaper Archive''  https://www.britishnewspaperarchive.co.uk/viewer/bl/0000250/18961125/067/0007. Print p. 7.&lt;/ref&gt;&lt;/blockquote&gt;

==December 1896==
===2 December 1896, Wednesday===
&lt;quote&gt;ACTORS' BENEVOLENT FUND. 

Mr LEOPOLD DE ROTHSCHILD presided over the sixth annual dinner of the Actors' Benevolent Fund, held at the Hotel Métrople on Wednesday evening. Actors have never shown themselves reluctant to assist in the cause of charity, and the profession was strongly represented on this festive occasion. The presence of Sir Henry Irving and other leading actors and managers gave special éclat to the festival, and the dinner proved in all respects to be one of the most successful held since the establishment of the fund. The claims of this admirable charity were pointedly put before the distinguished company by the chairman of the evening, Mr Leopold De Rothschild, who, in the course of his speech, mentioned that at the present time some seventy persons are in receipt of weekly grants from the fund. It is important to bear in mind that the direction of this excellent charity is in the hands of a committee of gentlemen of long and practical experience of the stage, and its modes of assistance are specially devised to meet the contingencies of theatrical life. The dinner took place in the Whitehall Rooms, and the following gentlemen accepted invitations:— [what follows is printed as a 3-column list which in this transcription reads across] Abrahams, Morris / Galer, Elliott / Millwood, W. / Alexander, Geore / Gatti, A. / Morgan, Ernest / Allen, W. E. / Gatti, S. / Mote, Henry / Armytage, H. T. / Gibson, Richard / Mundy, Luther / Asher, S. G. / Gleichen Count /  Nathan, L. / Baker, Col. W. H. / Griffith, Murray / Nathan, H. / Baker, E. / Grossmith, George / Nauheim, Carl / Baker, Ernest H. / Hague, Clarance / Nicholls, E. W. / Barnes, J. H. / Hallard, C. M. / Nicholson, G. J. / Benjamin, David H. / Hamilton, A. / Norman, Fredk. / Bell, H. / Hammond, G. J. / Ochs, James / Betty, H. / Harris, Herbert, A. / Ogilvie, R. A. / Bishop, Alfred / Harrison, Fredk. / Pallant, Walter / Blackley, Frank / Harvey, Edward / Paulton, H. / Blumenthal. M. A. / Heilbut, S. / Phipps, C. I. / Bolton, T. H. Henry, C. S. / Pittar, Parke M. / Bouverie, Hon. K. P. / Herring, George / Power, W. / Brown, G. V. / Herts, H. A. / Pyke, Joseph / Pull, E. H. / Hirsch, Adolph / Renwick, G. / Candy, George / Hollands, A. K. / Ritchie, Clement / Cawston, George / Honey, T. / Robinson, Fredk. / Chamberlain, Rich. / Holdsworth, J. / Roche, L. / Chinnery, H. J. / Howard Hon. K. / Samuel, Sir Saul / Chinnery, Ellis / Howson, Charles / Sarjeant, Arthur / Corgialegno, W. / Hurst, Joseph / Scudamore, F. A. / Cohn, M. / Irving, Sir Henry / Schmidt, H. / Cohen, A. L. / Joels, J. / Shade, A. R. / Cohen, Leonard / Joels, Woolf / Shaw, Sir E. M. / Cohen, J. / Johnson, Sam / Shone, R. V. / Coltson, C. L. / Kelly, C. A. / Silverthorne, E. C. / Cole, C. W. / King, A. P. / Skelly, Francis / Conguest [?], George / Kirchner, T. / Spalding, A. F. M. / Cooper, Frank / Langford, A. E. / Stern, James / Coster, Martin / Latham, T. / Stoker, George / Cruickshanks, C. / Lawrence, E. S. / Tapping, A. B. / Dam, H. J. W. / Lawrence, G. W. / Terry, Edward / Dawes, Richard / Lawrence, W. / Thorne, Thomas / Davidson, Louis / Ledger, Edward / Tidd, J. D. / Davies, Charles / Leign, J. H. / Tite, A. / Dornton, Charles / Lindo, Gabriel / Trevoe, F. M. / Duncannon Visent. [?] / Lockwood, Sir F. / Turner, G. H. / Durham, Frederick / Loewe, S. / Tyars, Frank / Edmonds, W. / Lowenfeld, H. / Villanueva, Dr. H. / Edmonds, W. jun. / Lukach, J. H. / Villiers. R. E. / Edwardes, George / Lumley, Ralph / Vincent, Henry / Ellis, Alfred / Macklin, F. H. / Waley, A. J. / Ellis, Granville / Maddick, E. D. / Waterlow, P. H. / Esmond. H. V. / Marks, H. H. / Wells, / Evill, Henry / Marsden, Peter / Westcott, W. / Fos, Raoul / Martin, Robert J. / Wingatr, H. L. / Forbes [?], Norman / Maskall, T. / Winter, M. / ?rece [?], J. De / Maunder. J. H. / Woolf, Lewis / Gabriel, Chas. S.M. / Mellish, Fuller / Wright, Rev. C. E. / Gabriel, S. / Middlemist, Dr. / Wyndham, Charles&lt;/quote&gt; (1896-12-05 Era)

===3 December 1896, Thursday===
[[Social Victorians/People/Horniman|Annie Horniman]]'s name was removed &quot;from the Roll of the Order&quot; of the Golden Dawn (Howe 136).

=== 9 December 1896, Wednesday ===
Christmas Dinner of the New Vagabonds, Bohemian Club:&lt;blockquote&gt;THE VAGABONDS' DINNER TO LORD ROBERTS, V.C.

S&lt;small&gt;PEECHES&lt;/small&gt; &lt;small&gt;BY&lt;/small&gt; L&lt;small&gt;ORD&lt;/small&gt; R&lt;small&gt;OBERTS&lt;/small&gt;, M&lt;small&gt;R&lt;/small&gt; J. K. J&lt;small&gt;EROME&lt;/small&gt;, D&lt;small&gt;R&lt;/small&gt; C&lt;small&gt;ONAN&lt;/small&gt; D&lt;small&gt;OYLE&lt;/small&gt;, AND S&lt;small&gt;IR&lt;/small&gt; J&lt;small&gt;OHN&lt;/small&gt; R. R&lt;small&gt;OBINSON&lt;/small&gt;.

&lt;big&gt;T&lt;/big&gt;&lt;small&gt;HE CHRISTMAS DINNER&lt;/small&gt; of the New Vagabonds, held on Dec. 9 at the King's Hall of the Holborn Restaurant, is one of the occasions on which this Bohemian club admits ladies. It may fairly be described an unqualified success. In the first place, one of the greatest of living Englishmen — Field Marshal Lord Roberts, V.C.— was courteous enough to come all the way from Ireland expressly to be its guest, and among those who accepted the invitation of the club to meet him were Sir John Robinson, manager of the ''Daily News''; Mr and Mrs Humphry Ward; and Messrs Frank Dicksee, R.A.; Henry Norman, of the ''Daily Chronicle''; Clement Shorter, of the ''Illustrated London News''; Sidney Low, of the ''St. James's Gazette''; J. K. Spender, of the ''Westminster Gazette''; while the editor of the ''Times'', who was kept away by a domestic bereavement, and Sir Edwin Arnold, of the ''Daily Telegraph'', Sir Douglas Straight, of the ''Pail Mall Gazette'', and Mr Alfred Harmsworth, of the ''Daily Mail'', and others, who were kept away by previous engagements, sent letters of regret. The chair was taken by Mr Jerome K. Jerome, and the vice-chairs by Messrs G. B. Burgin, Frankfort Moore, and Douglas Sladen, and amongst others [41, Col. 1a / 1b] present were Miss Helen Mathers, Miss Annie S. Swan, Miss Gertrude Kingston, Mrs Clement Shorter, Miss Marie Connor, Lady Cook, Miss Winifred Graham, Mr and Mrs George Grossmith, Sir James Linton, P.R.I., Mr Conan Doyle, Mr H. G. Wells, Mr Wm. Le Queux, Major Arthur Griffiths, editor of the ''World''; Mr J. G. Clarke, editor of the ''Christian World''; Mr Horace Cox, of the ''Field'' and ''Queen''; Mr F. H. Fisher, editor of the ''Literary World''; Mr Anthony Hope and his father, the rector of St. Bride's; Mr J. Penderel Brodhurst, editor of the ''St. James's Budget''; Mr M. H. Spielmann, editor of the ''Magazine of Art''; Mr John Coleman, author of &quot;The Duchess of Coolgardie;&quot; Mr Kenneth Grahame, author of &quot;The Golden Age;&quot; Mr and Mrs Coulson Kernahan, Mr and Mrs E. W. Hornung, Mr Silas K. Hocking, with Miss Hocking; Major and Mrs Nield, of Sydney, N.S.W.; Mr Neil Turner, Mr Robert Leighton, Mr and Mrs Robert Sauber, Col. Nisbet, Mr Alderman Treloar, Mr Frederick Whyte, Mr Percy White (author of &quot;Corruption&quot; and &quot;Mr Baily-Martin&quot;), Mr F. V. White, Mr and Mrs Will. Sharp, Hon. Mr and Mrs Gilbert Coleridge, Mr and Mrs Adam Black, Mr Reginald Cleaver, Mr I. N. Ford, London correspondent to the ''New York Tribune''; Mr McElway, editor of the ''Brooklyn Eagle''; Mr and Mrs A. S. Boyd, Mr Alfred Parsons, R.I.; Mr A. S. Hartrick, Miss Honner Morton, Mr Richard Le Gallienne, Mr Henry Blackburn, Mr Moncure D. Conway, Mr George Manville Fenn, Mr Zangwill, Mr Solomon, Mr Arthur Reed Ropes (the &quot;Adrian Ross&quot; of [41, Col. 1c / 2a] Gaiety librettos), Mr J. M. Dent, Miss Sarah Doudney, Mr James Greig, Mr G. P. R. Burgess, Mr W. W. Jacob, Dr Yorke Davies, Sir Henry and Lady Bergne, Mr A. C. Calmour, Mr Reginald Geard, Mrs T. P. O'Connor, Mr and Mrs W. B. Dalley, of Sydney; Mr H. W. Lowry, Mr H. Hartley and Mr Hart, of the Indian Exhibition; Mr Lewis Hind, editor of the ''Academy''; [[Social Victorians/People/Rook|Mr and Mrs Clarence Rook]]; Mr and Mrs Wilton Jones (Gertrude Warden), Mr Peter Keary, editor of ''Pearson's''; Mr C. F. Keary, Miss Norma Lorimer (author of &quot;A Sweet Disorder&quot;), Dr S. R. Keightley, author of &quot;The Crimson Sign;&quot; Mr and Mrs Max Pemberton, Mr Pett Ridge, Mr Roger Pocock, Mr Sullivan, [[Social Victorians/People/Todhunter|Dr and Mrs Todhunter]], Mr and Mrs C. N. Williamson, Mr and Mrs Morris Colles, Mr G. Herbert Thring, Mr and Mrs Joseph Hatton, Mr and Mrs Jopling Rowe (Louise Jopling), Mr Arthur Morrison, author of &quot;The Jago&quot; and &quot;Tales of Mean Streets.&quot; The dinner was held in the King's Hall of the Holborn Restaurant, and the company assembled in the Crown Room, the Entrance Hall, and the Lobby. One room could not contain everybody, for between six and seven hundred assembled to do honour to the guest of the evening, and not only was every inch of the floor used for dining tables, but all the galleries and the adjoining lobby. Grace was said by the chaplain, the Rev. St. Barbe S. Sladen; the company, as is usual at Vagabond dinners, having for the most part taken their places before the dinner was announced, and the guests, headed [41, Col. 2a / 2c] by Lord Roberts with Mrs Jerome and Mr Jerome with Mrs Humphry Ward, filed in. This, indeed, is almost the only trace of vagabondage about the club, unless one takes Vagabond and Bohemian in their most up-to-date sense. A Bohemian nowadays does not mean a man who has no bed to sleep on, or a man who wears no collar, but a spotted handkerchief in its place; it does not even imply drunkenness and disorderly conduct. A Bohemian in the London sense, is a person who does what he or she chooses, who conforms with conventionality and lives the ordinary Philistine life when it suits, and is equally prepared to go to a Covent Garden bal ''masqué''.

With one exception, there was not the slightest hitch over seating and arranging the vast assemblage — so skilfully and indefatigably had Messrs Moore and Burgin, who attended to these arrangements, done their work — to the exclusion of all their literary work for the best part of the past month. After the health of the Queen had been drunk with great enthusiasm, the chairman gave the toast of the evening, in a speech characterised by the happiest audacity, which completely captivated the audience, and no one more than its illustrious victim. ...

[43, Col. 1a] Mr Conway Dixon, who is Mr Hayden Coffin's understudy at Daly's, then sang, and his singing was characterised by its finish as well as by the beauty of the voice. Mr Dixon will undoubtedly be heard of in the future, for, in addition to the charm of his singing, he has a face and a figure just suited for a ''jeune premier''. He was vigorously encored, but it was getting on for eleven o'clock, and Mr George Grossmith was yet to come. Mr Grossmith gave his Irving and Beerbohm Tree speeches, and, fired by the brilliant and crowded audience, fairly excelled himself. Those who heard the improvisations which concluded Mr Beerbohm Tree's reply (in Mr Grossmith's imitation) are not likely to forget them. They were Tree to the life, and the great audience roared with laughter till tears ran down its cheeks. It also roared for an encore, but Mr Jerome was firm. There were so many people to be presented to Lord Roberts and Mrs Humphry Ward, that an adjournment had to be made that very moment to the Crown Room and the adjoining lobby, where the ''soirée'' which followed the dinner was to be held. Thus ended what the ''Westminster Gazette'' describes as the most brilliant gathering ever held at the Holborn Restaurant.&lt;ref&gt;&quot;The Vagabonds' Dinner to Lord Robert, V.C.&quot; The ''Queen'' 19 December 1896, Saturday: 41 [of 96], Col. 1a–3c – 43, Col. 1a. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002627/18961219/265/0041. Print pp. 1167–1169.&lt;/ref&gt;&lt;/blockquote&gt;

===16 December 1896, Wednesday===
Dolmetsch mentioned wanting to go to Florence. &quot;Dolmetsch was always keen to perform in Italy but was unable to afford such a trip on his own account. Horne, as usual, came to the rescue and used his influence to obtain a sponsor, but nowhere is the benefactor named. Although Dolmetsch was scrupulous in limiting his spending to the musical requirements of an undertaking, he was blissfully unconcerned as to the source of the funds so provided. All that occupied his thoughts at the moment was that at last he would be going to Italy — the land where culture pervaded everything and the very speech was music&quot; (Campbell ????).

===25 December 1896, Friday===
Christmas Day
===26 December 1896, Saturday===
Boxing Day
===26 December 1896, Saturday===
December Bank Holiday
==Works Cited==
*[1896-04-28 Aberdeen Journal] &quot;The Forbes-Erskine Marriage.&quot; Aberdeen Journal 28 April 1996, Tuesday: 5 [of 8], Col. 1c [of 8]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0000575/18960428/183/0004 (accessed 2019).
*[1896-05-23 Leamington Spa Courier] &quot;Personal Items.&quot; Leamington Spa Courier 23 May 1896, Saturday: 4 [of 8], Col. 6a [of 6]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0000319/18960523/025/0004 (accessed July 2019).
*[1896-06-12 The Courier] &quot;Fashionable Intelligence.” The Courier [in BNA Kent &amp; Sussex Courier] 12 June 1896, Friday: 6 [of 9], Col. 3b–4a [of 9]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0000483/18960612/175/0006 (accessed July 2019).
*[1896-06-30 Belfast News-Letter] &quot;Wedding of the Lord Lieutenant’s Daughter. A Brilliant Ceremony. From Our Own Correspondent. By Our Own Private Wire.&quot; Belfast News-Letter 30 June 1896, Tuesday: 5 [of 8], Col. 7a–9c [of 9]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0000038/18960630/023/0005 (accessed July 2019).
*[1896-07-14 Morning Post] &quot;The Queen’s Garden Party.” Morning Post 14 July 1896, Tuesday: 7, Col. 6a – 8, Col. 4a [of 12 pp and 7 cols]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18960714/099/0008 (accessed July 2019).
*[1896-12-05 Era] &quot;Actors' Benevolent Fund.&quot; The Era 5 December 1896, Saturday: 11 [of 32], Cols. 1a–3c [of 5]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0000053/18961205/016/0011 (accessed February 2020).
*Gibbs, Anthony Matthew. A Bernard Shaw Chronology. Author Chronologies, Ed. Norman Page. Houndmills, Basingstoke, Hampshire: Palgrave, 2001.
*Krout, Mary H., &quot;Women's Clubs,&quot; Chapter 9, A Looker-On in London. Rpt in Victorian London: Publications: Social Investigation/Journalism. Online: www.victorianlondon.org (August 2005).
*From a &quot;special announcement,&quot; quoted in &quot;Table Talk,&quot; The Literary World (3 Januray 1896), vol. 53, p. 13 [accessed 10 October 2009 in Google Books].
*&quot;Table Talk,&quot; The Literary World: Choice Readings from the Best New Books, and Critical Reviews, (London: James Clarke) 24 January 1896, vol. 53, p. 77, col. 1. (Accessed 9 October 2009 in Google Books.)
*&quot;Table Talk,&quot;The Literary World: Choice Readings from the Best New Books, and Critical Reviews, (London: James Clarke) 24 January 1896, vol. 53, p. 77, col. 2. (Accessed 9 October 2009 in Google Books.)
*&quot;Table Talk,&quot;The Literary World: Choice Readings from the Best New Books, and Critical Reviews, (London: James Clarke) 24 January 1896, vol. 53, p. 78, col. 1. (Accessed 9 October 2009 in Google Books.)
*&quot;Table Talk,&quot;The Literary World: Choice Readings from the Best New Books, and Critical Reviews, (London: James Clarke), 31 January 1896, vol. 53, p. 101, col. 3. (Accessed 9 October 2009 in Google Books.)
*&quot;Table Talk,&quot;The Literary World: Choice Readings from the Best New Books, and Critical Reviews, (London: James Clarke), 31 January 1896, vol. 53, p. 103, col. 2. (Accessed 9 October 2009 in Google Books.)
*&quot;Table Talk,&quot;The Literary World: Choice Readings from the Best New Books, and Critical Reviews, (London: James Clarke), 14 February 1896, vol. 53, p. 149, col. 1. (Accessed 9 October 2009 in Google Books.)
*&quot;Table Talk,&quot;The Literary World: Choice Readings from the Best New Books, and Critical Reviews, (London: James Clarke), 14 February 1896, vol. 53, p. 150, col. 1. (Accessed 9 October 2009 in Google Books.)
*&quot;Table Talk,&quot;The Literary World: Choice Readings from the Best New Books, and Critical Reviews, (London: James Clarke), 14 February 1896, vol. 53, p. 172, col. 3. (Accessed 9 October 2009 in Google Books.)
*&quot;Table Talk,&quot;The Literary World: Choice Readings from the Best New Books, and Critical Reviews, (London: James Clarke), 28 February 1896, vol. 53, p. 196, col. 2. (Accessed 9 October 2009 in Google Books.)
*&quot;Table Talk,&quot;The Literary World: Choice Readings from the Best New Books, and Critical Reviews, (London: James Clarke), 20 March 1896, vol. 53, p. 270, col. 1. (Accessed 10 October 2009 in Google Books.)
*&quot;Table Talk,&quot;The Literary World: Choice Readings from the Best New Books, and Critical Reviews, (London: James Clarke), 20 March 1896, vol. 53, p. 271, col. 2. (Accessed 10 October 2009 in Google Books.)
*&quot;Table Talk,&quot;The Literary World: Choice Readings from the Best New Books, and Critical Reviews, (London: James Clarke), 10 April 1896, vol. 53, p. 341, col. 2. (Accessed 13 October 2009 in Google Books.)
*&quot;Table Talk,&quot;The Literary World: Choice Readings from the Best New Books, and Critical Reviews, (London: James Clarke), 17 April 1896, vol. 53, p. 364, col. 2. (Accessed 13 October 2009 in Google Books.)
*&quot;Table Talk,&quot;The Literary World: Choice Readings from the Best New Books, and Critical Reviews, (London: James Clarke), 17 April 1896, vol. 53, p. 365, col. 1. (Accessed 13 October 2009 in Google Books.)
*&quot;Table Talk,&quot;The Literary World: Choice Readings from the Best New Books, and Critical Reviews, (London: James Clarke), 17 April 1896, vol. 53, p. 366, col. 2. (Accessed 13 October 2009 in Google Books.)
*&quot;Table Talk,&quot;The Literary World: Choice Readings from the Best New Books, and Critical Reviews, (London: James Clarke), 1 May 1896, vol. 53, p. 412, cols. 1-2. (Accessed 13 October 2009 in Google Books.)
*&quot;Table Talk,&quot;The Literary World: Choice Readings from the Best New Books, and Critical Reviews, (London: James Clarke), 1 May 1896, vol. 53, p. 412, col. 2. (Accessed 13 October 2009 in Google Books.)
*&quot;Table Talk,&quot;The Literary World: Choice Readings from the Best New Books, and Critical Reviews, (London: James Clarke), 1 May 1896, vol. 53, p. 415, col. 2. (Accessed 13 October 2009 in Google Books.)
*&quot;Table Talk,&quot;The Literary World: Choice Readings from the Best New Books, and Critical Reviews, (London: James Clarke), 1 May 1896, vol. 53, p. 415, col. 1. (Accessed 13 October 2009 in Google Books.)
*&quot;Table Talk,&quot;The Literary World: Choice Readings from the Best New Books, and Critical Reviews, (London: James Clarke), 8 May 1896, vol. 53, p. 436, col. 2. (Accessed 13 October 2009 in Google Books.)
*&quot;Table Talk,&quot;The Literary World: Choice Readings from the Best New Books, and Critical Reviews, (London: James Clarke), 15 May 1896, vol. 53, p. 461, col. 2. (Accessed 13 October 2009 in Google Books.)
*&quot;Table Talk,&quot;The Literary World: Choice Readings from the Best New Books, and Critical Reviews, (London: James Clarke), 15 May 1896, vol. 53, p. 462, col. 3. (Accessed 13 October 2009 in Google Books.)
*&quot;Table Talk,&quot;The Literary World: Choice Readings from the Best New Books, and Critical Reviews, (London: James Clarke), 22 May 1896, vol. 53, p. 484, cols. 1-2. (Accessed 13 October 2009 in Google Books.)
*&quot;Table Talk,&quot;The Literary World: Choice Readings from the Best New Books, and Critical Reviews, (London: James Clarke), 5 June 1896, vol. 53, p. 532, col. 1. (Accessed 13 October 2009 in Google Books.)
*&quot;Table Talk,&quot;The Literary World: Choice Readings from the Best New Books, and Critical Reviews, (London: James Clarke), 12 June 1896, vol. 53, p. 556, cols. 1-2. (Accessed 13 October 2009 in Google Books.)
*&quot;Table Talk,&quot;The Literary World: Choice Readings from the Best New Books, and Critical Reviews, (London: James Clarke), 12 June 1896, vol. 53, p. 556, col. 2. (Accessed 13 October 2009 in Google Books.)
*&quot;Table Talk,&quot;The Literary World: Choice Readings from the Best New Books, and Critical Reviews, (London: James Clarke), 19 June 1896, vol. 53, p. 581, cols. 1-3. (Accessed 13 October 2009 in Google Books.)
*&quot;Table Talk,&quot;The Literary World: Choice Readings from the Best New Books, and Critical Reviews, (London: James Clarke), 26 June 1896, vol. 53, p. 604, cols. 1-2. (Accessed 14 October 2009 in Google Books.)

== Footnotes ==
&lt;references /&gt;</text>
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  <page>
    <title>Social Victorians/Timeline/1899</title>
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      <contributor>
        <username>Scogdill</username>
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      <comment>/* January 1899 */</comment>
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      <text bytes="24590" sha1="nwbwv3toqxmykwudole5ecmyo5mvjoz" xml:space="preserve">[[Social Victorians/Timeline/1850s | 1850s]] [[Social Victorians/Timeline/1860s | 1860s]] [[Social Victorians/Timeline/1870s | 1870s]] [[Social Victorians/Timeline/1880s | 1880s Headlines]] [[Social Victorians/Timeline/1890s | 1890s Headlines]] [[Social Victorians/Timeline/1890 | 1890]] [[Social Victorians/Timeline/1891 | 1891]] [[Social Victorians/Timeline/1892 | 1892]] [[Social Victorians/Timeline/1893 | 1893]] [[Social Victorians/Timeline/1894 | 1894]] [[Social Victorians/Timeline/1895 | 1895]] [[Social Victorians/Timeline/1896 | 1896]] [[Social Victorians/Timeline/1897 | 1897]] [[Social Victorians/Timeline/1898 | 1898]] 1899 [[Social Victorians/Timeline/1900s | 1900s]] [[Social Victorians/Timeline/1910s | 1910s]] [[Social Victorians/Timeline/1920s-30s|1920s-30s]]

==Sometime in 1899==
In 1899 the International Council of Women was held in London.

The year 1899 was the celebration of Victoria's 80th birthday.

==January 1899==
===1 January 1899, Sunday, New Year's Day===

=== 11 January 1899, Wednesday ===
Princess Henry of Battenberg

===25 January 1899, Wednesday===
[[Social Victorians/People/Muriel Wilson|Muriel Wilson]] at the Holderness Hunt Ball:&lt;blockquote&gt;The Assembly Rooms at Beverley were on Tuesday the scene of much gaiety, the occasion being the bull in connection with the Holderness Hunt. The company numbered over 300, the county families being well represented. The spectacle was a brilliant one. The music was supplied by Wolfe's White Hungarian Band. The arrangements were chiefly in the hands of Mr. Clive Wilson and Mr. Harrison Broadley, whose efforts were certainly most successful. Among the guests were the Countess of Huntingdon, Lady Clementine Walsh, the Hon. Mary Hawke, [[Social Victorians/People/Arthur Stanley Wilson|Mr. and Mrs. Arthur Wilson]], Mrs. Lycett Green, Lady de Ramsey, Lady Beatrice Taylor, the Hon. Alexander Fellowes, Mr. and Mrs. Kenneth Wilson, Miss Muriel Wilson, Colonel and Mrs. Armytage, Colonel Haworth Booth and party, Colonel and Mrs. Grimston, Commander Bethell, M.P., the Hon. Dudley Majoribanks, Mr. Cecil Wellesley, Mr. and Mrs. Calverley-Rudston, Lord and Lady Herries and the Hon. Misses Maxwell, Mr. and Mrs. Gunter, Mr. Marco Wilson, Mr. and Mrs. Hall Watt, Captain Battine, Miss Parsons, Mr. and Mrs. George Duncombe, Mr. J. Hotham, Miss Bethell, Miss B. Walker, Mr. J. J. Harrison, Colonel Burstall, Mr. and Mrs. Harrison Broadley and party, Mr. and Mrs. Roland Heathcote Hacker, [[Social Victorians/People/Keppel|The Hon [sic] George Keppel]], Mr. Arthur Portman, the Hon. Harold Fitzclarence, Miss Daye Baker, Miss Helen Bower, Mr. and Mrs. Ellershaw, Major Macmullen, Sir Spencer Maryon Wilson, Mr. Prince, Sir Charles Hartopp, Mr. and Mrs. Kerr, Mr. Walter Burns, Mr. and Mrs. Wade and party, Mr. Harold Brassey, Miss Joan Wilson, Miss Enid Wilson, Mr. Henry Wilson, Lord Acheson, and many others. Dancing was kept up till the early hours of the morning.

The meet of the hounds next day was at the Beverley Grand Stand. Owing to the frost it was late before the pack arrived from the kennels. Many hundreds of people were present, tempted by the fine, crisp, bracing morning. Leaving Westwood the hounds proceeded to Broadedge Farm, where a fox was found, and he was hunted on to Bentley and Skidky to Cottingham. The company, having seen the throw-off, returned home.&lt;ref&gt;&quot;Holderness Hunt Ball.&quot; ''Yorkshire Herald'' 29 January 1899, Saturday: 10 [of 18], Col. 6c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000500/18990128/086/0010 (accessed July 2019).&lt;/ref&gt;&lt;/blockquote&gt;

==February 1899==
==March 1899==
===31 March 1899, Friday===
Good Friday

==April 1899==
===2 April 1899, Sunday===
Easter Sunday
==May 1899==
===2 May 1899, Tuesday===
The [[Social Victorians/People/Albert Edward, Prince of Wales|Prince of Wales]] visited Ruthin Castle, in Wales, for the Chester races. Ruthin Castle was the home of Cornwallis-West, and [[Social Victorians/People/Muriel Wilson|Muriel Wilson]] was a part of the house party that made up the reception for the prince (1899-05-03 Daily Telegraph). [[Social Victorians/People/Churchill|Jennie Spencer-Churchill]], Lady Randolph Churchill was there, as was [[Social Victorians/People/Cornwallis-West|George Cornwallis-West]]; their relationship was quite controversial and the Prince, normally quite warm to Jennie Churchill, was cold ().

===8 May 1899, Monday===
[[Social Victorians/People/Muriel Wilson|Muriel Wilson]] and [[Social Victorians/People/Arthur Stanley Wilson|Mrs. Arthur Wilson]] were at the opening of the Royal Opera, Covent Garden, Wagner’s ''Lohengrin''. The [[Social Victorians/People/Albert Edward, Prince of Wales|Prince of Wales]] was there, as were a number of notable celebrities (1899-05-13 Penny Illustrated Paper)

===15 May 1899, Monday===
&lt;quote&gt;Ipswich High School.— On Monday, May 15, the school was honoured by a visit of H.R.H. the Princess Louise, who came, faithful [386A/B] to a long-standing promise, to give away the prizes and certificates adjudged on the results of the Oxford and Cambridge Joint Board and the Cambridge Local Examinations of last year. Her Royal Highness was accompanied by the Marquis of Lome, K.G., and attended by Mr. E. B. Phipps, Assistant-Secretary to the G.P.D.S. Co., who acted as equerry in the place of Col. Collins. The Princess was received, on her arrival in Ipswich, by Mr. Bousfield, Chairman of the Council of the G P.D.S. Co., and by Lady Digby, Miss Gurney, Mr. Eve, and Mr. Buxton, members of the Council. The visit was of a semi-private character, and hence there was no official reception by the Mayor and Corporation of Ipswich. The High School was reached at two o'clock, and here the Princess was received by members of the Local Committee, with whom were Miss Youngman, the late Headmistress, Miss Kennett, the present Headmistress, and Mr. McDowall, Secretary to the G.P.D.S. Co. The girls, two hundred in number, were drawn up on either side of the Lower Hall, and presented an exceedingly bright appearance in their white dresses and sashes of crimson, the school colour. The Princess graciously consented to walk up the hall between the lines of girls and to receive a bouquet from Janet Stewaid, of Form II , the daughter of Mr. W. Steward, a member of the Local Committee. She then made the tour of the class-rooms, escorted by Mr. Bousfield, Miss Youngman, and Miss Kennett. Luncheon was served in the Upper Hall at 2.15. The number of invited guests included, in addition to those already mentioned, the Mayor of Ipswich, the Marquis of Bristol, Lord Lieutenant of the county, Sir Charles Dalrymple, M.P., Major Bond (in command of the Volunteer guard of honour), and Mr. John Farmer. At four o'clock the party adjourned to the Council Chamber of the Town Hall, where, after several songs by the pupils, under the conductorship of Mr. Farmer, and an exhibition of drill, Mr. Bousfield made a short speech, in which he explained the aims and ideals of the schools of the G.P.D.S. Co., and expressed the gratitude of the Council to Miss Youngman, who for twenty-one years had watched over and guided the development of the Ipswich School with so much energy and judgment. The Princess then distributed the prizes, after which a vote of thanks to her was proposed by Mr. Bousfield, and seconded by the Mayor. Lord Bristol also spoke to the resolution, which was unanimously carried. The Marquis of Lome having briefly responded, the proceedings were closed by the singing of &quot;Auld Lang Syne,&quot; the hymn &quot;O God, our help in ages past,&quot; and &quot;God save the Queen.&quot; The bouquet given to the Princess at the Town Hall was presented by Sybil Casley, of the Kindergarten, and the programme was handed to her Royal Highness by Judith Becher, of the Transition Class. At the conclusion of the afternoon's proceedings the Princess and her party partook of tea in the Mayor's parlour at the Town Hall, and left for London at six o'clock.&lt;/quote&gt; (The Journal of Education. Vol. 21, New Series (January to December 1899). Page 386 [June 1899].Cols. A-B. Google Books: http://books.google.com/books?id=jZFIAAAAYAAJ).

===20 May 1899, Saturday===
Emma Nevada at the Crystal Palace (https://www.msu.edu/~graye/emma/chronolo.html).

===27 May 1899, Saturday===
Emma Nevada at the Crystal Palace (https://www.msu.edu/~graye/emma/chronolo.html).

===31 May 1899, Wednesday===
Derby Day at Epsom Downs, so Luise Friederike Auguste Montagu  Duchess of Devonshire, hosted a ball that night?

==June 1899==
Summer 1899: [[Social Victorians/People/William Butler Yeats|William Butler Yeats]] summered with Lady Gregory at Coole Park 1897-1917 or so, until WBY bought the Tower at Ballylee. (I got this from Wade?).
===21 June 1899, Wednesday===
The ''Illustrated London News'' reported on a bazaar to raise money for the Charing Cross Hospital. Many celebrities were present. People whose portraits were drawn were Princess Henry of Pless, Marchioness of Granby, Duchess of Marlborough, [[Social Victorians/People/Muriel Wilson|Muriel Wilson]], Countess of Westmorland, Duchess of Sutherland, Mr. Burdett-Coutts, M.P., and mentioned in the story were the following: Princess Louise, Duchesses of Westminster, Portland, Abercorn, Sutherland, and Marlborough; Countesses of Westmorland, Cadogan, Chesterfield, Mrs. Choate (wife of American ambassador; Lord Glenesk (https://www.britishnewspaperarchive.co.uk/viewer/bl/0001578/18990701/054/0017). 

Another report: &lt;quote&gt;The Charing Cross Hospital Bazaar at the Albert Hall was great success. It was a very hot day, but the aristocracy were present in large numbers. There were many exquisite toilettes. [[Social Victorians/People/Arthur Stanley Wilson|Mrs Arthur Wilson]] wore black, with cream embroidery, covered with sequinned net; corn-coloured net toque, with sprays of gold and feathers. Mrs Kenneth Wilson’s costume was palest grey voile, tight fitting, with a lace yoke, and a white tulle hat with osprey in front. Mrs Menzies and [[Social Victorians/People/Muriel Wilson|Miss Muriel Wilson]] were at “Flowerland,&quot; with Ladies Marlborough, Westmoreland. Mar and Kellie, Chelsea, Craven, Juliette Lowther, and Norreys. Miss Muriel Wilson was in white silk and crepe gown, with bands of coarse cream lace and open neck, transparent sleeves of lace and crepe. She wore a large black chip hat of tulle fastened at the chin, and carried a large-handled basket of roses, tied with heliotrope satin ribbon. The three sisters from Warter Priory were at the refreshment stall attired in grey dresses, while fischus, and big white mob caps.&lt;/quote&gt; (1899-06-24 Beverley and East Riding Recorder)

===26 June 1899, Monday===
There was apparently a regular celebration of [[Social Victorians/People/Arthur Collins|Arthur Collins]]' birthday, 26 June, by Bret Harte, George Du Maurier, Arthur Sullivan, Alfred Cellier, Arthur Blunt, and John Hare (Nissen, Axel. Brent Harte: Prince and Pauper: 239. [http://books.google.com/books?id=WEDewmUnapcC]). Choosing 1885–1902 as the dates because those apparently are the dates of the close relationship between Harte and Collins, ending in Harte's death in 1902.

According to the ''Morning Post'' for 27 June 1899, Mr. Schreiber danced in the Gainesborough Quadrille at the annual Royal Caledonian Ball on Monday, 26 June 1899.&lt;blockquote&gt;THE ROYAL CALEDONIAN BALL.

The annual Ball, held last night at the Whitehall Rooms, Hôtel Métropole, in aid of the funds of the Royal Caledonian Asylum and the Royal Scottish Hospital proved a great success, nearly nine hundred persons being present. The interest ot the evening centred in the eightsome reels and the fancy quadrilles. The former were arranged by the Hon. Mrs. Baillie of Dochfour (who unfortunately through indisposition was prevented attending), and were danced as follows: 

1.

The Marquis of Tullibardine, D.S.O., Royal Horse Guards, and the Countess of Mar and Kellie.

Viscount Fincastle, V.C., 16th Lancers, and Laily Helen Stewart Murray.

The Hon. Alexander Ruthven, V.C., Cameron Highlanders, Miss Katharine Ramsay.

Mr. M'Neil, Seaforth Highlanders, and Miss Sibyl Murray.

2.

The Earl of Mar and Kellie and Lady Helen Graham.

Mr. Alastair Murray, younger, of Lochcarron, and Lady Hilda Keith-Falconer.

The Hon. Alexander Fraser and the Hon. Cecily Drummond.

Mr. M'Lean, Scots Guards, and Miss Baillie.

3.

Lord Lovat and Lady Grizel Cochrane.

Captain Greenhill-Gardyne, Gordon Highlanders, and the Hon. Ethel Fraser.

Mr. Baillie, Seaforth Highlanders, and the Hon. Daisy Fraser.

The Hon. Hugh Fraser, Scots Guards, and Miss Marvel MacGregor.

4.

Mr. Cameron, younger, of Lochiel, Grenadier Guards, and the Countess of Cromartie.

Mr. NIall Campbell and Miss Edith Chaplin.

Mr. Douglas Brodie and Miss Elspeth Campbell.

Mr. Alastair MacGregor of MacGregor and Miss Vere Brodie.

5.

Mr. Ramsay, Black Watch, and Lady Margaret Crichton Stewart.

Mr. M'Ray, Black Watch, and Lady Edith Montgomerie.

Mr. Matheson, Coldstream Guards, and the Hon. Beatrice Dalrymple.

The Hon. Kenneth Campbell and the Hon. Gwendolen Maxwell.

The gentlemen wore Highland dress, while the ladies were in white gowns with sashes formed of their respective tartans, the badges of their clans appearing in their hair and on their dresses.

The Countess of Hopetoun, one of the most energetic of the ladies patronesses, was responsible for the two fancy quadrilles. The undernamed took part in

THE ROMNEY QUADRILLE.

Viscount Crichton, Royal Horse Guards, and Countess Hopetoun.

Mr. C. C. de Crespigny, 2nd Life Guards, and Lady Constance Scott.

Hon. Claud Drummond Willoughby, Coldstream Guards, and Lady Florence Astley.

Hon. Gerald Ward, Ist Life Guards, and Lady Beatrice Herbert.

Mr. Tryon, Grenadier Guards, and Lady Mary Drummond Willoughby.

The Earl of Kerry, Grenadier Guards, and Lady Marjorie Carrington.

Mr. Trotter, Grenadier Guards, and the Hon Alice Grosvenor.

Mr. Hamilton, Grenadier Guards, and Miss Muriel White.

THE GAINSBOROUGH QUADRILLE.

Major Gordon-Gilmour, Grenadier Guards, and Lady Alice Shaw Stewart.

The Hon. Raymond de Montmorency, V.U., 21st Lancers, and Lady Sybil Primrose

Captain Brinton, 2nd Life Guards, and Lady Edith Villiers.

[[Social Victorians/People/Schreiber|Captain Schreiber]], 1st Life Guards, and Hon. Maud de Moleyna.[?]

Captain Heneage, Grenadier Guards, and Miss Long.

Mr. Stirling, Coldstream Guards, and Miss Cotton Jodrell.

Captain Green-Wilkinson, Rifle Brigade, and Miss Sibell Chaplin.

Mr. Vandeleur, D.S.O., Scots Guards, and Miss Muriel Chaplin.

The officers were in uniform, and their partners wera attired in gowns of white mousseline-de-soie over silk slips of different colours, those in the Romney quadrille wearing lace fichus, and those in the Gainsborough quadrille chiffon scarves, and all had their hair threaded with coloured chiffon or ribbon to match their sashes or scarves.

At eleven o'clock a procession was formed, headed by the Pipers, and the Duke of Atholl, Treasurer of the ball, and those taking part in the Reels and Quadrilles entered the ball-room, dancing immediately commencing to Herr Iff's orchestra. The Ladies Patronesses present included the Duchess of Buccleuch, the Duchess of Atholl, the Duchess of Montrose, the Marchioness of Bute, the Countess of Mar and Kellie, Mary Countess of Mar and Kellie, the Counters of Selkirk, the Countess of Dundonald, the Countess of Ancaster, Viscountess Strathallan, Viscountess Dalrymple, Lady Anne Murray, Lady Eleanor Brodie, Lady Herries, Lady Sinclair, Lady Middleton, Lady Ramsay of Bamff, Lady Maxwell of Monreith, Lady Macpherson Grant, Mrs. Munro, and Mrs. Murray of Polmaise. Before dancing became general the boys and girls of the Asylum, headed by their Pipers and band, marched round the ball-room. Much credit is due to the President and Vice-President of the ball, the Duke of Atholl and Marquis of Tullibardine, for their efforts in the cause of charity.&lt;ref&gt;&quot;The Royal Caledonian Ball.&quot; ''Morning Post'' 27 June 1899, Tuesday: 7 [of 12], Col. 7b–c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18990627/063/0007.&lt;/ref&gt;&lt;/blockquote&gt;

==July 1899==
July 1899, Emma Nevada sang for Queen Victoria at Osborne House (https://www.msu.edu/~graye/emma/chronolo.html).
===4 July 1899, Tuesday===
[[Social Victorians/People/Muriel Wilson|Muriel Wilson]] was at a garden party hosted by Lady Rothschild and Mrs. Leopold Rothschild after the end of the Women’s Congress: &lt;quote&gt;Not the least interesting features of the Women's Congress have been the social entertainments. On Tuesday, after the final sessions had been held, Lady Rothschild and Mrs. Leopold Rothschild invited the delegates to a garden party at Gunnersbury Park. Special trains conveyed the guests to Mill-hill Station. The guests were received by Lady and Mrs. Leopold Rothschild, the former in black lace over pale mauve silk, and the latter in blue and white muslin, and Lady Battersea, in a charming light grey and white frock and a little yellow bonnet that suited her to perfection. By five o'clock the grounds were crowded, and among the well known people to be seen walking about there were Lady Battersea, Lady Harcourt, Mr. and Lady Clementine Walsh, the latter in pale grey and white, Lord and Lady Gosford, and with them Lady Aldra Acheson; Lady Alice Stanley, in rose-pink; Mrs. Rolands [Ronalds?] in white; Mr. and Lady Barbara Smith, and Mrs. Maguire in a becoming frock of pale yellow. Lady Kilmorey, who came quite early in the afternoon wore white muslin, with a large straw hat with roses; Lady Chelsea, in mauve; and Miss Muriel Wilson, in a lovely dress of pale blue, with transparent lace sleeves, and large white hat with roses, looked particularly well. Lady Blandford was in pale grey; Mrs Arthur Sassoon was also in grey. Among some of the late arrivals were Lord and Lady Crewe, with Lady Annabel Milnes, Lady Crewe in a pretty white dress. From the terraces the scene was magnificent. The park stretched over a velvety green lawn, dotted with beds of of [sic] exquisite flowers; bridged over with roses, that gave them the appearance of great baskets. A fountain, with pond lilies nestling in its shadowy spots, lay at the left. At another side of the lawn was an artificial lake, with boats and boatmen at the disposal of visitors. A string band at the bank and a second marquee afforded rest, music, and refreshment to those who preferred to remain in a little world of their own. A three band played lively airs for the American bicycle polo team, who gave an exhibition of their skill on the lawn. And both circus and stage were utilised for the afternoon amusements.&lt;/quote&gt; (1899-07-08 Bridgnorth Journal)

=== 15 July 1899, Saturday ===
&lt;blockquote&gt;Lord Kenyon, Mr. Schomburg McDonnell (Lord Salisbury's private secretary), Colonel Dawson, Mr. H. Ridgway, Lady Gerard, the Hon. Miss Gerard, Mrs. Hartmann, and [[Social Victorians/People/De Jancourt|Mdlle Jancourt]] arrived at Broughton Castle on Saturday night on a weekend visit to Lord and Lady Algernon Gordon Lennox.&lt;ref&gt;&quot;Local Town and Country Notes.&quot; ''Banbury Guardian'' 20 July 1899 Thursday: 8 [of 8], Col. 1b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001523/18990720/116/0008.&lt;/ref&gt;&lt;/blockquote&gt;

==August 1899==
===28 August 1899===
Summer Bank Holiday

==September 1899==
==October 1899==
===31 October 1899, Tuesday===
Halloween.
==November 1899==
===5 November 1899, Sunday===
Guy Fawkes Day

=== 23 November 1899, Thursday ===
Captain C.S. Schreiber attended a Royal and Imperial Dinner Party at Windsor Castle:&lt;blockquote&gt;The Imperial and Royal dinner party included their Imperial Majesties the German Emperor and Empress, their Royal Highnesses the Prince and Princess of Wales, the Duke and Duchess of Connaught, Prince and Princess Christian of Schleswig-Holstein, her Royal Highness Princess Louise Marchioness of Lorne and Marquis of Lorne, their Royal Highnesses Princess Henry of Battenberg, Princess Victoria of Wales, his Serene Highness and her Grand Ducal Highness Prince and Princess Louis of Battenberg, his Highness Prince Albert of Schleswig-Holstein, the Duchess of Buccleuch, Mistress of the Robes; Fraulein von Gersdorff, the Countess Stollberg, the Dowager Lady Ampthill, the Danish Minster, Mons. de Bille; the Belgian Minister, Baron Whettnall; the Portuguese Minister, Mons. de Soveral ; the Greek Chargé d'Affaires, Mons. Metaxas; the Lord Steward, the Lord Chamberlain, his Excellency Count von Bülow, his Excellency Count Eulenburg, his Excellency General von Plessen, Lord Suffield, the Right Hon. Sir Frank Lascelles, Lord Colville of Culross, Sir Francis Knollys, Vice-Admiral Sir John Fullerton, Major-General Swaine, commanding North-Western District, and Signor de Martino.
The band of the Royal Artillery, conducted by Cavaliere L. Zavertal, played the following selection of music in St. George's Hall in the evening:

March from tlie Suite &quot;Sylvia&quot; Delibes.

Vorspiel &quot;Das Heimchen am Herd&quot; Goldmark.

Three Dances from the music to &quot;Henry VIII.&quot; Ed. German.

1, Morris; 2. Shepherd's; 3. Torch.

(a) Adagietto from the Suite &quot;L'Arlésienae&quot; Bizet.

(b) &quot;La Chaise-à-Porteurs&quot; Chaminade.

Ballet Music, &quot;Der Damon&quot; Rubinstein.

&quot;Abendruhe &quot; Loeschhorn.

Angelus from the Suite &quot;Scènes Pittoresques&quot; Massenet.

Overture, &quot;Cleopatra&quot; Mancinelli.

York March.

Her Majesty's guests invited to dine at the Castle, together with the Ladies and Gentlemen of the Royal Household and the suites in attendance on the Queen's Imperial and Royal guests, had the honour of joining the Royal Circle in St. George's Hall.

The following had the honour ot receiving invitations to be present: Lord and Lady Esher, Lady Edwards, Lady and Miss Victoria Bigge, Mr. and Lady Emily Van de Weyer, Miss Loch, Miss Emily Loch, and Miss Catherine Loch, Sir Walter Parratt, the Head Master at Eton and Mrs. Warre, the Provost of Eton and Miss Hornby, Mr. E. C. Austen Leigh, M.A., Mr. A. C. Benson, M.A., Baron and Baroness Campbell von Laurentz, Lieutenant-Colonel C. N. Miles, Captain G. F. Milner, and [[Social Victorians/People/Schreiber|Captain C. S. Schreiber]], 1st Life Guards; Major the Hon. J. St. Aubyn, Captain the Hon. W. Cavendish, and Lieutenant and Adjutant E. Gascoigne, 1st Battalion Grenadier Guards; and Colonel Swinfen, Major Bolton, and Lieutenant-Colonel Tighe, Military Knights of Windsor.

The Queen did not attend the dinner or the concert in St. George's Hall, owing to having so recently received the news of the death of her Grand Ducal Highness the Princess of Leiningen, her Majesty's niece.&lt;ref&gt;&quot;Court Circular.&quot; ''Morning Post'' 24 November 1899, Friday: 5 [of 10], Col. 5a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18991124/045/0005.&lt;/ref&gt;&lt;/blockquote&gt;

==December 1899==
===25 December 1899, Monday===
Christmas Day
===26 December 1899, Tuesday===
Boxing Day
===30 December 1899, Saturday===
[[Social Victorians/People/Arthur Conan Doyle| Arthur Conan Doyle]]'s New Year's Eve party at Hindhead, [[Social Victorians/Haslemere | Haslemere]].

==Works Cited==
*[1899-05-03 Daily Telegraph] &quot;Prince of Wales at Chester Races. Visit to Ruthin Castle. From Our Own Correspondent.&quot; Daily Telegraph &amp; Courier 3 May 1899, Wednesday: 10 [of 16], Col. 3a–c [of 7]. British Newspaper Archive (accessed July 2019).
*[1899-05-13 Penny Illustrated Paper] &quot;At the Opening of the Royal Opera, Covent Garden.&quot; Penny Illustrated Paper 13 May 1899, Saturday: 2 [of 16], Col. 2a, 3a [of 4]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0000693/18990513/008/0002 (accessed July 2019).
*[1899-06-24 Beverley and East Riding Recorder] &quot;East Riding Ladies at a London Bazaar.&quot; Beverley and East Riding Recorder 24 June 1899, Saturday: 5 [of 8], Col. 6b [of 6]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0001565/18990624/074/0005 (accessed July 2019).
*[1899-07-08 Bridgnorth Journal] &quot;Women’s Congress Ended.&quot; Bridgnorth Journal 8 July 1899, Saturday: 8 [of 8], Col. 3c [of 6]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0001961/18990708/138/0008 (accessed July 2019).
*Gray, Eugene F. &quot;Chronology of Events in the Life of Emma Nevada.&quot; Emma Nevada: An American Diva. https://www.msu.edu/~graye/emma/chronolo.html (retrieved 14 April 2010).

== Footnotes ==
&lt;references /&gt;</text>
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      <text bytes="24593" sha1="3qrrgvqyr3o5632b59m9tuu3to683h2" xml:space="preserve">[[Social Victorians/Timeline/1850s | 1850s]] [[Social Victorians/Timeline/1860s | 1860s]] [[Social Victorians/Timeline/1870s | 1870s]] [[Social Victorians/Timeline/1880s | 1880s Headlines]] [[Social Victorians/Timeline/1890s | 1890s Headlines]] [[Social Victorians/Timeline/1890 | 1890]] [[Social Victorians/Timeline/1891 | 1891]] [[Social Victorians/Timeline/1892 | 1892]] [[Social Victorians/Timeline/1893 | 1893]] [[Social Victorians/Timeline/1894 | 1894]] [[Social Victorians/Timeline/1895 | 1895]] [[Social Victorians/Timeline/1896 | 1896]] [[Social Victorians/Timeline/1897 | 1897]] [[Social Victorians/Timeline/1898 | 1898]] 1899 [[Social Victorians/Timeline/1900s | 1900s]] [[Social Victorians/Timeline/1910s | 1910s]] [[Social Victorians/Timeline/1920s-30s|1920s-30s]]

==Sometime in 1899==
In 1899 the International Council of Women was held in London.

The year 1899 was the celebration of Victoria's 80th birthday.

==January 1899==
===1 January 1899, Sunday, New Year's Day===

=== 11 January 1899, Wednesday ===
Princess Henry of Battenberg

===25 January 1899, Wednesday===
[[Social Victorians/People/Muriel Wilson|Muriel Wilson]] at the Holderness Hunt Ball:&lt;blockquote&gt;The Assembly Rooms at Beverley were on Tuesday the scene of much gaiety, the occasion being the bull in connection with the Holderness Hunt. The company numbered over 300, the county families being well represented. The spectacle was a brilliant one. The music was supplied by Wolfe's White Hungarian Band. The arrangements were chiefly in the hands of Mr. Clive Wilson and Mr. Harrison Broadley, whose efforts were certainly most successful. Among the guests were the Countess of Huntingdon, Lady Clementine Walsh, the Hon. Mary Hawke, [[Social Victorians/People/Arthur Stanley Wilson|Mr. and Mrs. Arthur Wilson]], Mrs. Lycett Green, Lady de Ramsey, Lady Beatrice Taylor, the Hon. Alexander Fellowes, Mr. and Mrs. Kenneth Wilson, Miss Muriel Wilson, Colonel and Mrs. Armytage, Colonel Haworth Booth and party, Colonel and Mrs. Grimston, Commander Bethell, M.P., the Hon. Dudley Majoribanks, Mr. Cecil Wellesley, Mr. and Mrs. Calverley-Rudston, Lord and Lady Herries and the Hon. Misses Maxwell, Mr. and Mrs. Gunter, Mr. Marco Wilson, Mr. and Mrs. Hall Watt, Captain Battine, Miss Parsons, Mr. and Mrs. George Duncombe, Mr. J. Hotham, Miss Bethell, Miss B. Walker, Mr. J. J. Harrison, Colonel Burstall, Mr. and Mrs. Harrison Broadley and party, Mr. and Mrs. Roland Heathcote Hacker, [[Social Victorians/People/Keppel|The Hon [sic] George Keppel]], Mr. Arthur Portman, the Hon. Harold Fitzclarence, Miss Daye Baker, Miss Helen Bower, Mr. and Mrs. Ellershaw, Major Macmullen, Sir Spencer Maryon Wilson, Mr. Prince, Sir Charles Hartopp, Mr. and Mrs. Kerr, Mr. Walter Burns, Mr. and Mrs. Wade and party, Mr. Harold Brassey, Miss Joan Wilson, Miss Enid Wilson, Mr. Henry Wilson, Lord Acheson, and many others. Dancing was kept up till the early hours of the morning.&lt;p&gt;

The meet of the hounds next day was at the Beverley Grand Stand. Owing to the frost it was late before the pack arrived from the kennels. Many hundreds of people were present, tempted by the fine, crisp, bracing morning. Leaving Westwood the hounds proceeded to Broadedge Farm, where a fox was found, and he was hunted on to Bentley and Skidky to Cottingham. The company, having seen the throw-off, returned home.&lt;ref&gt;&quot;Holderness Hunt Ball.&quot; ''Yorkshire Herald'' 29 January 1899, Saturday: 10 [of 18], Col. 6c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000500/18990128/086/0010 (accessed July 2019).&lt;/ref&gt;&lt;/blockquote&gt;

==February 1899==
==March 1899==
===31 March 1899, Friday===
Good Friday

==April 1899==
===2 April 1899, Sunday===
Easter Sunday
==May 1899==
===2 May 1899, Tuesday===
The [[Social Victorians/People/Albert Edward, Prince of Wales|Prince of Wales]] visited Ruthin Castle, in Wales, for the Chester races. Ruthin Castle was the home of Cornwallis-West, and [[Social Victorians/People/Muriel Wilson|Muriel Wilson]] was a part of the house party that made up the reception for the prince (1899-05-03 Daily Telegraph). [[Social Victorians/People/Churchill|Jennie Spencer-Churchill]], Lady Randolph Churchill was there, as was [[Social Victorians/People/Cornwallis-West|George Cornwallis-West]]; their relationship was quite controversial and the Prince, normally quite warm to Jennie Churchill, was cold ().

===8 May 1899, Monday===
[[Social Victorians/People/Muriel Wilson|Muriel Wilson]] and [[Social Victorians/People/Arthur Stanley Wilson|Mrs. Arthur Wilson]] were at the opening of the Royal Opera, Covent Garden, Wagner’s ''Lohengrin''. The [[Social Victorians/People/Albert Edward, Prince of Wales|Prince of Wales]] was there, as were a number of notable celebrities (1899-05-13 Penny Illustrated Paper)

===15 May 1899, Monday===
&lt;quote&gt;Ipswich High School.— On Monday, May 15, the school was honoured by a visit of H.R.H. the Princess Louise, who came, faithful [386A/B] to a long-standing promise, to give away the prizes and certificates adjudged on the results of the Oxford and Cambridge Joint Board and the Cambridge Local Examinations of last year. Her Royal Highness was accompanied by the Marquis of Lome, K.G., and attended by Mr. E. B. Phipps, Assistant-Secretary to the G.P.D.S. Co., who acted as equerry in the place of Col. Collins. The Princess was received, on her arrival in Ipswich, by Mr. Bousfield, Chairman of the Council of the G P.D.S. Co., and by Lady Digby, Miss Gurney, Mr. Eve, and Mr. Buxton, members of the Council. The visit was of a semi-private character, and hence there was no official reception by the Mayor and Corporation of Ipswich. The High School was reached at two o'clock, and here the Princess was received by members of the Local Committee, with whom were Miss Youngman, the late Headmistress, Miss Kennett, the present Headmistress, and Mr. McDowall, Secretary to the G.P.D.S. Co. The girls, two hundred in number, were drawn up on either side of the Lower Hall, and presented an exceedingly bright appearance in their white dresses and sashes of crimson, the school colour. The Princess graciously consented to walk up the hall between the lines of girls and to receive a bouquet from Janet Stewaid, of Form II , the daughter of Mr. W. Steward, a member of the Local Committee. She then made the tour of the class-rooms, escorted by Mr. Bousfield, Miss Youngman, and Miss Kennett. Luncheon was served in the Upper Hall at 2.15. The number of invited guests included, in addition to those already mentioned, the Mayor of Ipswich, the Marquis of Bristol, Lord Lieutenant of the county, Sir Charles Dalrymple, M.P., Major Bond (in command of the Volunteer guard of honour), and Mr. John Farmer. At four o'clock the party adjourned to the Council Chamber of the Town Hall, where, after several songs by the pupils, under the conductorship of Mr. Farmer, and an exhibition of drill, Mr. Bousfield made a short speech, in which he explained the aims and ideals of the schools of the G.P.D.S. Co., and expressed the gratitude of the Council to Miss Youngman, who for twenty-one years had watched over and guided the development of the Ipswich School with so much energy and judgment. The Princess then distributed the prizes, after which a vote of thanks to her was proposed by Mr. Bousfield, and seconded by the Mayor. Lord Bristol also spoke to the resolution, which was unanimously carried. The Marquis of Lome having briefly responded, the proceedings were closed by the singing of &quot;Auld Lang Syne,&quot; the hymn &quot;O God, our help in ages past,&quot; and &quot;God save the Queen.&quot; The bouquet given to the Princess at the Town Hall was presented by Sybil Casley, of the Kindergarten, and the programme was handed to her Royal Highness by Judith Becher, of the Transition Class. At the conclusion of the afternoon's proceedings the Princess and her party partook of tea in the Mayor's parlour at the Town Hall, and left for London at six o'clock.&lt;/quote&gt; (The Journal of Education. Vol. 21, New Series (January to December 1899). Page 386 [June 1899].Cols. A-B. Google Books: http://books.google.com/books?id=jZFIAAAAYAAJ).

===20 May 1899, Saturday===
Emma Nevada at the Crystal Palace (https://www.msu.edu/~graye/emma/chronolo.html).

===27 May 1899, Saturday===
Emma Nevada at the Crystal Palace (https://www.msu.edu/~graye/emma/chronolo.html).

===31 May 1899, Wednesday===
Derby Day at Epsom Downs, so Luise Friederike Auguste Montagu  Duchess of Devonshire, hosted a ball that night?

==June 1899==
Summer 1899: [[Social Victorians/People/William Butler Yeats|William Butler Yeats]] summered with Lady Gregory at Coole Park 1897-1917 or so, until WBY bought the Tower at Ballylee. (I got this from Wade?).
===21 June 1899, Wednesday===
The ''Illustrated London News'' reported on a bazaar to raise money for the Charing Cross Hospital. Many celebrities were present. People whose portraits were drawn were Princess Henry of Pless, Marchioness of Granby, Duchess of Marlborough, [[Social Victorians/People/Muriel Wilson|Muriel Wilson]], Countess of Westmorland, Duchess of Sutherland, Mr. Burdett-Coutts, M.P., and mentioned in the story were the following: Princess Louise, Duchesses of Westminster, Portland, Abercorn, Sutherland, and Marlborough; Countesses of Westmorland, Cadogan, Chesterfield, Mrs. Choate (wife of American ambassador; Lord Glenesk (https://www.britishnewspaperarchive.co.uk/viewer/bl/0001578/18990701/054/0017). 

Another report: &lt;quote&gt;The Charing Cross Hospital Bazaar at the Albert Hall was great success. It was a very hot day, but the aristocracy were present in large numbers. There were many exquisite toilettes. [[Social Victorians/People/Arthur Stanley Wilson|Mrs Arthur Wilson]] wore black, with cream embroidery, covered with sequinned net; corn-coloured net toque, with sprays of gold and feathers. Mrs Kenneth Wilson’s costume was palest grey voile, tight fitting, with a lace yoke, and a white tulle hat with osprey in front. Mrs Menzies and [[Social Victorians/People/Muriel Wilson|Miss Muriel Wilson]] were at “Flowerland,&quot; with Ladies Marlborough, Westmoreland. Mar and Kellie, Chelsea, Craven, Juliette Lowther, and Norreys. Miss Muriel Wilson was in white silk and crepe gown, with bands of coarse cream lace and open neck, transparent sleeves of lace and crepe. She wore a large black chip hat of tulle fastened at the chin, and carried a large-handled basket of roses, tied with heliotrope satin ribbon. The three sisters from Warter Priory were at the refreshment stall attired in grey dresses, while fischus, and big white mob caps.&lt;/quote&gt; (1899-06-24 Beverley and East Riding Recorder)

===26 June 1899, Monday===
There was apparently a regular celebration of [[Social Victorians/People/Arthur Collins|Arthur Collins]]' birthday, 26 June, by Bret Harte, George Du Maurier, Arthur Sullivan, Alfred Cellier, Arthur Blunt, and John Hare (Nissen, Axel. Brent Harte: Prince and Pauper: 239. [http://books.google.com/books?id=WEDewmUnapcC]). Choosing 1885–1902 as the dates because those apparently are the dates of the close relationship between Harte and Collins, ending in Harte's death in 1902.

According to the ''Morning Post'' for 27 June 1899, Mr. Schreiber danced in the Gainesborough Quadrille at the annual Royal Caledonian Ball on Monday, 26 June 1899.&lt;blockquote&gt;THE ROYAL CALEDONIAN BALL.

The annual Ball, held last night at the Whitehall Rooms, Hôtel Métropole, in aid of the funds of the Royal Caledonian Asylum and the Royal Scottish Hospital proved a great success, nearly nine hundred persons being present. The interest ot the evening centred in the eightsome reels and the fancy quadrilles. The former were arranged by the Hon. Mrs. Baillie of Dochfour (who unfortunately through indisposition was prevented attending), and were danced as follows: 

1.

The Marquis of Tullibardine, D.S.O., Royal Horse Guards, and the Countess of Mar and Kellie.

Viscount Fincastle, V.C., 16th Lancers, and Laily Helen Stewart Murray.

The Hon. Alexander Ruthven, V.C., Cameron Highlanders, Miss Katharine Ramsay.

Mr. M'Neil, Seaforth Highlanders, and Miss Sibyl Murray.

2.

The Earl of Mar and Kellie and Lady Helen Graham.

Mr. Alastair Murray, younger, of Lochcarron, and Lady Hilda Keith-Falconer.

The Hon. Alexander Fraser and the Hon. Cecily Drummond.

Mr. M'Lean, Scots Guards, and Miss Baillie.

3.

Lord Lovat and Lady Grizel Cochrane.

Captain Greenhill-Gardyne, Gordon Highlanders, and the Hon. Ethel Fraser.

Mr. Baillie, Seaforth Highlanders, and the Hon. Daisy Fraser.

The Hon. Hugh Fraser, Scots Guards, and Miss Marvel MacGregor.

4.

Mr. Cameron, younger, of Lochiel, Grenadier Guards, and the Countess of Cromartie.

Mr. NIall Campbell and Miss Edith Chaplin.

Mr. Douglas Brodie and Miss Elspeth Campbell.

Mr. Alastair MacGregor of MacGregor and Miss Vere Brodie.

5.

Mr. Ramsay, Black Watch, and Lady Margaret Crichton Stewart.

Mr. M'Ray, Black Watch, and Lady Edith Montgomerie.

Mr. Matheson, Coldstream Guards, and the Hon. Beatrice Dalrymple.

The Hon. Kenneth Campbell and the Hon. Gwendolen Maxwell.

The gentlemen wore Highland dress, while the ladies were in white gowns with sashes formed of their respective tartans, the badges of their clans appearing in their hair and on their dresses.

The Countess of Hopetoun, one of the most energetic of the ladies patronesses, was responsible for the two fancy quadrilles. The undernamed took part in

THE ROMNEY QUADRILLE.

Viscount Crichton, Royal Horse Guards, and Countess Hopetoun.

Mr. C. C. de Crespigny, 2nd Life Guards, and Lady Constance Scott.

Hon. Claud Drummond Willoughby, Coldstream Guards, and Lady Florence Astley.

Hon. Gerald Ward, Ist Life Guards, and Lady Beatrice Herbert.

Mr. Tryon, Grenadier Guards, and Lady Mary Drummond Willoughby.

The Earl of Kerry, Grenadier Guards, and Lady Marjorie Carrington.

Mr. Trotter, Grenadier Guards, and the Hon Alice Grosvenor.

Mr. Hamilton, Grenadier Guards, and Miss Muriel White.

THE GAINSBOROUGH QUADRILLE.

Major Gordon-Gilmour, Grenadier Guards, and Lady Alice Shaw Stewart.

The Hon. Raymond de Montmorency, V.U., 21st Lancers, and Lady Sybil Primrose

Captain Brinton, 2nd Life Guards, and Lady Edith Villiers.

[[Social Victorians/People/Schreiber|Captain Schreiber]], 1st Life Guards, and Hon. Maud de Moleyna.[?]

Captain Heneage, Grenadier Guards, and Miss Long.

Mr. Stirling, Coldstream Guards, and Miss Cotton Jodrell.

Captain Green-Wilkinson, Rifle Brigade, and Miss Sibell Chaplin.

Mr. Vandeleur, D.S.O., Scots Guards, and Miss Muriel Chaplin.

The officers were in uniform, and their partners wera attired in gowns of white mousseline-de-soie over silk slips of different colours, those in the Romney quadrille wearing lace fichus, and those in the Gainsborough quadrille chiffon scarves, and all had their hair threaded with coloured chiffon or ribbon to match their sashes or scarves.

At eleven o'clock a procession was formed, headed by the Pipers, and the Duke of Atholl, Treasurer of the ball, and those taking part in the Reels and Quadrilles entered the ball-room, dancing immediately commencing to Herr Iff's orchestra. The Ladies Patronesses present included the Duchess of Buccleuch, the Duchess of Atholl, the Duchess of Montrose, the Marchioness of Bute, the Countess of Mar and Kellie, Mary Countess of Mar and Kellie, the Counters of Selkirk, the Countess of Dundonald, the Countess of Ancaster, Viscountess Strathallan, Viscountess Dalrymple, Lady Anne Murray, Lady Eleanor Brodie, Lady Herries, Lady Sinclair, Lady Middleton, Lady Ramsay of Bamff, Lady Maxwell of Monreith, Lady Macpherson Grant, Mrs. Munro, and Mrs. Murray of Polmaise. Before dancing became general the boys and girls of the Asylum, headed by their Pipers and band, marched round the ball-room. Much credit is due to the President and Vice-President of the ball, the Duke of Atholl and Marquis of Tullibardine, for their efforts in the cause of charity.&lt;ref&gt;&quot;The Royal Caledonian Ball.&quot; ''Morning Post'' 27 June 1899, Tuesday: 7 [of 12], Col. 7b–c [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18990627/063/0007.&lt;/ref&gt;&lt;/blockquote&gt;

==July 1899==
July 1899, Emma Nevada sang for Queen Victoria at Osborne House (https://www.msu.edu/~graye/emma/chronolo.html).
===4 July 1899, Tuesday===
[[Social Victorians/People/Muriel Wilson|Muriel Wilson]] was at a garden party hosted by Lady Rothschild and Mrs. Leopold Rothschild after the end of the Women’s Congress: &lt;quote&gt;Not the least interesting features of the Women's Congress have been the social entertainments. On Tuesday, after the final sessions had been held, Lady Rothschild and Mrs. Leopold Rothschild invited the delegates to a garden party at Gunnersbury Park. Special trains conveyed the guests to Mill-hill Station. The guests were received by Lady and Mrs. Leopold Rothschild, the former in black lace over pale mauve silk, and the latter in blue and white muslin, and Lady Battersea, in a charming light grey and white frock and a little yellow bonnet that suited her to perfection. By five o'clock the grounds were crowded, and among the well known people to be seen walking about there were Lady Battersea, Lady Harcourt, Mr. and Lady Clementine Walsh, the latter in pale grey and white, Lord and Lady Gosford, and with them Lady Aldra Acheson; Lady Alice Stanley, in rose-pink; Mrs. Rolands [Ronalds?] in white; Mr. and Lady Barbara Smith, and Mrs. Maguire in a becoming frock of pale yellow. Lady Kilmorey, who came quite early in the afternoon wore white muslin, with a large straw hat with roses; Lady Chelsea, in mauve; and Miss Muriel Wilson, in a lovely dress of pale blue, with transparent lace sleeves, and large white hat with roses, looked particularly well. Lady Blandford was in pale grey; Mrs Arthur Sassoon was also in grey. Among some of the late arrivals were Lord and Lady Crewe, with Lady Annabel Milnes, Lady Crewe in a pretty white dress. From the terraces the scene was magnificent. The park stretched over a velvety green lawn, dotted with beds of of [sic] exquisite flowers; bridged over with roses, that gave them the appearance of great baskets. A fountain, with pond lilies nestling in its shadowy spots, lay at the left. At another side of the lawn was an artificial lake, with boats and boatmen at the disposal of visitors. A string band at the bank and a second marquee afforded rest, music, and refreshment to those who preferred to remain in a little world of their own. A three band played lively airs for the American bicycle polo team, who gave an exhibition of their skill on the lawn. And both circus and stage were utilised for the afternoon amusements.&lt;/quote&gt; (1899-07-08 Bridgnorth Journal)

=== 15 July 1899, Saturday ===
&lt;blockquote&gt;Lord Kenyon, Mr. Schomburg McDonnell (Lord Salisbury's private secretary), Colonel Dawson, Mr. H. Ridgway, Lady Gerard, the Hon. Miss Gerard, Mrs. Hartmann, and [[Social Victorians/People/De Jancourt|Mdlle Jancourt]] arrived at Broughton Castle on Saturday night on a weekend visit to Lord and Lady Algernon Gordon Lennox.&lt;ref&gt;&quot;Local Town and Country Notes.&quot; ''Banbury Guardian'' 20 July 1899 Thursday: 8 [of 8], Col. 1b [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001523/18990720/116/0008.&lt;/ref&gt;&lt;/blockquote&gt;

==August 1899==
===28 August 1899===
Summer Bank Holiday

==September 1899==
==October 1899==
===31 October 1899, Tuesday===
Halloween.
==November 1899==
===5 November 1899, Sunday===
Guy Fawkes Day

=== 23 November 1899, Thursday ===
Captain C.S. Schreiber attended a Royal and Imperial Dinner Party at Windsor Castle:&lt;blockquote&gt;The Imperial and Royal dinner party included their Imperial Majesties the German Emperor and Empress, their Royal Highnesses the Prince and Princess of Wales, the Duke and Duchess of Connaught, Prince and Princess Christian of Schleswig-Holstein, her Royal Highness Princess Louise Marchioness of Lorne and Marquis of Lorne, their Royal Highnesses Princess Henry of Battenberg, Princess Victoria of Wales, his Serene Highness and her Grand Ducal Highness Prince and Princess Louis of Battenberg, his Highness Prince Albert of Schleswig-Holstein, the Duchess of Buccleuch, Mistress of the Robes; Fraulein von Gersdorff, the Countess Stollberg, the Dowager Lady Ampthill, the Danish Minster, Mons. de Bille; the Belgian Minister, Baron Whettnall; the Portuguese Minister, Mons. de Soveral ; the Greek Chargé d'Affaires, Mons. Metaxas; the Lord Steward, the Lord Chamberlain, his Excellency Count von Bülow, his Excellency Count Eulenburg, his Excellency General von Plessen, Lord Suffield, the Right Hon. Sir Frank Lascelles, Lord Colville of Culross, Sir Francis Knollys, Vice-Admiral Sir John Fullerton, Major-General Swaine, commanding North-Western District, and Signor de Martino.
The band of the Royal Artillery, conducted by Cavaliere L. Zavertal, played the following selection of music in St. George's Hall in the evening:

March from tlie Suite &quot;Sylvia&quot; Delibes.

Vorspiel &quot;Das Heimchen am Herd&quot; Goldmark.

Three Dances from the music to &quot;Henry VIII.&quot; Ed. German.

1, Morris; 2. Shepherd's; 3. Torch.

(a) Adagietto from the Suite &quot;L'Arlésienae&quot; Bizet.

(b) &quot;La Chaise-à-Porteurs&quot; Chaminade.

Ballet Music, &quot;Der Damon&quot; Rubinstein.

&quot;Abendruhe &quot; Loeschhorn.

Angelus from the Suite &quot;Scènes Pittoresques&quot; Massenet.

Overture, &quot;Cleopatra&quot; Mancinelli.

York March.

Her Majesty's guests invited to dine at the Castle, together with the Ladies and Gentlemen of the Royal Household and the suites in attendance on the Queen's Imperial and Royal guests, had the honour of joining the Royal Circle in St. George's Hall.

The following had the honour ot receiving invitations to be present: Lord and Lady Esher, Lady Edwards, Lady and Miss Victoria Bigge, Mr. and Lady Emily Van de Weyer, Miss Loch, Miss Emily Loch, and Miss Catherine Loch, Sir Walter Parratt, the Head Master at Eton and Mrs. Warre, the Provost of Eton and Miss Hornby, Mr. E. C. Austen Leigh, M.A., Mr. A. C. Benson, M.A., Baron and Baroness Campbell von Laurentz, Lieutenant-Colonel C. N. Miles, Captain G. F. Milner, and [[Social Victorians/People/Schreiber|Captain C. S. Schreiber]], 1st Life Guards; Major the Hon. J. St. Aubyn, Captain the Hon. W. Cavendish, and Lieutenant and Adjutant E. Gascoigne, 1st Battalion Grenadier Guards; and Colonel Swinfen, Major Bolton, and Lieutenant-Colonel Tighe, Military Knights of Windsor.

The Queen did not attend the dinner or the concert in St. George's Hall, owing to having so recently received the news of the death of her Grand Ducal Highness the Princess of Leiningen, her Majesty's niece.&lt;ref&gt;&quot;Court Circular.&quot; ''Morning Post'' 24 November 1899, Friday: 5 [of 10], Col. 5a [of 7]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18991124/045/0005.&lt;/ref&gt;&lt;/blockquote&gt;

==December 1899==
===25 December 1899, Monday===
Christmas Day
===26 December 1899, Tuesday===
Boxing Day
===30 December 1899, Saturday===
[[Social Victorians/People/Arthur Conan Doyle| Arthur Conan Doyle]]'s New Year's Eve party at Hindhead, [[Social Victorians/Haslemere | Haslemere]].

==Works Cited==
*[1899-05-03 Daily Telegraph] &quot;Prince of Wales at Chester Races. Visit to Ruthin Castle. From Our Own Correspondent.&quot; Daily Telegraph &amp; Courier 3 May 1899, Wednesday: 10 [of 16], Col. 3a–c [of 7]. British Newspaper Archive (accessed July 2019).
*[1899-05-13 Penny Illustrated Paper] &quot;At the Opening of the Royal Opera, Covent Garden.&quot; Penny Illustrated Paper 13 May 1899, Saturday: 2 [of 16], Col. 2a, 3a [of 4]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0000693/18990513/008/0002 (accessed July 2019).
*[1899-06-24 Beverley and East Riding Recorder] &quot;East Riding Ladies at a London Bazaar.&quot; Beverley and East Riding Recorder 24 June 1899, Saturday: 5 [of 8], Col. 6b [of 6]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0001565/18990624/074/0005 (accessed July 2019).
*[1899-07-08 Bridgnorth Journal] &quot;Women’s Congress Ended.&quot; Bridgnorth Journal 8 July 1899, Saturday: 8 [of 8], Col. 3c [of 6]. British Newspaper Archive https://www.britishnewspaperarchive.co.uk/viewer/bl/0001961/18990708/138/0008 (accessed July 2019).
*Gray, Eugene F. &quot;Chronology of Events in the Life of Emma Nevada.&quot; Emma Nevada: An American Diva. https://www.msu.edu/~graye/emma/chronolo.html (retrieved 14 April 2010).

== Footnotes ==
&lt;references /&gt;</text>
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  <page>
    <title>Cells</title>
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      <text bytes="5930" sha1="r377keo9vchmqxlv7xp19cvkdasn1fa" xml:space="preserve">Cells are the basis of all life on Earth, producing every living organism, from a single cellular bacteria, through to trillions of cells making up a human. All living species on Earth have cells.

== Sub-Cellular Structures ==

=== Animal Cells ===
[[File:Endomembrane system diagram en.svg|thumb|A diagram showing the organelles of an animal cell]]
Nucleus — contains the cells' genetic material. Visible as an opaque area through a light microscope. The nucleus controls all activities in the cell.

Cell membrane — the membrane encasing the cell. Flexible, allowing movement. Ensures the cell stays together

Mitochondria — release energy, through respiration, typically from glucose. Allows the cell to carry out metabolic functions

Ribosomes — carry out protein synthesis and the site of DNA transcription


=== Plant Cell ===
[[File:Plant cell structure-en.svg|thumb|A diagram showing organelles of a plant cell]]
Nucleus — contains the cell's genetic material. Visible as an opaque area through a light microscope.

Cell membrane — the membrane encasing the cell. Flexible, allowing movement. Ensures the cell's organelles do not spill out.

Cell wall — a strong layer of cellulose and other polysaccharides, making the cell rigid. Ensures that plants can stand upright

Chloroplasts — allow sunlight, water, and carbon dioxide to be converted to glucose and oxygen. Provide energy for the cell through photosynthesis.

Mitochondria — release energy, through respiration, typically from glucose (made by photosynthesis). Allows the cell to carry out metabolic functions

Vacuole — a fluid-filled sac within the cell containing cell sap, mineral ions, and other chemicals essential for the cell to function 

Ribosomes — carry out protein synthesis and the site of DNA transcription

=== Prokaryotic (bacterial) Cell ===
Prokaryotic cells do not have a nucleus, unlike eukaryotic cells which have a nucleus.[[File:Prokaryote cell.svg|thumb|A diagram showing the organelles of a prokaryotic cell]]
Chromosomal DNA — DNA arranged in chromosomes. Contains most of a prokaryotic cells’ genetic material

Plasmid DNA — small rings of DNA within the cytoplasm of the cell. Allows bacteria to share genetic material throughout a population

Cell membrane — the membrane encasing the cell. Flexible, allowing movement. Ensures the cells organelles do not spill out.

Ribosomes — carry out protein synthesis and are the site of DNA transcription

Flagella — whip-like structures on bacteria, allowing them to move

== Some Cells are Specialised to their Function ==

=== Sperm Cells ===
Sperm cells are the male gamete, used during sexual reproduction.
[[File:Prefertilization Sperm cell.PNG|thumb|A diagram showing a sperm cell]]
Acrosome — the region in the head of the sperm cell containing digestive enzymes. Used to ‘burrow’ through the cell membrane of the egg during fertilisation

Haploid nucleus — the nucleus of a sperm cell contains 23 chromosomes, as it is made by the process of meiosis. This creates haploid cells, which have ½ the number of chromosomes of the parent cell.

Mitochondria — sperm cells contain many mitochondrion, to release energy to move the flagellum. These are located in the mid piece

Tail — sperm cells have a tail (flagellum), a whip-like structure, allowing the cell to propel itself during fertilisation

=== Egg Cells ===
Egg cells are the female gamete, used during sexual reproduction.
[[File:Egg cell fertilization - Zygote.png|thumb|A diagram showing the fertilisation of an egg cell with a sperm cell]]
Nutrients in the Cytoplasm — egg cells contain many nutrients in the cytoplasm to provide sufficient nutrition should the egg be fertilised. Allows the cell division and specialisation immediately after fertilisation to begin

Haploid nucleus — the nucleus of an egg cell contains 23 chromosomes (along with the sperm cell), as it is made through the process of meiosis

Changes in the Cell Membrane after Fertilisation — when an egg cell is fertilised with a sperm cell, the cell membrane changes, releasing chemicals that prevent any further sperm from fertilising the cell

=== Ciliated Epithelial Cell ===
[[File:Bronchiolar epithelium 4 - SEM.jpg|thumb|A scanning electron micrograph of respiratory epithelial cells]]
Ciliated epithelial cells have many, small, hair-like projections coming out of their cell membrane.

These projections are used to ‘waft’ waste products out of the trachea (wind pipe) and bronchioles. Mucus is transported out of the lungs with cilia, allowing it to enter the stomach, where it is digested.

These cells have a large proportion of mitochondria, as they use active methods (energy using) to ‘waft’ their hair-like projections.

=== Corona radiarta ===
The corona radiarta is a layer of cells covering an unfertilised egg or egg cell. During fertilisation, sperm must push through the layer to reach the outer layer of the egg, the zona pellucida.&lt;ref&gt;Information given by science based company Nucleus.&lt;/ref&gt;

== Cell death ==

=== Cell decomposition ===
When somebody has died, the cells in the human body stop receiving oxygen, and continue to function only for a few minutes. [[Carbon dioxide]] begins to build up, rupturing sacs inside the cells. The sacs contain enzymes that begin to digest the cells from the inside out. Cell death also occurs because of Necrosis, along with the work of bacteria in the body. A third example of cell death: The cell membranes break, releasing the inside fluids into the surrounding tissue.&lt;ref&gt;Information given by scientific company Seeker, also known as D News.&lt;/ref&gt;

=== Original explanation of cell death ===
A person’s cells die every day, but the body is protecting itself. In a process called apoptosis, cells commit controlled suicide, if infected by viruses or harmful cancerous mutations.

== References ==

[[Category:Biology]]</text>
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    <title>Workings of gcc and ld in plain view</title>
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      <contributor>
        <username>Young1lim</username>
        <id>21186</id>
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      <comment>/* Linking Libraries */</comment>
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      <text bytes="3346" sha1="kyeoksve8ykqobf6e9oniexyc9xwc5p" xml:space="preserve">=== Workings of the GNU Compiler for IA-32 ===

==== Overview ====
* Overview ([[Media:Overview.20200211.pdf |pdf]])

==== Data Processing ====
* Access ([[Media:Access.20200409.pdf |pdf]])
* Operators ([[Media:Operator.20200427.pdf |pdf]])

==== Control ====
* Conditions ([[Media:Condition.20230630.pdf |pdf]])
* Control ([[Media:Control.20220616.pdf |pdf]])

==== Function calls ==== 
* Procedure ([[Media:Procedure.20220412.pdf |pdf]])
* Recursion ([[Media:Recursion.20210824-2.pdf |pdf]])

==== Pointer and Aggregate Types ==== 
* Arrays ([[Media:Array.20211018.pdf |pdf]])
* Structures ([[Media:Structure.20220101.pdf |pdf]])
* Alignment ([[Media:Alignment.20201117.pdf |pdf]])
* Pointers ([[Media:Pointer.20201106.pdf |pdf]])

==== Integer Arithmetic ==== 
* Overview ([[Media:gcc.1.Overview.20240813.pdf |pdf]])
* Carry Flag ([[Media:gcc.2.Carry.20241204.pdf |pdf]])
* Overflow Flag ([[Media:gcc.3.Overflow.20241205.pdf |pdf]])
* Examples ([[Media:gcc.4.Examples.20240724.pdf |pdf]])
* Borrow ([[Media:Borrow.20241214.pdf |pdf]])

==== Floating point Arithmetic ====
&lt;/br&gt;

=== Workings of the GNU Linker for IA-32 ===

==== Linking Libraries ====

* Static Libraries ([[Media:LIB.1A.Static.20241128.pdf |pdf]])

* Shared Libraries ([[Media:LIB.2A.Shared.20241217.pdf |pdf]])

==== Managing Libraries ====

* Shared Library Names ([[Media:MNG.1A.Names.20241214.pdf |pdf]])

==== Library Search Path ====

* Using -L and -l only ([[Media:Link.4A.LibSearch-withLl.20240807.pdf |A.pdf]], [[Media:Link.4B.LibSearch-withLl.20240705.pdf |B.pdf]])

* Using RPATH ([[Media:Link.5A.LibSearch-RPATH.20241101.pdf |A.pdf]], [[Media:Link.5B.LibSearch-RPATH.20240705.pdf |B.pdf]])

==== Linking Process ====

* Object Files ([[Media:Link.3.A.Object.20190121.pdf |A.pdf]], [[Media:Link.3.B.Object.20190405.pdf |B.pdf]])

* Symbols ([[Media:Link.4.A.Symbol.20190312.pdf |A.pdf]], [[Media:Link.4.B.Symbol.20190312.pdf |B.pdf]])

* Relocation ([[Media:Link.5.A.Relocation.20190320.pdf |A.pdf]], [[Media:Link.5.B.Relocation.20190322.pdf |B.pdf]])

* Loading ([[Media:Link.6.A.Loading.20190501.pdf |A.pdf]], [[Media:Link.6.B.Loading.20190126.pdf |B.pdf]])

* Static Linking ([[Media:Link.7.A.StaticLink.20190122.pdf |A.pdf]], [[Media:Link.7.B.StaticLink.20190128.pdf |B.pdf]], [[Media:LNK.5C.StaticLinking.20241128.pdf |C.pdf]])

* Dynamic Linking ([[Media:Link.8.A.DynamicLink.20190207.pdf |A.pdf]], [[Media:Link.8.B.DynamicLink.20190209.pdf |B.pdf]], [[Media:LNK.6C.DynamicLinking.20241128.pdf |C.pdf]])

* Position Independent Code ([[Media:Link.9.A.PIC.20190304.pdf |A.pdf]], [[Media:Link.9.B.PIC.20190309.pdf |B.pdf]])

==== Example I ====
* Vector addition ([[Media:Eg1.1A.Vector.20190121.pdf |A.pdf]], [[Media:Eg1.1B.Vector.20190121.pdf |B.pdf]])
* Swapping array elements ([[Media:Eg1.2A.Swap.20190302.pdf |A.pdf]], [[Media:Eg1.2B.Swap.20190121.pdf |B.pdf]])
* Nested functions ([[Media:Eg1.3A.Nest.20190121.pdf |A.pdf]], [[Media:Eg1.3B.Nest.20190121.pdf |B.pdf]])

==== Examples II ====
* analysis of static linking ([[Media:Ex1.A.StaticLinkEx.20190121.pdf |A.pdf]], [[Media:Ex2.B.StaticLinkEx.20190121.pdf |B.pdf]])

* analysis of dynamic linking ([[Media:Ex2.A.DynamicLinkEx.20190121.pdf |A.pdf]])

* analysis of PIC ([[Media:Ex3.A.PICEx.20190121.pdf |A.pdf]])


&lt;/br&gt;
go to [ [[C programming in plain view]] ]

[[Category:C programming language]]</text>
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  <page>
    <title>Social Victorians/Terminology</title>
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      <text bytes="109801" sha1="mdj7rkawcj64htxwj2y6t0v4twr6ise" xml:space="preserve">Especially with respect to fashion, the newspapers at the end of the 19th century in the UK often used specialized terminology. The definitions on this page are to provide a sense of what someone in the late 19th century might have meant by the term rather than a definition of what we might mean by it today. In the absence of a specialized glossary from the end of the 19th century in the U.K., we use the ''Oxford English Dictionary'' because the senses of a word are illustrated with examples that have dates so we can be sure that the senses we pick are appropriate for when they are used in the quotations we have.

We also sometimes use the French ''Wikipédia'' to define a word because many technical terms of fashion were borrowings from the French. Also, often the French ''Wikipédia'' provides historical context for the uses of a word similar to the way the OED does.

== Articles or Parts of Clothing: Non-gender-specific ==

=== Mantle, Cloak, Cape ===
In 19th-century newspaper accounts, these terms are sometimes used without precision as synonyms. These are all outer garments.

'''Mantle'''

A mantle — often a long outer garment — might have elements like a train, sleeves, collars, revers, fur, and a cape. A late-19th-century writer making a distinction between a mantle and a cloak might use ''mantle'' if the garment is more voluminous.

'''Cloak'''

'''Cape'''

=== Peplum ===
According to the French ''Wiktionnaire'', a peplum is a &quot;Short skirt or flared flounce layered at the waist of a jacket, blouse or dress&quot; [translation by Google Translate].&lt;ref&gt;{{Cite journal|date=2021-07-02|title=péplum|url=https://fr.wiktionary.org/w/index.php?title=p%C3%A9plum&amp;oldid=29547727|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/p%C3%A9plum.&lt;/ref&gt; The ''Oxford English Dictionary'' has a fuller definition, although, it focuses on women's clothing because the sense is written for the present day:&lt;blockquote&gt;''Fashion''. ... a kind of overskirt resembling the ancient peplos (''obsolete''). Hence (now usually) in modern use: a short flared, gathered, or pleated strip of fabric attached at the waist of a woman's jacket, dress, or blouse to create a hanging frill or flounce.&lt;ref name=&quot;:5&quot;&gt;“peplum, n.”. ''Oxford English Dictionary'', Oxford University Press, September 2023, &lt;https://doi.org/10.1093/OED/1832614702&gt;.&lt;/ref&gt;&lt;/blockquote&gt;Men haven't worn peplums since the 18th century, except when wearing costumes based on historical portraits. The ''Daily News'' reported in 1896 that peplums had been revived as a fashion item for women.&lt;ref name=&quot;:5&quot; /&gt;

=== Revers ===
According to the ''Oxford English Dictionary'', ''revers'' are the &quot;edge[s] of a garment turned back to reveal the undersurface (often at the lapel or cuff) (chiefly in ''plural''); the material covering such an edge.&quot;&lt;ref&gt;&quot;revers, n.&quot; ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/164777. Accessed 17 April 2023.&lt;/ref&gt; The term is French and was used this way in the 19th century (according to the ''Wiktionnaire'').&lt;ref&gt;{{Cite journal|date=2023-03-07|title=revers|url=https://fr.wiktionary.org/w/index.php?title=revers&amp;oldid=31706560|journal=Wiktionnaire|language=fr}} https://fr.wiktionary.org/wiki/revers.&lt;/ref&gt;

== Articles or Parts of Clothing: Men's ==
[[Social Victorians/Terminology#Military|Men's military uniforms]] are discussed below.

=== À la Romaine ===
[[File:Johann Baptist Straub - Mars um 1772-1.jpg|thumb|left|alt=Old and damaged marble statue of a Roman god of war with flowing cloak, big helmet with a plume on top, and armor|Johann Baptist Straub's 1772 ''à la romaine'' ''Mars'']]
A few people who attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's fancy-dress ball in 1897]] personated Roman gods or people. They were dressed not as Romans, however, but ''à la romaine'', which was a standardized style of depicting Roman figures that was used in paintings, sculpture and the theatre for historical dress from the 17th until the 20th century. The codification of the style was developed in France in the 17th century for theatre and ballet, when it became popular for masked balls.

Women as well as men could be dressed ''à la romaine'', but much sculpture, portraiture and theatre offered opportunities for men to dress in Roman style — with armor and helmets — and so it was most common for men. In large part because of the codification of the style as well as the painting and sculpture, the style persisted and remained influential into the 20th century and can be found in museums and galleries and on monuments.

For example, Johann Baptist Straub's 1772 statue of Mars (left), now in the Bayerisches Nationalmuseum, Munich, missing part of an arm, shows Mars ''à la romaine''. In London, an early 17th-century example of a figure of Mars ''à la romaine'', with a helmet, '''was''' &quot;at the foot of the Buckingham tomb in Henry VII's Chapel at Westminster Abbey.&quot;&lt;ref&gt;Webb, Geoffrey. “Notes on Hubert Le Sueur-II.” ''The Burlington Magazine for Connoisseurs'' 52, no. 299 (1928): 81–89. http://www.jstor.org/stable/863535.&lt;/ref&gt;{{rp|81, Col. 2c}}

=== Cavalier ===
[[File:Sir-Anthony-van-Dyck-Lord-John-Stuart-and-His-Brother-Lord-Bernard-Stuart.jpg|thumb|alt=Old painting of 2 men flamboyantly and stylishly dressed in colorful silk, with white lace, high-heeled boots and long hair|Van Dyck's c. 1638 painting of cavaliers Lord John Stuart and his brother Lord Bernard Stuart]]
As a signifier in the form of clothing of a royalist political and social ideology begun in France in the early 17th century, the cavalier established France as the leader in fashion and taste. Adopted by [[Social Victorians/Terminology#Military|wealthy royalist British military officers]] during the time of the Restoration, the style signified a political and social position, both because of the loyalty to Charles I and II as well the wealth required to achieve the cavalier look. The style spread beyond the political, however, to become associated generally with dress as well as a style of poetry.&lt;ref&gt;{{Cite journal|date=2023-04-25|title=Cavalier poet|url=https://en.wikipedia.org/w/index.php?title=Cavalier_poet&amp;oldid=1151690299|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Cavalier_poet.&lt;/ref&gt;

Van Dyck's 1638 painting of two brothers (right) emphasizes the cavalier style of dress.

=== Coats ===

==== Doublet ====
* In the 19th-century newspaper accounts we have seen that use this word, doublet seems always to refer to a garment worn by a man, but historically women may have worn doublets. In fact, a doublet worn by Queen Elizabeth I exists and '''is somewhere'''.
* Technically doublets were long sleeved, although we cannot be certain what this or that Victorian tailor would have done for a costume. For example, the [[Social Victorians/People/Spencer Compton Cavendish#Costume at the Duchess of Devonshire's 2 July 1897 Fancy-dress Ball|Duke of Devonshire's costume as Charles V]] shows long sleeves that may be part of the surcoat but should be the long sleeves of the doublet.

==== Pourpoint ====
A padded doublet worn under armor to protect the warrior from the metal chafing. A pourpoint could also be worn without the armor.

==== Surcoat ====
Sometimes just called ''coat''.

[[File:Oscar Wilde by Sarony 1882 18.jpg|thumb|alt=Old photograph of a young man wearing a velvet jacket, knee breeches, silk hose and shiny pointed shoes with bows, seated on a sofa and leaning on his left hand and holding a book in his right| Oscar Wilde, 1882, by Napoleon Sarony]]
=== Hose, Stockings and Tights ===
Newspaper accounts from the late 19th century of men's clothing use the term ''hose'' for what we might call stockings or tights.

In fact, the terminology is specific. ''Stockings'' is the more general term and could refer to hose or tights. With knee breeches men wore hose, which ended above the knee, and women wore hose under their dresses.

The ''Oxford English Dictionary'' defines tights as &quot;Tight-fitting breeches, worn by men in the 18th and early 19th centuries, and still forming part of court-dress.&quot;&lt;ref&gt;“Tights, N.” ''Oxford English Dictionary'', Oxford UP, July 2023, https://doi.org/10.1093/OED/2693287467.&lt;/ref&gt; By 1897, the term was in use for women's stockings, which may have come up only to the knee. Tights were also worn by dancers and acrobats. This general sense of ''tights'' does not assume that they were knitted.

''Clocking'' is decorative embroidery on hose, usually, at the ankles on either the inside or the outside of the leg. It started at the ankle and went up the leg, sometimes as far as the knee. On women's hose, the clocking could be quite colorful and elaborate, while the clocking on men's hose was more inconspicuous.

In many photographs men's hose are wrinkled, especially at the ankles and the knees, because they were shaped from woven fabric. Silk hose were knitted instead of woven, which gave them elasticity and reduced the wrinkling.

The famous Sarony carte de visite photograph of Oscar Wilde (right) shows him in 1882 wearing knee breeches and silk hose, which are shiny and quite smoothly fitted although they show a few wrinkles at the ankles and knees. In the portraits of people in costume at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]], the men's hose are sometimes quite smooth, which means they were made of knitted silk and may have been smoothed for the portrait.

In painted portraits the hose are almost always depicted as smooth, part of the artist's improvement of the appearance of the subject.

=== Shoes and Boots ===

== Articles or Parts of Clothing: Women's ==

=== '''Chérusque''' ===
According to the French ''Wikipedia'', ''chérusque'' is a 19th-century term for the kind of standing collar like the ones worn by ladies in the Renaissance.&lt;ref&gt;{{Cite journal|date=2021-06-26|title=Collerette (costume)|url=https://fr.wikipedia.org/w/index.php?title=Collerette_(costume)&amp;oldid=184136746|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Collerette_(costume)#Au+xixe+siècle+:+la+Chérusque.&lt;/ref&gt;

=== Corsage ===
According to the ''Oxford English Dictionary'', the corsage is the &quot;'body' of a woman's dress; a bodice.&quot;&lt;ref&gt;&quot;corsage, n.&quot; ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/42056. Accessed 7 February 2023.&lt;/ref&gt; This sense is well documented in the ''OED'' for the mid and late 19th-century, used this way in fiction as well as in a publication like ''Godey's Lady's Book'', which would be expected to use appropriate terminology associated with fashion and dress making.

The sense of &quot;a bouquet worn on the bodice&quot; is, according to the ''OED'', American.

=== Décolletage ===

=== Girdle ===

=== Mancheron ===
According to the ''Oxford English Dictionary'', a ''mancheron'' is a &quot;historical&quot; word for &quot;A piece of trimming on the upper part of a sleeve on a woman's dress.&quot;&lt;ref&gt;&quot;mancheron, n.&quot; ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/113251. Accessed 17 April 2023.&lt;/ref&gt; At the present, in French, a ''mancheron'' is a cap sleeve &quot;cut directly on the bodice.&quot;&lt;ref&gt;{{Cite journal|date=2022-11-28|title=Manche (vêtement)|url=https://fr.wikipedia.org/w/index.php?title=Manche_(v%C3%AAtement)&amp;oldid=199054843|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Manche_(v%C3%AAtement).&lt;/ref&gt;

=== Petticoat ===

According to the ''O.E.D.'', a petticoat is a &lt;blockquote&gt;skirt, as distinguished from a bodice, worn either externally or showing beneath a dress as part of the costume (often trimmed or ornamented); an outer skirt; a decorative underskirt. Frequently in ''plural'': a woman's or girl's upper skirts and underskirts collectively. Now ''archaic'' or ''historical''.&lt;ref&gt;“petticoat, n., sense 2.b”.  ''Oxford English Dictionary'', Oxford University Press,  September 2023, &lt;https://doi.org/10.1093/OED/1021034245&gt;&lt;/ref&gt; &lt;/blockquote&gt;This sense is, according to the ''O.E.D.'', &quot;The usual sense between the 17th and 19th centuries.&quot; However, while petticoats belong in both outer- and undergarments — that is, meant to be seen or hidden, like underwear — they were always under another garment, for example, underneath an open overskirt. The primary sense seems to have shifted through the 19th century so that, by the end, petticoats were underwear and the term ''underskirt'' was used to describe what showed under an open overskirt. 

=== Stomacher ===
According to the ''O.E.D.'', a stomacher is &quot;An ornamental covering for the chest (often covered with jewels) worn by women under the lacing of the bodice,&quot;&lt;ref&gt;“stomacher, n.¹, sense 3.a”. ''Oxford English Dictionary'', Oxford University Press, September 2023, &lt;https://doi.org/10.1093/OED/1169498955&gt;&lt;/ref&gt; although by the end of the 19th century, the bodice did not often have visible laces. Some stomachers were so decorated that they were thought of as part of the jewelry.

=== Train ===
A train is

The Length of the Train

'''For the monarch [or a royal?]'''

According to Debrett's,&lt;blockquote&gt;A peeress's coronation robe is a long-trained crimson velvet mantle, edged with miniver pure, with a miniver pure cape. The length of the train varies with the rank of the wearer:

* Duchess: for rows of ermine; train to be six feet
* Marchioness: three and a half rows of ermine; train to be three and three-quarters feet
* Countess: three rows of ermine; train to be three and a half feet
* Viscountess: two and a half rows of ermine; train to be three and a quarter feet
* Baroness: two rows of ermine; train to be three feet&lt;ref name=&quot;:2&quot;&gt;{{Cite web|url=https://debretts.com/royal-family/dress-codes/|title=Dress Codes|website=debretts.com|language=en-US|access-date=2023-07-27}} https://debretts.com/royal-family/dress-codes/.&lt;/ref&gt;
&lt;/blockquote&gt;The pattern on the coronet worn was also quite specific, similar but not exactly the same for peers and peeresses. Debrett's also distinguishes between coronets and tiaras, which were classified more like jewelry, which was regulated only in very general terms.

Peeresses put on their coronets after the Queen or Queen Consort has been crowned. ['''peers?''']

=== Foundation Garments ===
Unlike undergarments, Victorian women's foundation garments created the distinctive silhouette. Victorian undergarments included the chemise, the bloomers, the corset cover — articles that are not structural. 

The corset was an important element of the understructure of foundation garments — hoops, bustles, petticoats and so on — but it has never been the only important element. 

=== Corset ===
[[File:Corset - MET 1972.209.49a, b.jpg|thumb|alt=Photograph of an old silk corset on a mannequin, showing the closure down the front, similar to a button, and channels in the fabric for the boning. It is wider at the top and bottom, creating smooth curves from the bust to the compressed waist to the hips, with a long point below the waist in front.|French 1890s corset, now in the Metropolitan Museum of Art, NYC]]
The understructure of Victorian women's clothing is what makes the costumes worn by the women at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] so distinctly Victorian in appearance. An example of a corset that has the kind of structure often worn by fashionably dressed women in 1897 is the one at right. 

This corset exaggerated the shape of the women's bodies and made possible a bodice that looked and was fitted in the way that is so distinctive of the time — very controlled and smooth. And, as a structural element, this foundation garment carried the weight of all those layers and all that fabric and decoration on the gowns, trains and mantles. (The trains and mantles could be attached directly to the corset itself.)

* This foundation emphasizes the waist and the bust in particular, in part because of the contrast between the very small waist and the rounded fullness of the bust and hips.
* The idealized waist is defined by its small span and the sexualizing point at the center-bottom of the bodice, which directs the eye downwards. Interestingly, the pointed waistline worn by Elizabethan men has become level in the Victorian age. Highly fashionable Victorian women wearing the traditional style, however, had extremely pointed waists.
* The busk (a kind of boning in the front of a corset that is less flexible than the rest) smoothed the bodice, flattened the abdomen and prevented the point on the bodice from curling up.
* The sharp definition of the waist was caused by
** length of the corset (especially on the sides)
** the stiffness of the boning
** the layers of fabric
** the lacing (especially if the woman used tightlacing)
** the over-all shape, which was so much wider at the top and the bottom
** the contrast between the waist and the wider top and bottom
* The late-19th-century corset was long, ending below the waist even on the sides and back.
* The boning and the top edge of the late 19th-century fashion corset pushed up the bust, rounding (rather than flattening, as in earlier styles) the breasts, drawing attention to their exposed curves and creating cleavage.
* The exaggerated bust was larger than the hips, whenever possible, an impression reinforced by the A-line of the skirt and the inverted Vs in the decorative trim near the waist and on the skirt.
* This corset made the bodice very smooth with a very precise fit, that had no wrinkles, folds or loose drapery. The bodice was also trimmed or decorated, but the base was always a smooth bodice. More formal gowns would still have the fitted bodice and more elaborate trim made from lace, embroidery, appliqué, beading and possibly even jewels.
The advantages and disadvantages of corseting and especially tight lacing were the subject of thousands of articles and opinions in the periodical press for a great part of the century, but the fetishistic and politicized tight lacing was practiced by very few women. And no single approach to corsetry was practiced by all women all the time. Most of the women at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 ball]] were not tightly laced, but the progressive style does not dominate either, even though all the costumes are technically historical dress. Part of what gives most of the costumes their distinctive 19th-century &quot;look&quot; is the more traditional corset beneath them. Even though this highly fashionable look was widely present in the historical costumes at the ball, some women's waists were obviously very small and others were hardly '''emphasized''' at all. Women's waists are never mentioned in the newspaper coverage of the ball — or, indeed, of any of the social events attended by the network at the ball — so it is only in photographs that we can see the effects of how they used their corsets.

=== Hoops ===
'''This section is under construction right now'''.

''Hoops'' is a mid-19th-century term for a cage-like structure worn under a skirt to hold it away from the body. '''Striking''' for how long they lasted and '''the ways''' they evolved, hoops were the foundation undergarment for the bottom half of a woman's body, for a skirt and petticoat.

Women wore this cage-like structure from the '''15th century''' through the late 19th century. The 16th-century Katherine of Aragon is credited with making it fashionable outside Spain.

The cage caused the silhouette of skirts to change shape over time and enabled the extreme distortions of 17th-and-18th-century panniers and the late 19th-century bustle. Early hoops circled the body in a bell, cone or drum shape, then were moved to the sides with panniers, then ballooned around the body like the top half of a sphere, and finally were pulled to the rear with a bustle.

That is, the distorted shapes of high fashion were made possible by hoops. High fashion demanded these shapes, which disguised women's bodies, especially below the waist, while [[Social Victorians/Terminology#Corsets|corsets]] did their work above it. 

Besides the shape, the structure used to construct hoops evolved — from cane and wood to whalebone, then steel '''bands''' and wire. Add fabric structural stuff: tabs, wires inserted into casings in a linen, muslin or, later, crinoline underskirt

[[File:Pedro García de Benabarre St John Retable Detail.jpg|thumb|alt=Old oil painting of a woman wearing a dress from the 1400s holding the decapitated head of a man with a halo before a table of people at a dinner party|Pedro García de Benabarre, Detail from St. John Altarpiece, c. 1470, Showing Visible Hoops]]
[[File:Alonso Sánchez Coello 011.jpg|thumb|alt=Old painting of a princess wearing a richly jeweled outfit|Infanta Isabel Clara Eugenia Wearing a Vertugado, or Spanish Farthingale]]
==== 15th Century ====
Hoops first appeared in Spain in the 15th century and influenced European fashion for '''many years'''.

A detail (right) from Pedro García de Benabarre's c. 1470 larger altarpiece painting shows women wearing a style of hoops that predates the farthingale but marks the beginning point of the development of that fashion. Salome (holding John the Baptist's head) is wearing a dress with what looks like visible wooden hoops attached to the outside of the skirt, which also appears to have padding at the hips underneath it.

De Benabarre was &quot;active in Aragon and in Catalonia, between 1445–1496,&quot;&lt;ref&gt;{{Cite web|url=https://www.mfab.hu/artworks/10528/|title=Saint Peter|website=Museum of Fine Arts, Budapest|language=en-US|access-date=2024-12-11}} https://www.mfab.hu/artworks/10528/.&lt;/ref&gt; so perhaps he saw the styles worn by people like Katharine of Aragon.

==== 16th Century ====
Styles in personal adornment and architectural decoration: The &quot;Golden Age&quot; in '''England''', the Elizabethan Age.
[[File:Queen Elizabeth I ('The Ditchley portrait') by Marcus Gheeraerts the YoungerFXD.jpg|thumb|alt=Old oil painting of a queen in a white dress with shoulders and hips exaggerated by her dress|Queen Elizabeth I in a French Cartwheel Farthingale]]
In the 16th century, the garment we call ''hoops'' was called a farthingale.&lt;blockquote&gt;''&quot;FARTHINGALE:  Renaissance (1450-1550 C.E. to Elizabethan (1550-1625 C.E.). Linen underskirt with '''wire supports''' which, when shaped, produced a variety of dome, bell, and oblong shapes.&quot;&lt;ref name=&quot;:7&quot; /&gt;''{{rp|105}} ['''our emphasis''']&lt;/blockquote&gt;''Vertugadin'' is a French term for ''farthingale'' — &quot;un élément essentiel de la mode Tudor en Angleterre [an essential element of Tudor fashion in England].&quot;&lt;ref name=&quot;:0&quot;&gt;{{Cite journal|date=2022-03-12|title=Vertugadin|url=https://fr.wikipedia.org/w/index.php?title=Vertugadin&amp;oldid=191825729|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Vertugadin.&lt;/ref&gt; ''Farthingale'' is the term in English; in French, it's ''vertugadin'', and in Spanish ''vertugado''. The hoops in the Pedro García de Benabarre painting (above right) predate what would technically be a vertugado.&lt;p&gt;
Blanche Payne says,&lt;blockquote&gt;Katherine of Aragon is reputed to have introduced the Spanish farthingale ... into England early in the century. The result was to convert the columnar skirt of the fifteenth century into the cone shape of the sixteenth. ...&lt;p&gt;
Spanish influence had introduced the hoop-supported skirt, smooth in contour, '''which was quite generally worn'''.&lt;ref name=&quot;:11&quot; /&gt;{{rp|291}} ['''our emphasis''']&lt;/blockquote&gt;

In fact, &quot;The Spanish princess Catherine of Aragon brought the fashion to England for her marriage to Prince Arthur, eldest son of Henry VII in 1501 [La princesse espagnole Catherine d'Aragon amena la mode en Angleterre pour son mariage avec le prince Arthur, fils aîné d'Henri VII en 1501].&quot;&lt;ref name=&quot;:0&quot; /&gt; Catherine of Aragon, of course, married Henry VIII after Arthur's death.

The vertugado was &quot;quite generally worn&quot; among the ruling and culturally elite classes in Spain, and not by working-class women, which was enforced by sumptuary laws.

By the end of the 16th century the French and Spanish farthingales were not identical.

The Spanish vertugado shaped the skirt into an A-line with a graduated series of hoops sewn to an undergarment. Alonso Sánchez Coello's c. 1584&lt;ref name=&quot;:11&quot; /&gt;{{rp|316}} portrait (right) shows infanta Isabel Clara Eugenia wearing a vertugado, with its &quot;typically Spanish smooth cone-shaped contour.&quot;&lt;ref name=&quot;:11&quot; /&gt;{{rp|315–316}}

The French vertugadin was a flattish &quot;cartwheel&quot; '''in which a''' platter of hoops worn below the waist and above the hips held the skirt out more or less horizontally. Once past the vertugadin, the skirt then fell straight to the floor, shaping it into a kind of drum. Marcus Gheeraerts the Younger's portrait (right) of Queen Elizabeth I shows an English queen wearing a French drum-shaped farthingale. The skirt over a cartwheel farthingale did not touch the floor in front, so the dress flowed and the women's feet would show as they walked. Interestingly, shoes often appear in portraits of women wearing the vertugadin, as Elizabeth's do in Gheeraerts' image. 

The shoes do not show in the portraits of women wearing the Spanish vertugado. The round hoops stayed in place in front, giving their feet enough room to take steps.

By the end of the 16th century France had become the arbiter of fashion for the western world, which it still is.

==== 17th Century ====
Styles in personal adornment and architectural decoration: The Cavaliers, the Baroque Age[[File:Турнюр.jpg|thumb|Турнюр]]
[[File:Panniers 1.jpg|thumb|alt=Photograph of the wooden and fabric skeleton of an 18th-century women's foundation garment|Panniers 1]]
People associate bustles with late-19th-century styles, but in fact the bustle existed in the 17th century, sometimes as padding rather than a structural cage. Panniers are associated with 18th-century styles, but they first began in the 17th century as well.

Generally, panniers were a kind of undergarment worn in the 17th and 18th centuries. Their design evolved during the century. Made of hoops of wood, they were &quot;baskets&quot; or cages worn on either side of the waist to broaden the skirts to the sides.

bum rolls, padding

Illustration

Payne says, &quot;The bustle was a continuation of the 1690 mode.&quot;&lt;ref name=&quot;:11&quot; /&gt;{{rp|411}}

==== 18th Century ====

Styles in personal adornment and architectural decoration: Rococo, post French Revolution, Empire

By the 18th century, the farthingale was called hoops, which were at this point made of wood.


Blanche Payne outlines the evolution of hoops, and thus the shape of the skirt, in the 18th century:&lt;blockquote&gt;SKIRT FASHIONS. Since skirts experienced the greatest alterations, a brief summary of the successive silhouettes should help to place individual costumes in their proper niches. Six basic forms appeared during the century, in the following order:

# The bustle was a continuation of the 1690 mode.
# The bell or dome shape resulted from the reintroduction of hoops; in England by 1710, in France by 1720.
# The ellipse, the second phase of the hoop skirt, was achieved by broadening the support from side to side and compressing it from front to back. It had a long run of popularity, from 1740 to 1770, the extreme width being retained in court costumes. In France it persisted until the revolution, except that skirts were allowed to curve outward in [the] back again. English court costume [411/413] followed this fashion well into the nineteenth century.
# The dairy maid, or polonaise, style could be achieved either by pulling the lower part of the overskirt through its own pocket holes, thus creating a bouffant effect, or by planned control of the overskirt, through the cut or by means of draw cords, ribbons, or loops and buttons, which were used to form the three great ‘poufs’ known as the polonaise .... These diversions appeared in the late sixties and became prevalent in the seventies. They were much like the familiar styles of our own [American] Revolutionary War period.
# The return of the bustle in the 1780s.
# The tubular form, drawn from classic art, in the 1790s.&lt;ref name=&quot;:11&quot; /&gt;{{rp|411, 413}}
&lt;/blockquote&gt;While we think of the bustle as a 19th-century look, it can be found in the 18th century, as Payne says.&lt;p&gt;
The Polonaise was a late-Georgian or late-18th-century style, dating in written English, according to the ''Oxford English Dictionary'', from 1773:&lt;blockquote&gt;A woman's dress consisting of a tight, unboned bodice and a skirt open from the waist downwards to reveal a decorative underskirt. Now historical.&lt;ref name=&quot;:13&quot;&gt;“Polonaise, N. &amp; Adj.” ''Oxford English Dictionary'', Oxford UP, September 2024, https://doi.org/10.1093/OED/2555138986.&lt;/ref&gt;&lt;/blockquote&gt;Even though it looks ''à la français'', the term itself does not appear as a term used to describe clothing by the French, either now or in the past.&lt;p&gt;
Payne says,&lt;blockquote&gt;The dairy maid, or polonaise, style could be achieved either by pulling the lower part of the overskirt through its own pocket holes, thus creating a bouffant effect, or by planned control of the overskirt, through the cut or by means of draw cords, ribbons, or loops and buttons, [or, later, buckles] which were used to form the three great ‘poufs’ known as the polonaise .... These diversions appeared in the late sixties and became prevalent in the seventies. They were much like the familiar styles of our own [American] Revolutionary War period.&lt;ref name=&quot;:11&quot; /&gt;{{rp|413}}&lt;/blockquote&gt;

==== 19th Century ====
Possible images:

* File:Crinoline era3.gif
* Crinoline (6795291959).jpg
* Elisabeth Franziska wearing a crinoline and feathered hat.jpg
* HM Queen Victoria. Photograph by C. Clifford of Madrid, 1861 Wellcome V0027547.jpg
* Queen Victoria photographed by Mayall.JPG
* Her Majesty the Queen Victoria.JPG

[[File:Cutaway sketch of crinoline.gif|thumb|Cutaway sketch of crinoline]]
[[File:Paris voulant englober la banlieue.JPG|thumb|Paris voulant englober la banlieue]]
[[File:Vrouw moet haar hoepelrok uitdoen om de tram te betreden New Omnibus Regulation. Werry sorry'm, but yer l'av to leave yer Krinerline outside (Vide Punch) (titel op object), RP-F-F10723.jpg|thumb|Vrouw moet haar hoepelrok uitdoen om de tram te betreden New Omnibus Regulation. Werry sorry'm, but yer l'av to leave yer Krinerline outside (Vide Punch) (titel op object), RP-F-F10723]]
Styles in personal adornment and architectural decoration: Romantic, Victorian (at least in '''the UK'''), &quot;New Woman,&quot; [[Social Victorians/Terminology#Traditional vs Progressive Style|Traditional vs Progressive Style]], Crinoline 

In the 19th century, the hoops were made of wire and became lighter. By the 1860s, hoops caused skirts to be huge and round.

By the 19th century, fashion had begun to move down the social classes so that hoops (and, for example, top hats) were worn by the middle and sometimes working classes.

'''''1880s'''''


Laura Ingalls Wilder wrote about the hoops her fictionalized self wore the century before. In ''These Happy Golden Years'' (1943), she gives a detailed description of the clothing under her dress:&lt;blockquote&gt;
“Then carefully over her under-petticoats she put on her hoops. She liked these new hoops. They were the very latest style in the East, and these were the first of the kind that Miss Bell had got. Instead of wires, there were wide tapes across the front, almost to her knees, holding the petticoats so that her dress would lie flat. These tapes held the wire bustle in place at the back, and it was an adjustable bustle. Short lengths of tape were fastened either end of it; these could be buckled together underneath the bustle to puff it out, either large or small. Or they could be buckled together in front, drawing the bustle down close in back so that a dress rounded smoothly over it. Laura did not like a large bustle, so she buckled the tapes in front.

&quot;Then carefully over all she buttoned her best petticoat, and over all the starched petticoats she put on the underskirt of her new dress. It was of brown cambric, fitting smoothly around the top over the bustle, and gored to flare smoothly down over the hoops. At the bottom, just missing the floor, was a twelve-inch-wide flounce of the brown poplin, bound with an inch-wide band of plain brown silk. The poplin was not plain poplin, but striped with an openwork silk stripe.

&quot;Then over this underskirt and her starched white corset-cover, Laura put on the polonaise. Its smooth, long sleeves fitted her arms perfectly to the wrists, where a band of the plain silk ended them. The neck was high with a smooth band of the plain silk around the throat. The polonaise fitted tightly and buttoned all down the front with small round buttons covered with the plain brown silk. Below the smooth hips it flared and rippled down and covered the top of the flounce on the underskirt. A band of the plain silk finished the polonaise at the bottom.&quot;&lt;ref&gt;Wilder, Laura Ingalls. ''These Happy Golden Years.'' Harper &amp; Row, Publishers, 1943. Pp. 161–163.&lt;/ref&gt;&lt;/blockquote&gt;
When a 20th-century Laura Ingalls Wilder calls her character's late-19th-century dress a polonaise, she is probably referring to the &quot;tight, unboned bodice&quot;&lt;ref name=&quot;:13&quot; /&gt; and perhaps the simple, modest look of a dairy maid. 

In Wilder's 1941 ''Little Town on the Prairie'', she provides an interesting story about how the wind could affect hoops:&lt;blockquote&gt;“Well,” Laura began; then she stopped and spun round and round, for the strong wind blowing against her always made the wires of her hoop skirt creep slowly upward under her skirts until they bunched around her knees. Then she must whirl around and around until the wires shook loose and spiraled down to the bottom of her skirts where they should be.

“As she and Carrie hurried on she began again. “I think it was silly, the way they dressed when Ma was a girl, don’t you? Drat this wind!” she exclaimed as the hoops began creeping upward again.

“Quietly Carrie stood by while Laura whirled. “I’m glad I’m not old enough to have to wear hoops,” she said. “They’d make me dizzy.”

“They are rather a nuisance,” Laura admitted. “But they are stylish, and when you’re my age you’ll want to be in style.”&lt;ref&gt;Wilder, Laura Ingalls. ''Little Town on the Prairie.'' Harper and Row, 1941. Pp. 272–273.&lt;/ref&gt;&lt;/blockquote&gt;This moment is set in 1883.&lt;ref&gt;Hill, Pamela Smith, ed. ''Pioneer Girl: The Annotated Autobiography''.&lt;/ref&gt; The 16-year-old Laura makes the comment that she wants to be in style, but she lives on the prairie, far from a large city, and would not necessarily wear the latest Parisian style. This description of the way the wind could make hoops creep — and the solution of spinning to get the hoops to go back down — is very unusual. It must have been happening to other women wearing hoops at the time, but no other writer addresses this.

== '''Traditional vs Progressive Style''' ==

=== Progressive Style ===
The terms ''artistic dress'' and ''aesthetic dress'' are not synonymous and were in use at different times to refer to different groups of people in different contexts, but we recognize them as referring to a similar kind of personal style in clothing, a style we call progressive dress or the progressive style. Used in a very precise way, ''artistic dress'' is associated with the Pre-Raphaelite artists and the women in their circle beginning in the 1860s. Similarly, ''aesthetic dress'' is associated with the 1880s and 1890s and dress reform movements. In general, the progressive style is characterized by its resistance to the highly structured fashion of its day, especially corseting, aniline dyes and an extremely close fit.

=== Traditional Style ===
By the end of the century designs from the [[Social Victorians/People/Dressmakers and Costumiers#The House of Worth|House of Worth]] (or Maison Worth) define what we think of as the traditional Victorian look, which was very stylish and expensive. Blanche Payne describes an example of the 1895 &quot;high style&quot; in a gown by Worth with &quot;the idiosyncrasies of the [1890s] full blown&quot;:&lt;blockquote&gt;The dress is white silk with wine-red stripes. Sleeves, collars, bows, bag, hat, and hem border match the stripes. The sleeve has reached its maximum volume; the bosom full and emphasized with added lace; the waistline is elongated, pointed, and laced to the point of distress; the skirt is smooth over the hips, gradually swinging out to sweep the floor. This is the much vaunted hourglass figure.&lt;ref name=&quot;:11&quot;&gt;Payne, Blanche. ''History of Costume from the Ancient Egyptians to the Twentieth Century''. Harper &amp; Row, 1965.&lt;/ref&gt;{{rp|530}}&lt;/blockquote&gt;

The Victorian-looking gowns at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] are stylish in a way that recalls the designs of the House of Worth. The elements that make their look so Victorian are anachronisms on the costumes representing fashion of earlier eras. The women wearing these gowns preferred the standards of beauty from their own day to a more-or-less historically accurate look. The style competing at the very end of the century with the Worth look was not the historical, however, but a progressive style called at the time ''artistic'' or ''aesthetic''.

William Powell Frith's 1883 painting ''A Private View at the Royal Academy, 1881'' (discussion below) pits this kind of traditional style against the progressive or artistic style.

=== The Styles ===
[[File:Frith A Private View.jpg|thumb|William Powell Frith, ''A Private View at the Royal Academy, 1881'']]
We typically think of the late-Victorian silhouette as universal but, in the periods in which corsets dominated women's dress, not all women wore corsets and not all corsets were the same, as William Powell Frith's 1883 ''A Private View at the Royal Academy, 1881'' (right) illustrates. Frith is clear in his memoir that this painting — &quot;recording for posterity the aesthetic craze as regards dress&quot; — deliberately contrasts what he calls the &quot;folly&quot; of the Artistic Dress movement and the look of the traditional corseted waist.&lt;ref&gt;Frith, William Powell. ''My Autobiography and Reminiscences''. 1887.&lt;/ref&gt; Frith considered the Artistic Movement and Artistic Dress &quot;ephemeral,&quot; but its rejection of corsetry looks far more consequential to us in hindsight than it did in the 19th century.

As Frith sees it, his painting critiques the &quot;craze&quot; associated with the women in this set of identifiable portraits who are not corseted, but his commitment to realism shows us a spectrum, a range, of conservatism and if not political then at least stylistic progressivism among the women. The progressives, oddly, are the women wearing artistic (that is, somewhat historical) dress, because they’re not corseted. It is a misreading to see the presentation of the women’s fashion as a simple opposition. Constance, Countess of Lonsdale — situated at the center of this painting with Frederick Leighton, president of the Royal Academy of Art — is the most conservatively dressed of the women depicted, with her narrow sleeves, tight waist and almost perfectly smooth bodice, which tells us that her corset has eyelets so that it can be laced precisely and tightly, and it has stays (or &quot;bones&quot;) to prevent wrinkles or natural folds in the overclothing. Lillie Langtry, in the white dress, with her stylish narrow sleeves, does not have such a tightly bound waist or smooth bodice, suggesting she may not be corseted at all, as we know she sometimes was not.['''citation'''] Jenny Trip, a painter’s model, is the woman in the green dress in the aesthetic group being inspected by Anthony Trollope, who may be taking notes. She looks like she is not wearing a corset. Both Langtry and Trip are toward the middle of this spectrum: neither is dressed in the more extreme artistic dress of, say, the two figures between Trip and Trollope.

A lot has been written about the late-Victorian attraction to historical dress, especially in the context of fancy-dress balls and the Gothic revival in social events as well as art and music. Part of the appeal has to have been the way those costumes could just be beautiful clothing beautifully made. Historical dress provided an opportunity for some elite women to wear less-structured but still beautiful and influential clothing. ['''Calvert'''&lt;ref&gt;Calvert, Robyne Erica. ''Fashioning the Artist: Artistic Dress in Victorian Britain 1848-1900''. Ph.D. thesis, University of Glasgow, 2012. &lt;nowiki&gt;https://theses.gla.ac.uk/3279/&lt;/nowiki&gt;&lt;/ref&gt;] The standards for beauty, then, with historical dress were Victorian, with the added benefit of possibly less structure. So, at the Duchess of Devonshire's ball, &quot;while some attendees tried to hew closely to historical precedent, many rendered their historical or mythological personage in the sartorial vocabulary they knew best. The [photographs of people in their costumes at the ball offer] a glimpse into how Victorians understood history, not a glimpse into the costume of an authentic historical past.&quot;&lt;ref&gt;Mitchell, Rebecca N. &quot;The Victorian Fancy Dress Ball, 1870–1900.&quot; ''Fashion Theory'' 2017 (21: 3): 291–315. DOI: 10.1080/1362704X.2016.1172817.&lt;/ref&gt; (294)

* historical dress: beautiful clothing.

* the range at the ball, from Minnie Paget to Gwladys
* &quot;In light of such efforts, the ball remains to this day one of the best documented outings of the period, and a quick glance at the album shows that ...&quot;

Women had more choices about their waists than the simple opposition between no corset and tightlacing can accommodate. The range of choices is illustrated in Frith's painting, with a woman locating herself on it at a particular moment for particular reasons. Much analysis of 19th-century corsetry focuses on its sexualizing effects — corsets dominated Victorian photographic pornography ['''citations'''] and at the same time, the absence of a corset was sexual because it suggested nudity.['''citations'''] A great deal of analysis of 19th-century corsetry, on the other hand, assumes that women wore corsets for the male gaze ['''citations'''] or that they tightened their waists to compete with other women.['''citations'''] 

But as we can see in Frith's painting, the sexualizing effect was not universal or sweeping, and these analyses do not account for the choices women had in which corset to wear or how tightly to lace it. Especially given the way that some photographic portraits were mechanically altered to make the waist appear smaller, the size of a woman's waist had to do with how she was presenting herself to the world. That is, the fact that women made choices about the size of or emphasis on their waists suggests that they had agency that needs to be taken into account. 

As they navigated the complex social world, women's fashion choices had meaning. Society or political hostesses had agency not only in their clothing but generally in that complex social world. They had roles managing social events of the upper classes, especially of the upper aristocracy and oligarchy, like the Duchess of Devonshire's ball. Their class and rank, then, were essential to their agency, including to some degree their freedom to choose what kind of corset to wear and how to wear it. Also, by the end of the century lots of different kinds of corsets were available for lots of different purposes. Special corsets existed for pregnancy, sports (like tennis, bicycling, horseback riding, golf, fencing, archery, stalking and hunting), theatre and dance and, of course, for these women corsets could be made to support the special dress worn over it.  

Women's choices in how they presented themselves to the world included more than just their foundation garments, of course. &quot;Every cap, bow, streamer, ruffle, fringe, bustle, glove,&quot; that is, the trim and decorations on their garments, their jewelry and accessories — which Davidoff calls &quot;elaborations&quot;&lt;ref name=&quot;:1&quot;&gt;Davidoff, Leonore. ''The Best Circles: Society Etiquette and the Season''. Intro., Victoria Glendinning. The Cressett Library (Century Hutchinson), 1986 (orig 1973).&lt;/ref&gt;{{rp|93}} — pointed to a host of status categories, like class, rank, wealth, age, marital status, engagement with the empire, how sexual they wanted to seem, political alignment and purpose at the social event. For example, when women were being presented to the monarch, they were expected to wear three ostrich plumes, often called the [[Social Victorians/Terminology#Prince of Wales's Feathers or White Plumes|Prince of Wales's feathers]].  

Like all fashions, the corset, which was quite long-lasting in all its various forms, eventually went out of style. Of the many factors that might have influenced its demise, perhaps most important was the women's movement, in which women's rights, freedom, employment and access to their own money and children were less slogan-worthy but at least as essential as votes for women. The activities of the animal-rights movements drew attention not only to the profligate use of the bodies and feathers of birds but also to the looming extinction of the baleen whale, which made whale bone scarce and expensive. Perhaps the century's debates over corseting and especially tightlacing were relevant to some decisions not to be corseted. And, of course, perhaps no other reason is required than that the nature of fashion is to change.  

== Cinque Cento ==
According to the ''Oxford English Dictionary'', ''Cinque Cento'' is a shortening of ''mil cinque cento'', or 1500.&lt;ref&gt;&quot;cinquecento, n.&quot; ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/33143. Accessed 7 February 2023.&lt;/ref&gt; The term, then would refer, perhaps informally, to the sixteenth century.

== Crevé ==
''Creve'', without the accent, is an old word in English (c. 1450) for burst or split.&lt;ref&gt;&quot;creve, v.&quot; ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/44339. Accessed 8 February 2023.&lt;/ref&gt; ['''With the acute accent, it looks like a past participle in French.''']

== Elastic ==
Elastic had been invented and was in use by the end of the 19th century. For the sense of &quot;Elastic cord or string, usually woven with india-rubber,&quot;&lt;ref name=&quot;:6&quot;&gt;“elastic, adj. &amp; n.”.  ''Oxford English Dictionary'', Oxford University Press,  September 2023, &lt;https://doi.org/10.1093/OED/1199670313&gt;.&lt;/ref&gt; the ''Oxford English Dictionary'' has usage examples beginning in 1847. The example for 1886 is vivid: &quot;The thorough-going prim man will always place a circle of elastic round his hair previous to putting on his college cap.&quot;&lt;ref name=&quot;:6&quot; /&gt;

== Elaborations ==
In her 1973 ''The Best Circles: Society, Etiquette and the Season'', Leonore Davidoff notes that women’s status was indicated by dress and especially ornament: “Every cap, bow, streamer, ruffle, fringe, bustle, glove and other elaboration,” she says, “symbolised some status category for the female wearer.”&lt;ref name=&quot;:1&quot; /&gt;{{rp|93}}

Looking at these elaborations as meaningful rather than dismissing them as failed attempts at &quot;historical accuracy&quot; reveals a great deal about the individual women who wore or carried them — and about the society women and political hostesses in their roles as managers of the social world. In her review of ''The House of Worth: Portrait of an Archive'', Mary Frances Gormally says,&lt;blockquote&gt;In a socially regulated year, garments custom made with a Worth label provided women with total reassurance, whatever the season, time of day or occasion, setting them apart as members of the “Best Circles” dressed in luxurious, fashionable and always appropriate attire (Davidoff 1973). The woman with a Worth wardrobe was a woman of elegance, lineage, status, extreme wealth and faultless taste.&lt;ref&gt;Gormally, Mary Frances. Review essay of ''The House of Worth: Portrait of an Archive'', by Amy de la Haye and Valerie D. Mendes (V&amp;A Publishing, 2014). ''Fashion Theory'' 2017 (21, 1): 109–126. DOI: 10.1080/1362704X.2016.1179400.&lt;/ref&gt; (117)&lt;/blockquote&gt;

[[File:Aglets from Spanish portraits - collage by shakko.jpg|thumb|alt=A collage of 12 different ornaments typically worn by elite people from Spain in the 1500s and later|Aglets — Detail from Spanish Portraits]]
=== Aglet, Aiglet ===
Historically, an aglet is a &quot;point or metal piece that capped a string [or ribbon] used to attach two pieces of the garment together, i.e., sleeve and bodice.&quot;&lt;ref name=&quot;:7&quot; /&gt;{{rp|4}} Although they were decorative, they were not always visible on the outside of the clothing. They were often stuffed inside the layers at the waist (for example, attaching the bodice to a skirt or breeches).

Alonso Sánchez Coello's c. 1584 (316) portrait (above right, in the [[Social Victorians/Terminology#16th Century|Hoops section]]) shows infanta Isabel Clara Eugenia wearing a vertugado, with its &quot;typically Spanish smooth cone-shaped contour,&quot; with &quot;handsome aiglets cascad[ing] down center front.&quot;&lt;ref name=&quot;:11&quot; /&gt; (315)

=== Frou-frou ===
In French, ''frou-frou'' or, spelled as ''froufrou'', is the sound of the rustling of silk or sometimes of fabrics in general.&lt;ref&gt;{{Cite journal|date=2023-07-25|title=frou-frou|url=https://fr.wiktionary.org/w/index.php?title=frou-frou&amp;oldid=32508509|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/frou-frou.&lt;/ref&gt; The first use the French ''Wiktionnaire'' lists is Honoré Balzac, ''La Cousine Bette'', 1846.&lt;ref&gt;{{Cite journal|date=2023-06-03|title=froufrou|url=https://fr.wiktionary.org/w/index.php?title=froufrou&amp;oldid=32330124|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/froufrou.&lt;/ref&gt;

''Frou-frou'' is a term clothing historians use to describe decorative additions to an article of clothing; often the term has a slight negative connotation, suggesting that the additions are superficial.

=== Pouf, Puff, Poof ===
According to the French ''Wikipédia'', a pouf was, beginning in 1744, a &quot;kind of women's hairstyle&quot;:&lt;blockquote&gt;The hairstyle in question, known as the “pouf”, had launched the reputation of the enterprising Rose Bertin, owner of the Grand Mogol, a very prominent fashion accessories boutique on Rue Saint-Honoré in Paris in 1774. Created in collaboration with the famous hairdresser, Monsieur Léonard, the pouf was built on a scaffolding of wire, fabric, gauze, horsehair, fake hair, and the client's own hair held up in an almost vertical position. — (Marie-Antoinette, ''Queen of Fashion'', translated from the American by Sylvie Lévy, in ''The Rules of the Game'', n° 40, 2009)&lt;/blockquote&gt;''Puff'' and ''poof'' are used to describe clothing.

=== Shirring ===
''Shirring'' is the gathering of fabric to make poufs or puffs. The 19th century is known for its use of this decorative technique. Even men's clothing had shirring: at the shoulder seam.

=== Sequins ===
Sequins, paillettes, spangles

Sequins — or paillettes — are &quot;small, scalelike glittering disks.&quot;&lt;ref name=&quot;:7&quot; /&gt;(216) The French ''Wiktionnaire'' defines ''paillette'' as &quot;Lamelle de métal, brillante, mince, percée au milieu, ordinairement ronde, et qu’on applique sur une étoffe pour l’orner [A strip of metal, shiny, thin, pierced in the middle, usually round, and which is applied to a fabric in order to decorate it.]&quot;&lt;ref name=&quot;:8&quot;&gt;{{Cite journal|date=2024-03-18|title=paillette|url=https://fr.wiktionary.org/w/index.php?title=paillette&amp;oldid=33809572|journal=Wiktionnaire, le dictionnaire libre|language=fr}} https://fr.wiktionary.org/wiki/paillette.&lt;/ref&gt;

According to the ''OED'', the use of ''sequin'' as a decorative device for clothing (as opposed to gold coins minted and used for international trade) goes back to the 1850s.&lt;ref&gt;“Sequin, N.” ''Oxford English Dictionary'', Oxford UP, September 2023,  https://doi.org/10.1093/OED/4074851670.&lt;/ref&gt; The first instance of ''spangle'' as &quot;A small round thin piece of glittering metal (usually brass) with a hole in the centre to pass a thread through, used for the decoration of textile fabrics and other materials of various sorts&quot; is from c. 1420.&lt;ref&gt;“Spangle, N. (1).” ''Oxford English Dictionary'', Oxford UP, July 2023,  https://doi.org/10.1093/OED/4727197141.&lt;/ref&gt; The first use of ''paillette'' listed in the French ''Wiktionnaire'' is in Jules Verne in 1873 to describe colored spots on icy walls.&lt;ref name=&quot;:8&quot; /&gt;

Currently many distinguish between sequins (which are smaller) and paillettes (which are larger).

Before the 20th century, sequins were metal discs or foil leaves, and so of course if they were silver or copper, they tarnished. It is not until well into the 20th century that plastics were invented and used for sequins.

=== Trim and Lace ===
''A History of Feminine Fashion'', published sometime before 1927 and probably commissioned by [[Social Victorians/People/Dressmakers and Costumiers#Worth, of Paris|the Maison Worth]], describes Charles Frederick Worth's contributions to the development of embroidery and [[Social Victorians/Terminology#Passementerie|passementerie]] (trim) from about the middle of the 19th century:&lt;blockquote&gt;For it must be remembered that one of M. Worth's most important and lasting contributions to the prosperity of those who cater for women's needs, as well as to the variety and elegance of his clients' garments, was his insistence on new fabrics, new trimmings, new materials of every description. In his endeavours to restore in Paris the splendours of the days of La Pompadour, and of Marie Antoinette, he found himself confronted at the outset with a grave difficulty, which would have proved unsurmountable to a man of less energy, resource and initiative. The magnificent materials of those days were no longer to be had! The Revolution had destroyed the market for beautiful materials of this, type, and the Restoration and regime of Louis Philippe had left a dour aspect in the City of Light. ... On parallel lines [to his development of better [[Social Victorians/Terminology#Satin|satin]]], [Worth] stimulated also the manufacture of embroidery and ''passementerie''. It was he who first started the manufacture of laces copied from the designs of the real old laces. He was the / first dressmaker to use fur in the trimming of light materials — but he employed only the richer furs, such as sable and ermine, and had no use whatever for the inferior varieties of skins.&lt;ref name=&quot;:9&quot; /&gt;{{rp|6–7}}&lt;/blockquote&gt;

==== Gold and Silver Fabric and Lace ====

The ''Encyclopaedia Britannica'' (9th edition) has an article on gold and silver fabric, threads and lace attached to the article on gold. (This article is based on knowledge that would have been available toward the end of the 19th century and does not, obviously, reflect current knowledge or ways of talking.)&lt;blockquote&gt;GOLD AND SILVER LACE. Under this heading a general account may be given of the use of the precious metals in textiles of all descriptions into which they enter. That these metals were used largely in the sumptuous textiles of the earliest periods of civilization there is abundant testimony; and to this day, in the Oriental centres whence a knowledge and the use of fabrics inwoven, ornamented, and embroidered with gold and silver first spread, the passion for such brilliant and costly textiles is still most strongly and generally prevalent. The earliest mention of the use of gold in a woven fabric occurs in the description of the ephod made for Aaron (Exod. xxxix. 2, 3) — &quot;And he made the ephod of gold, blue, and purple, and scarlet, and fine twined linen. And they did beat the gold into thin plates, and cut it into wires (strips), to work it in the blue, and in the purple, and in the scarlet, and in the fine linen, with cunning work.&quot; In both the ''Iliad'' and the ''Odyssey'' distinct allusion is frequently made to inwoven and embroidered golden textiles. Many circumstances point to the conclusion that the art of weaving and embroidering with gold and silver originated in India, where it is still principally prosecuted, and that from one great city to another the practice travelled westward, — Babylon, Tarsus, Baghdad, Damascus, the islands of Cyprus and Sicily, Con- / stantinople and Venice, all in the process of time becoming famous centres of these much prized manufactures. Alexander the Great found Indian kings and princes arrayed in robes of gold and purple; and the Persian monarch Darius, we are told, wore a war mantle of cloth of gold, on which were figured two golden hawks as if pecking at each other. There is reason, according to Josephus, to believe that the “royal apparel&quot; worn by Herod on the day of his death (Acts xii. 21) was a tissue of silver. Agrippina, the wife of the emperor Claudius, had a robe woven entirely of gold, and from that period downwards royal personages and high ecclesiastical dignitaries used cloth and tissues of gold and silver for their state and ceremonial robes, as well as for costly hangings and decorations. In England, at different periods, various names were applied to cloths of gold, as ciclatoun, tartarium, naques or nac, baudekiu or baldachin, Cyprus damask, and twssewys or tissue. The thin flimsy paper known as tissue paper, is so called because it originally was placed between the folds of gold &quot;tissue&quot; to prevent the contiguous surfaces from fraying each other. At what time the drawing of gold wire for the preparation of these textiles was first practised is not accurately known. The art was probably introduced and applied in different localities at widely different dates, but down till mediaeval times the method graphically described in the Pentateuch continued to be practised with both gold and silver.

Fabrics woven with gold and silver continue to be used on the largest scale to this day in India; and there the preparation of the varieties of wire, and the working of the various forms of lace, brocade, and embroidery, is at once an important and peculiar art. The basis of all modern fabrics of this kind is wire, the &quot;gold wire&quot; of the manufacturer being in all cases silver gilt wire, and silver wire being, of course, composed of pure silver. In India the wire is drawn by means of simple draw-plates, with rude and simple appliances, from rounded bars of silver, or gold-plated silver, as the case may be. The wire is flattened into the strip or ribbon-like form it generally assumes by passing it, fourteen or fifteen strands simultaneously, over a fine, smooth, round-topped anvil, and beating it as it passes with a heavy hammer having a slightly convex surface. From wire so flattened there is made in India soniri, a tissue or cloth of gold, the web or warp being composed entirely of golden strips, and ruperi, a similar tissue of silver. Gold lace is also made on a warp of thick yellow silk with a weft of flat wire, and in the case of ribbons the warp or web is composed of the metal. The flattened wires are twisted around orange (in the case of silver, white) coloured silk thread, so as completely to cover the thread and present the appearance of a continuous wire; and in this form it is chiefly employed for weaving into the rich brocades known as kincobs or kinkhábs. Wires flattened, or partially flattened, are also twisted into exceedingly fine spirals, and in this form they are the basis of numerous ornamental applications. Such spirals drawn out till they present a waved appearance, and in that state flattened, are much used for rich heavy embroideries termed karchobs. Spangles for embroideries, &amp;c., are made from spirals of comparatively stout wire, by cutting them down ring by ring, laying each C-like ring on an anvil, and by a smart blow with a hammer flattening it out into a thin round disk with a slit extending from the centre to one edge. Fine spirals are also used for general embroidery purposes. The demand for various kinds of loom-woven and embroidered gold and silver work in India is immense; and the variety of textiles so ornamented is also very great. &quot;Gold and silver,&quot; says Dr Birdwood in his ''Handbook to the British-Indian Section, Paris Exhibition'', 1878, &quot;are worked into the decoration of all the more costly loom-made garments and Indian piece goods, either on the borders only, or in stripes throughout, or in diapered figures. The gold-bordered loom embroideries are made chiefly at Sattara, and the gold or silver striped at Tanjore; the gold figured ''mashrus'' at Tanjore, Trichinopoly, and Hyderabad in the Deccau; and the highly ornamented gold-figured silks and gold and silver tissues principally at Ahmedabad, Benares, Murshedabad, and Trichinopoly.&quot;

Among the Western communities the demand for gold and silver lace and embroideries arises chiefly in connexion with naval and military uniforms, court costumes, public and private liveries, ecclesiastical robes and draperies, theatrical dresses, and the badges and insignia of various orders. To a limited extent there is a trade in gold wire and lace to India and China. The metallic basis of the various fabrics is wire round and flattened, the wire being of three kinds — 1st, gold wire, which is invariably silver gilt wire; 2d, copper gilt wire, used for common liveries and theatrical purposes; and 3d, silver wire. These wires are drawn by the ordinary processes, and the flattening, when done, is accomplished by passing the wire between a pair of revolving rollers of fine polished steel. The various qualities of wire are prepared and used in precisely the same way as in India, — round wire, flat wire, thread made of flat gold wire twisted round orange-coloured silk or cotton, known in the trade as &quot;orris,&quot; fine spirals and spangles, all being in use in the West as in the East. The lace is woven in the same manner as ribbons, and there are very numerous varieties in richness, pattern, and quality.  Cloth of gold, and brocades rich in gold and silver, are woven for ecclesiastical vestments and draperies.

The proportions of gold and silver in the gold thread for the lace trade varies, but in all cases the proportion of gold is exceedingly small. An ordinary gold lace wire is drawn from a bar containing 90 parts of silver and 7 of copper, coated with 3 parts of gold. On an average each ounce troy of a bar so plated is drawn into 1500 yards of wire; and therefore about 16 grains of gold cover a mile of wire. It is estimated that about 250,000 ounces of gold wire are made annually in Great Britain, of which about 20 per cent, is used for the headings of calico, muslin, &amp;c., and the remainder is worked up in the gold lace trade.&lt;ref&gt;William Chandler Roberts-Austen and H. Bauerman [W.C.R. — H.B.]. &quot;Gold and Silver Lace.&quot; In &quot;Gold.&quot; ''Encyclopaedia Britannica'', 9th Edition (1875–1889). Vol. 10 (X). Adam and Charles Black (Publisher). https://archive.org/details/encyclopaedia-britannica-9ed-1875/Vol%2010%20%28G-GOT%29%20193592738.23/page/753/mode/1up (accessed January 2023): 753, Col. 2c – 754, Cols. 1a–b – 2a–b.&lt;/ref&gt;&lt;/blockquote&gt;

==== Honiton Lace ====

Kate Stradsin says,&lt;blockquote&gt;Honiton lace was the finest English equivalent of Brussels bobbin lace and was constructed in small ‘sprigs, in the cottages of lacemakers[.'] These sprigs were then joined together and bleached to form the large white flounces that were so sought after in the mid-nineteenth century.&lt;ref&gt;Strasdin, Kate. &quot;Rediscovering Queen Alexandra’s Wardrobe: The Challenges and Rewards of Object-Based Research.&quot; ''The Court Historian'' 24.2 (2019): 181-196. Rpt  http://repository.falmouth.ac.uk/3762/15/Rediscovering%20Queen%20Alexandra%27s%20Wardrobe.pdf: 13, and (for the little quotation) n. 37, which reads &quot;Margaret Tomlinson, ''Three Generations in the Honiton Lace Trade: A Family History'', self-published, 1983.&quot;&lt;/ref&gt;&lt;/blockquote&gt;

[[File:Strook in Alençon naaldkant, 1750-1775.jpg|thumb|alt=A long piece of complex white lace with garlands, flowers and bows|Point d'Alençon lace, 1750-1775]]
==== Passementerie ====
''Passementerie'' is the French term for trim on clothing or furniture. The 19th century (especially during the First and Second Empire) was a time of great &quot;''exubérance''&quot; in passementerie in French design, including the development and widespread use of the Jacquard loom.&lt;ref&gt;{{Cite journal|date=2023-06-10|title=Passementerie|url=https://fr.wikipedia.org/w/index.php?title=Passementerie&amp;oldid=205068926|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Passementerie.&lt;/ref&gt;

==== Point d'Alençon Lace ====
A lace made by hand using a number of complex steps and layers. The lacemakers build the point d'Alençon design on some kind of mesh and sometimes leave some of the mesh in as part of the lace and perhaps to provide structure.

Elizabeth Lewandowski defines point d'Alençon lace and Alençon lace separately. Point lace is needlepoint lace,&lt;ref name=&quot;:7&quot;&gt;Lewandowski, Elizabeth J. ''The Complete Costume Dictionary''. Scarecrow Press, 2011.&lt;/ref&gt;{{rp|233}} so Alençon point is &quot;a two thread [needlepoint] lace.&quot;&lt;ref name=&quot;:7&quot; /&gt;{{rp|7}} Alençon lace has a &quot;floral design on [a] fine net ground [and is] referred to as [the] queen of French handmade needlepoint laces. The original handmade Alençon was a fine needlepoint lace made of linen thread.&quot;&lt;ref name=&quot;:7&quot; /&gt;{{rp|7}}

The sample of point d'Alençon lace (right), from 1750–1775, shows the linen mesh that the lace was constructed on.&lt;ref&gt;{{Cite web|url=http://openfashion.momu.be/#9ce5f00e-8a06-4dab-a833-05c3371f3689|title=MoMu - Open Fashion|website=openfashion.momu.be|access-date=2024-02-26}} ModeMuseum Antwerpen. http://openfashion.momu.be/#9ce5f00e-8a06-4dab-a833-05c3371f3689.&lt;/ref&gt; The consistency in this sample suggests it may have been made by machine.

== Fabric ==

=== Brocatelle ===
Brocatelle is a kind of brocade, more simple than most brocades because it uses fewer warp and weft threads and fewer colors to form the design. The article in the French ''Wikipédia'' defines it like this:&lt;blockquote&gt;La '''brocatelle''' est un type de tissu datant du &lt;abbr&gt;xvi&lt;sup&gt;e&lt;/sup&gt;&lt;/abbr&gt; siècle qui comporte deux chaînes et deux trames, au minimum. Il est composé pour que le dessin ressorte avec un relief prononcé, grâce à la chaîne sur un fond en sergé. Les brocatelles les plus anciennes sont toujours fabriquées avec une des trames en lin.&lt;ref&gt;{{Cite journal|date=2023-06-01|title=Brocatelle|url=https://fr.wikipedia.org/w/index.php?title=Brocatelle&amp;oldid=204796410|journal=Wikipédia|language=fr}} https://fr.wikipedia.org/wiki/Brocatelle.&lt;/ref&gt;&lt;/blockquote&gt;Which translates to this:&lt;blockquote&gt;Brocatelle is a type of fabric dating from the 16th century that has two warps and two wefts, at a minimum. It is composed so that the design stands out with a pronounced relief, thanks to the weft threads on a twill background. The oldest brocades were always made with one of the wefts being linen.&lt;/blockquote&gt;The ''Oxford English Dictionary'' says, brocatelle is an &quot;imitation of brocade, usually made of silk or wool, used for tapestry, upholstery, etc., now also for dresses. Both the nature and the use of the stuff have changed&quot; between the late 17th century and 1888, the last time this definition was revised.&lt;ref&gt;&quot;brocatelle, n.&quot; ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/23550. Accessed 4 July 2023.&lt;/ref&gt;

=== Broché ===

=== Ciselé ===

=== Crépe de Chine ===
The ''Oxford English Dictionary'' distinguishes the use of ''crêpe'' (using a circumflex rather than an acute accent over the first ''e'') from ''crape'' in textiles, saying ''crêpe'' is &quot;often borrowed [from the French] as a term for all crapy fabrics other than ordinary black mourning crape,&quot;&lt;ref&gt;&quot;crêpe, n.&quot; ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/44242. Accessed 10 February 2023.&lt;/ref&gt; with usage examples ranging from 1797 to the mid 20th century. Crêpe de chine, it says is &quot;a white or other coloured crape made of raw silk.&quot;

=== Épinglé Velvet ===
Often spelled ''épingle'' rather than ''épinglé'', this term appears to have been used for a fabric made of wool, or at least wool along with linen or cotton, that was heavier and stiffer than silk velvet. It was associated with outer garments and men's clothing. Nowadays, épinglé velvet is an upholstery fabric in which the pile is cut into designs and patterns, and the portrait of [[Social Victorians/People/Douglas-Hamilton Duke of Hamilton|Mary, Duchess of Hamilton]] shows a mantle described as épinglé velvet that does seem to be a velvet with a woven pattern perhaps cut into the pile. 

=== Lace ===
While lace also functioned sometimes as fabric — at the décolletage, for example, on the stomacher or as a veil — here we organize it as a [[Social Victorians/Terminology#Trim and Lace|part of the elaboration of clothing]].

=== Liberty Fabrics ===

=== Lisse ===
According to the ''Oxford English Dictionary'', the term ''lisse'' as a &quot;kind of silk gauze&quot; was used in the 19th-century UK and US.&lt;ref&gt;&quot;lisse, n.1.&quot; ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/108978. Accessed 4 July 2023.&lt;/ref&gt;

=== Satin ===
The pre-1927 ''History of Feminine Fashion'', probably commissioned by Charles Frederick Worth's sons, describes Worth's &quot;insistence on new fabrics, new trimmings, new materials of every description&quot; at the beginning of his career in the mid 19th century:&lt;blockquote&gt;When Worth first entered the business of dressmaking, the only materials of the richer sort used for woman's dress were velvet, faille, and watered silk. Satin, for example, was never used. M. Worth desired to use satin very extensively in the gowns he designed, but he was not satisfied with what could be had at the time; he wanted something very much richer than was produced by the mills at Lyons. That his requirements entailed the reconstruction of mills mattered little — the mills were reconstructed under his directions, and the Lyons looms turned out a richer satin than ever, and the manufacturers prospered accordingly.&lt;ref name=&quot;:9&quot;&gt;[Worth, House of.] {{Cite book|url=http://archive.org/details/AHistoryOfFeminineFashion|title=A History Of Feminine Fashion (1800s to 1920s)}} Before 1927. [Likely commissioned by Worth. Link is to Archive.org; info from Wikimedia Commons: https://commons.wikimedia.org/wiki/File:Worth_Biarritz_salon.jpg.]&lt;/ref&gt;{{rp|6 in printed, 26 in digital book}}&lt;/blockquote&gt;

=== Selesia ===
According to the ''Oxford English Dictionary'', ''silesia'' is &quot;A fine linen or cotton fabric originally manufactured in Silesia in what is now Germany (''Schlesien'').&lt;ref&gt;&quot;Silesia, n.&quot; ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/179664. Accessed 9 February 2023.&lt;/ref&gt; It may have been used as a lining — for pockets, for example — in garments made of more luxurious or more expensive cloth. The word ''sleazy'' — &quot;Of textile fabrics or materials: Thin or flimsy in texture; having little substance or body.&quot;&lt;ref&gt;&quot;sleazy, adj.&quot; ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/181563. Accessed 9 February 2023.&lt;/ref&gt; — may be related.

=== Shot Fabric ===
According to the ''Oxford English Dictionary'', &quot;Of a textile fabric: Woven with warp-threads of one colour and weft-threads of another, so that the fabric (usually silk) changes in tint when viewed from different points.&quot;&lt;ref&gt;“Shot, ''Adj.''”  ''Oxford English Dictionary'', Oxford UP,  July 2023,  https://doi.org/10.1093/OED/2977164390.&lt;/ref&gt; A shot fabric might also be made of silk and cotton fibers.

=== Tissue ===
A lightly woven fabric like gauze or chiffon. The light weave can make the fabric translucent and make pleating and gathering flatter and less bulky. Tissue can be woven to be shot, sheer, stiff or soft.

Historically, the term in English was used for a &quot;rich kind of cloth, often interwoven with gold or silver&quot; or &quot;various rich or fine fabrics of delicate or gauzy texture.&quot;&lt;ref&gt;“Tissue, N.” ''Oxford English Dictionary'', Oxford UP, March 2024, https://doi.org/10.1093/OED/5896731814.&lt;/ref&gt;

== Fan ==

The ''Encyclopaedia Britannica'' (9th edition) has an article on the fan. (This article is based on knowledge that would have been available toward the end of the 19th century and does not, obviously, reflect current knowledge or ways of talking.)&lt;blockquote&gt;FAN (Latin, ''vannus''; French, ''éventail''), a light implement used for giving motion to the air. ''Ventilabrum'' and ''flabellum'' are names under which ecclesiastical fans are mentioned in old inventories. Fans for cooling the face have been in use in hot climates from remote ages. A bas-relief in the British Museum represents Sennacherib with female figures carrying feather fans. They were attributes of royalty along with horse-hair fly-flappers and umbrellas. Examples may be seen in plates of the Egyptian sculptures at Thebes and other places, and also in the ruins of Persepolis. In the museum of Boulak, near Cairo, a wooden fan handle showing holes for feathers is still preserved. It is from the tomb of Amen-hotep, of the 18th dynasty, 17th century &lt;small&gt;B&lt;/small&gt;.&lt;small&gt;C&lt;/small&gt;. In India fans were also attributes of men in authority, and sometimes sacred emblems. A heartshaped fan, with an ivory handle, of unknown age, and held in great veneration by the Hindus, was given to the prince of Wales. Large punkahs or screens, moved by a servant who does nothing else, are in common use by Europeans in India at this day.

Fans were used in the early Middle Ages to keep flies from the sacred elements during the celebrations of the Christian mysteries. Sometimes they were round, with bells attached — of silver, or silver gilt. Notices of such fans in the ancient records of St Paul’s, London, Salisbury cathedral, and many other churches, exist still. For these purposes they are no longer used in the Western church, though they are retained in some Oriental rites. The large feather fans, however, are still carried in the state processions of the supreme pontiff in Rome, though not used during the celebration of the mass. The fan of Queen Theodolinda (7th century) is still preserved in the treasury of the cathedral of Monza. Fans made part of the bridal outfit, or ''mundus muliebris'', of ancient Roman ladies.

Folding fans had their origin in Japan, and were imported thence to China. They were in the shape still used—a segment of a circle of paper pasted on a light radiating frame-work of bamboo, and variously decorated, some in colours, others of white paper on which verses or sentences are written. It is a compliment in China to invite a friend or distinguished guest to write some sentiment on your fan as a memento of any special occasion, and this practice has continued. A fan that has some celebrity in France was presented by the Chinese ambassador to the Comtesse de Clauzel at the coronation of Napoleon I. in 1804. When a site was given in 1635, on an artificial island, for the settlement of Portuguese merchants in Nippo in Japan, the space was laid out in the form of a fan as emblematic of an object agreeable for general use. Men and women of every rank both in China and Japan carry fans, even artisans using them with one hand while working with the other. In China they are often made of carved ivory, the sticks being plates very thin and sometimes carved on both sides, the intervals between the carved parts pierced with astonishing delicacy, and the plates held together by a ribbon. The Japanese make the two outer guards of the stick, which cover the others, occasionally of beaten iron, extremely thin and light, damascened with gold and other metals.

Fans were used by Portuguese ladies in the 14th century, and were well known in England before the close of the reign of Richard II. In France the inventory of Charles V. at the end of the 14th century mentions a folding ivory fan. They were brought into general use in that country by Catherine de’ Medici, probably from Italy, then in advance of other countries in all matters of personal luxury. The court ladies of Henry VIII.’s reign in England were used to handling fans, A lady in the Dance of Death by Holbein holds a fan. Queen Elizabeth is painted with a round leather fan in her portrait at Gorhambury; and as many as twenty-seven are enumerated in her inventory (1606). Coryat, an English traveller, in 1608 describes them as common in Italy. They also became of general use from that time in Spain. In Italy, France, and Spain fans had special conventional uses, and various actions in handling them grew into a code of signals, by which ladies were supposed to convey hints or signals to admirers or to rivals in society. A paper in the ''Spectator'' humorously  proposes to establish a regular drill for these purposes.

The chief seat of the European manufacture of fans during the 17th century was Paris, where the sticks or frames, whether of wood or ivory, were made, and the decorations painted on mounts of very carefully prepared vellum (called latterly ''chicken skin'', but not correctly), — a material stronger and tougher than paper, which breaks at the folds. Paris makers exported fans unpainted to Madrid and other Spanish cities, where they were decorated by native artists. Many were exported complete; of old fans called Spanish a great number were in fact made in France. Louis XIV. issued edicts at various times to regulate the manufacture. Besides fans mounted with parchment, Dutch fans of ivory were imported into Paris, and decorated by the heraldic painters in the process called “Vernis Martin,” after a famous carriage painter and inventor of colourless lac varnish. Fans of this kind belonging to the Queen and to the late baroness de Rothschild were exhibited in 1870 at Kensington. A fan of the date of 1660, representing sacred subjects, is attributed to Philippe de Champagne, another to Peter Oliver in England in the / 17th century. Cano de Arevalo, a Spanish painter of the 17th century devoted himself to fan painting. Some harsh expressions of Queen Christina to the young ladies of the French court are said to have caused an increased ostentation in the splendour of their fans, which were set with jewels and mounted in gold. Rosalba Carriera was the name of a fan painter of celebrity in the 17th century. Lebrun and Romanelli were much employed during the same period. Klingstet, a Dutch artist, enjoyed a considerable reputation for his fans from the latter part of the 17th and the first thirty years of the 18th century.

The revocation of the edict of Nantes drove many fan-makers out of France to Holland and England. The trade in England was well established under the Stuart sovereigns. Petitions were addressed by the fan-makers to Charles II. against the importation of fans from India, and a duty was levied upon such fans in consequence. This importation of Indian fans, according to Savary, extended also to France. During the reign of Louis XV. carved Indian and China fans displaced to some extent those formerly imported from Italy, which had been painted on swanskin parchment prepared with various perfumes.

During the 18th century all the luxurious ornamentation of the day was bestowed on fans as far as they could display it. The sticks were made of mother-of-pearl or ivory, carved with extraordinary skill in France, Italy, England, and other countries. They were painted from designs of Boucher, Watteau, Lancret, and other &quot;genre&quot; painters, Hébert, Rau, Chevalier, Jean Boquet, Mad. Verité, are known as fan painters. These fashions were followed in most countries of Europe, with certain national differences. Taffeta and silk, as well as fine parchment, were used for the mounts. Little circles of glass were let into the stick to be looked through, and small telescopic glasses were sometimes contrived at the pivot of the stick. They were occasionally mounted with the finest point lace. An interesting fan (belonging to Madame de Thiac in France), the work of Le Flamand, was presented by the municipality of Dieppe to Marie Antoinette on the birth of her son the dauphin. From the time of the Revolution the old luxury expended on fans died out. Fine examples ceased to be exported to England and other countries. The painting on them represented scenes or personages connected with political events. At a later period fan mounts were often prints coloured by hand. The events of the day mark the date of many examples found in modern collections. Amongst the fanmakers of the present time the names of Alexandre, Duvelleroy, Fayet, Vanier, may be mentioned as well known in Paris. The sticks are chiefly made in the department of Oise, at Le Déluge, Crèvecœur, Méry, Ste Geneviève, and other villages, where whole families are engaged in preparing them; ivory sticks are carved at Dieppe. Water-colour painters of distinction often design and paint the mounts, the best designs being figure subjects. A great impulse has been given to the manufacture and painting of fans in England since the exhibition which took place at South Kensington in 1870. Other exhibitions have since been held, and competitive prizes offered, one of which was gained by the Princess Louise. Modern collections of fans take their date from the emigration of many noble families from France at the time of the Revolution. Such objects were given as souvenirs and occasionally sold by families in straitened circumstances. A large number of fans of all sorts, principally those of the 18th century, French, English, German, Italian Spanish, &amp;c., have been lately bequeathed to the South Kensington Museum.

Regarding the different parts of folding fans it may be well to state that the sticks are called in French ''brins'', the two outer guards ''panaches'', and the mount ''feuille''.&lt;ref&gt;J. H. Pollen [J.H.P.]. &quot;Fan.&quot; ''Encyclopaedia Britannica'', 9th Edition (1875–1889). Vol. '''10''' ('''X'''). Adam and Charles Black (Publisher). https://archive.org/details/encyclopaedia-britannica-9ed-1875/Vol%209%20%28FAL-FYZ%29%20193323016.23/page/26/mode/2up (accessed January 2023): 27, Col. 1b – 28, Col. 1c.&lt;/ref&gt;&lt;/blockquote&gt;

== Fancy-dress Ball ==
Fancy-dress (or costume) balls were popular and frequent in the U.K. and France as well as the rest of Europe during the 19th century. The themes and styles of the fancy-dress balls influenced those that followed. 

At the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]], the guests came dressed in costume from times before 1820, as instructed on '''the invitation''', but their clothing was much more about late-Victorian standards of beauty and fashion than the standards of whatever time period the portraits they were copying or basing their costumes on. 

''The Queen'' published dress and fashion information and advice under the byline of Ardern Holt, who regularly answered questions from readers about fashion as well as about fancy dress. (More about Ardern Holt, which is almost certainly a pseudonym, can be found on the [[Social Victorians/People/Working in Publishing#Journalists|People Working in Publishing]] page.) Holt also ran wrote entire articles with suggestions for what might make an appealing fancy-dress costume as well as pointing readers away from costumes that had been worn too frequently. The suggestions for costumes are based on familiar types or portraits available to readers, similar to Holt's books on fancy dress, which ran through a number of editions in the 1880s and 1890s. Fancy-dress questions sometimes asked for details about costumes worn in theatrical or operatic productions, which Holt provides. 

In November 1897, Holt refers to the Duchess of Devonshire's 2 July ball: &quot;Since the famous fancy ball, given at Devonshire House during this year, historical fancy dresses have assumed a prominence that they had not hitherto known.&quot;&lt;ref&gt;Holt, Ardern. &quot;Fancy Dress a la Mode.&quot; The ''Queen'' 27 November 1897, Saturday: 94 [of 145 in BNA; print p. 1026], Col. 1a [of 3]. ''British Newspaper Archive''  https://www.britishnewspaperarchive.co.uk/viewer/bl/0002627/18971127/459/0094.&lt;/ref&gt; Holt goes on to provide a number of ideas for costumes for historical fancy dress, as always with a strong leaning toward Victorian standards of beauty and style and away from any concern for historical accuracy.

Ardern Holt published books on fancy dress as well as writing for the ''Queen'' and other periodicals, but not all of them were about fancy dress.

# ''Gentlemen's Fancy Dress: How to Choose It''. Wyman &amp; Sons, 1882. (''Google Books'' has this: https://books.google.com/books/about/Gentlemen_s_Fancy_Dress.html?id=ED8CAAAAQAAJ.) Later editions: 1898 (HathiTrust)
# ''Fancy Dresses Described; Or, What to Wear at Fancy Balls''. Debenham &amp; Freebody, 1882. Illustr., Lillian Young. (HathiTrust has this.) Later editions: 4th ed — 1884; 1887 (HathiTrust); 6th ed. — 1896 (HathiTrust)


As Leonore Davidoff says, &quot;Every cap, bow, streamer, ruffle, fringe, bustle, glove and other elaboration symbolised some status category for the female wearer.&quot;&lt;ref name=&quot;:1&quot; /&gt;{{rp|93}} [handled under Elaborations]

=== Historical Accuracy ===
Many of the costumes at the ball were based on portraits, especially when the guest was dressed as a historical figure. If possible, we have found the portraits likely to have been the originals, or we have found, if possible, portraits that show the subjects from the two time periods at similar ages.

The way clothing was cut changed quite a bit between the 18th and 19th centuries. We think of Victorian clothing — particularly women's clothing, and particularly at the end of the century — as inflexible and restrictive, especially compared to 20th- and 21st-century customs permitting freedom of movement. The difference is generally evolutionary rather than absolute — that is, as time has passed since the 18th century, clothing has allowed an increasingly greater range of movement, especially for people who did not do manual labor.

By the end of the 19th century, garments like women's bodices and men's coats were made fitted and smooth by attention to the grain of the fabric and by the use of darts (rather than techniques that assembled many small, individual pieces of fabric).

* clothing construction and flat-pattern techniques
* Generally, the further back in time we go, the more 2-dimensional the clothing itself was. 

==== Women's Versions of Historical Accuracy at the Ball ====
As always with this ball, whatever historical accuracy might be present in a woman's costume is altered so that the wearer is still a fashionable Victorian lady. What makes the costumes look &quot;Victorian&quot; to our eyes is the line of the silhouette caused by the foundation undergarments as well as the many &quot;elaborations&quot;&lt;ref name=&quot;:1&quot; /&gt;{{rp|93}}, mostly in the decorations, trim and accessories. 

Also, the clothing hangs and drapes differently because the fabric was cut on grain and the shoulders were freed by the way the sleeves were set in. 

==== Men's Versions of Historical Accuracy at the Ball ====
Because men were not wearing a Victorian foundation garment at the end of the century, the men's costumes at the ball are more historically accurate in some ways.

* Trim
* Mixing neck treatments
* Hair
* Breeches
* Shoes and boots
* Military uniforms, arms, gloves, boots

== Feathers and Plumes ==

=== Aigrette ===
Elizabeth Lewandowski defines ''aigrette'' as &quot;France. Feather or plume from an egret or heron.&quot;&lt;ref name=&quot;:7&quot; /&gt;(5) Sometimes the newspapers use the term to refer to an accessory (like a fan or ornament on a hat) that includes such a feather or plume. The straight and tapered feathers in an aigrette are in a bundle.

=== Prince of Wales's Feathers or White Plumes ===
The feathers in an aigrette came from egrets and herons; Prince of Wales's feathers came from ostriches. A fuller discussion of Prince of Wales's feathers and the white ostrich plumes worn at court appears on [[Social Victorians/Victorian Things#Ostrich Feathers and Prince of Wales's Feathers|Victorian Things]].

For much of the late 18th and 19th centuries, white ostrich plumes were central to fashion at court, and at a certain point in the late 18th century they became required for women being presented to the monarch and for their sponsors. Our purpose here is to understand why women were wearing plumes at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] as part of their costumes.

First published in 1893, [[Social Victorians/People/Lady Colin Campbell|Lady Colin Campbell]]'s ''Manners and Rules of Good Society'' (1911 edition) says that&lt;blockquote&gt;It was compulsory for both Married and Unmarried Ladies to Wear Plumes. The married lady’s Court plume consisted of three white feathers. An unmarried lady’s of two white feathers. The three white feathers should be mounted as a Prince of Wales plume and worn towards the left hand side of the head. Colored feathers may not be worn. In deep mourning, white feathers must be worn, black feathers are inadmissible.&lt;p&gt;
White veils or lace lappets must be worn with the feathers. The veils should not be longer than 45 inches.&lt;ref&gt;{{Cite web|url=https://www.edwardianpromenade.com/etiquette/the-court-presentation/|title=The Court Presentation|last=Holl|first=Evangeline|date=2007-12-07|website=Edwardian Promenade|language=en-US|access-date=2022-12-18}} https://www.edwardianpromenade.com/etiquette/the-court-presentation/.&lt;/ref&gt;&lt;/blockquote&gt;[[Social Victorians/Victorian Things#Ostrich Feathers and Prince of Wales's Feathers|This fashion was imported from France]] in the mid 1770s.&lt;ref&gt;&quot;Abstract&quot; for Blackwell, Caitlin. &quot;'&lt;nowiki/&gt;''The Feather'd Fair in a Fright''': The Emblem of the Feather in Graphic Satire of 1776.&quot; ''Journal for Eighteenth-Century Studies'' 20 January 2013 (Vol. 36, Issue 3): 353-376. ''Wiley Online'' DOI: https://doi.org/10.1111/j.1754-0208.2012.00550.x (accessed November 2022).&lt;/ref&gt;

Separately, a secondary heraldic emblem of the Prince of Wales has been a specific arrangement of 3 ostrich feathers in a gold coronet&lt;ref&gt;{{Cite journal|date=2022-11-07|title=Prince of Wales's feathers|url=https://en.wikipedia.org/w/index.php?title=Prince_of_Wales%27s_feathers&amp;oldid=1120556015|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Prince_of_Wales's_feathers.&lt;/ref&gt; since King Edward III (1312–1377&lt;ref&gt;{{Cite journal|date=2022-12-14|title=Edward III of England|url=https://en.wikipedia.org/w/index.php?title=Edward_III_of_England&amp;oldid=1127343221|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Edward_III_of_England.&lt;/ref&gt;).

Some women at the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball]] wore white ostrich feathers in their hair, but most of them are not Prince of Wales's feathers. Most of the plumes in these portraits are arrangements of some kind of headdress to accompany the costume. A few, wearing what looks like the Princes of Wales's feathers, might be signaling that their character is royal or has royal ancestry. '''One of the women [which one?] was presented to the royals at this ball?'''

Here is the list of women who are wearing white ostrich plumes in their portraits in the [[Social Victorians/1897 Fancy Dress Ball/Photographs|''Diamond Jubilee Fancy Dress Ball'' album of 286 photogravure portraits]]:

# Kathleen Pelham-Clinton, the [[Social Victorians/People/Newcastle|Duchess of Newcastle]]
# [[Social Victorians/People/Louisa Montagu Cavendish|Luise Cavendish]], the Duchess of Devonshire
# Jesusa Murrieta del Campo Mello y Urritio (née Bellido), [[Social Victorians/People/Santurce|Marquisa de Santurce]]
# Lady [[Social Victorians/People/Farquhar|Emilie Farquhar]]
# Princess (Laura Williamina Seymour) Victor of  [[Social Victorians/People/Gleichen#Laura%20Williamina%20Seymour%20of%20Hohenlohe-Langenburg|Hohenlohe Langenburg]]
# Louisa Acheson, [[Social Victorians/People/Gosford|Lady Gosford]]
# Alice Emily White Coke, [[Social Victorians/People/Leicester|Viscountess Coke]]
# Lady Mary Stewart, Helen Mary Theresa [[Social Victorians/People/Londonderry|Vane-Tempest-Stewart]]
#[[Social Victorians/People/Consuelo Vanderbilt Spencer-Churchill|Consuelo Vanderbilt Spencer-Churchill]], Duchess of [[Social Victorians/People/Marlborough|Marlborough]], dressed as the wife of the French Ambassador at the Court of Catherine of Russia (not white, but some color that reads dark in the black-and-white photograph)
#Mrs. Mary [[Social Victorians/People/Chamberlain|Chamberlain]] (at 491), wearing white plumes, as Madame d'Epinay
#Lady Clementine [[Social Victorians/People/Tweeddale|Hay]] (at 629), wearing white plumes, as St. Bris (''Les Huguenots'')
#[[Social Victorians/People/Meysey-Thompson|Lady Meysey-Thompson]] (at 391), wearing white plumes, as Elizabeth, Queen of Bohemia
#Mrs. [[Social Victorians/People/Grosvenor|Algernon (Catherine) Grosvenor]] (at 510), wearing white plumes, as Marie Louise
#Lady [[Social Victorians/People/Ancaster|Evelyn Ewart]], at 401), wearing white plumes, as the Duchess of Ancaster, Mistress of the Robes to Queen Charlotte, 1757, after a picture by Hudson
#[[Social Victorians/People/Lyttelton|Edith Sophy Balfour Lyttelton]] (at 580), wearing what might be white plumes on a large-brimmed white hat, after a picture by Romney
#[[Social Victorians/People/Yznaga|Emilia Yznaga]] (at 360), wearing what might be white plumes, as Cydalise of the Comedie Italienne from the time of Louis XV
#Lady [[Social Victorians/People/Ilchester|Muriel Fox Strangways]] (at 403), wearing what might be two smallish white plumes, as Lady Sarah Lennox, one of the bridesmaids of Queen Charlotte A.D. 1761
#Lady [[Social Victorians/People/Lucan|Violet Bingham]] (at 586), wearing perhaps one white plume in a headdress not related to the Prince of Wales's feathers
#Rosamond Fellowes, [[Social Victorians/People/de Ramsey|Lady de Ramsey]] (at 329), wearing a headdress that includes some white plumes, as Lady Burleigh
#[[Social Victorians/People/Dupplin|Agnes Blanche Marie Hay-Drummond]] (at 682), in a big headdress topped with white plumes, as Mademoiselle Andrée de Taverney A.D. 1775
#Florence Canning, [[Social Victorians/People/Garvagh|Lady Garvagh]] (at 336), wearing what looks like Prince of Wales's plumes
#[[Social Victorians/People/Suffolk|Marguerite Hyde &quot;Daisy&quot; Leiter]] (at 684), wearing what looks like Prince of Wales's plumes
#Lady [[Social Victorians/People/Spicer|Margaret Spicer]] (at 281), wearing one smallish white and one black plume, as Countess Zinotriff, Lady-in-Waiting to the Empress Catherine of Russia
#Mrs. [[Social Victorians/People/Cavendish Bentinck|Arthur James]] (at 318), wearing what looks like Prince of Wales's plumes, as Elizabeth Cavendish, daughter of Bess of Hardwick
#Nellie, [[Social Victorians/People/Kilmorey|Countess of Kilmorey]] (at 207), wearing three tall plumes, 2 white and one dark, as Comtesse du Barri
#Daisy, [[Social Victorians/People/Warwick|Countess of Warwick]] (at 53), wearing at least 1 white plume, as Marie Antoinette

More men than women were wearing plumes reminiscent of the Prince of Wales's feathers:

*

==== Bibliography for Plumes and Prince of Wales's Feathers ====
* Blackwell, Caitlin. &quot;'''The Feather'd Fair in a Fright'&lt;nowiki/&gt;'': The Emblem of the Feather in Graphic Satire of 1776.&quot; Journal for ''Eighteenth-Century Studies'' 20 January 2013 (Vol. 36, Issue 3): 353-376. Wiley Online DOI: https://doi.org/10.1111/j.1754-0208.2012.00550.x.
* &quot;Prince of Wales's feathers.&quot; ''Wikipedia'' https://en.wikipedia.org/wiki/Prince_of_Wales%27s_feathers (accessed November 2022). ['''Add women to this page''']
* Simpson, William. &quot;On the Origin of the Prince of Wales' Feathers.&quot; ''Fraser's magazine'' 617 (1881): 637-649. Hathi Trust https://babel.hathitrust.org/cgi/pt?id=chi.79253140&amp;view=1up&amp;seq=643&amp;q1=feathers (accessed December 2022). Deals mostly with use of feathers in other cultures and in antiquity; makes brief mention of feathers and plumes in signs and pub names that may not be associated with the Prince of Wales. No mention of the use of plumes in women's headdresses or court dress.

== Honors ==

=== The Bath ===
The Most Honourable Order of the Bath (GCB, Knight or Dame Grand Cross; KCB or DCB, Knight or Dame Commander; CB, Companion)

=== The Garter ===
The Most Noble Order of the Knights of the Garter (KG, Knight Companion; LG, Lady Companion)

[[File:The Golden Fleece - collar exhibited at MET, NYC.jpg|thumb|The Golden Fleece collar and pendant for the 2019 &quot;Last Knight&quot; exhibition at the MET, NYC.|alt=Recent photograph of a gold necklace on a wide band, with a gold skin of a sheep hanging from it as a pendant]]
=== The Golden Fleece ===
To wear the golden fleece is to wear the insignia of the Order of the Golden Fleece, said to be &quot;the most prestigious and historic order of chivalry in the world&quot; because of its long history and strict limitations on membership.&lt;ref name=&quot;:10&quot;&gt;{{Cite journal|date=2020-09-25|title=Order of the Golden Fleece|url=https://en.wikipedia.org/w/index.php?title=Order_of_the_Golden_Fleece&amp;oldid=980340875|journal=Wikipedia|language=en}}&lt;/ref&gt; The monarchs of the U.K. were members of the originally Spanish order, as were others who could afford it, like the Duke of Wellington,&lt;ref name=&quot;:12&quot;&gt;Thompson, R[obert]. H[ugh]. &quot;The Golden Fleece in Britain.&quot; Publication of the ''British Numismatic Society''. 2009 https://www.britnumsoc.org/publications/Digital%20BNJ/pdfs/2009_BNJ_79_8.pdf (accessed January 2023).&lt;/ref&gt; the first Protestant to be admitted to the order.&lt;ref name=&quot;:10&quot; /&gt; Founded in 1429/30 by Philip the Good, Duke of Burgundy, the order separated into two branches in 1714, one Spanish and the other Austrian, still led by the House of Habsburg.&lt;ref name=&quot;:10&quot; /&gt;

[[File:Prince Albert - Franz Xaver Winterhalter 1842.jpg|thumb|1842 Winterhalter portrait of Prince Albert wearing the insignia of the Order of the Golden Fleece, 1842|left|alt=1842 Portrait of Prince Albert by Winterhalter, wearing the insignia of the Golden Fleece]]
The photograph (upper right) is of a Polish badge dating from the &quot;turn of the XV and XVI centuries.&quot;&lt;ref&gt;{{Citation|title=Polski:  Kolana orderowa orderu Złotego Runa, przełom XV i XVI wieku.|url=https://commons.wikimedia.org/wiki/File:The_Golden_Fleece_-_collar_exhibited_at_MET,_NYC.jpg|date=2019-11-10|accessdate=2023-01-10|last=Wulfstan}}. https://commons.wikimedia.org/wiki/File:The_Golden_Fleece_-_collar_exhibited_at_MET,_NYC.jpg.&lt;/ref&gt; The collar to this Golden Fleece might be similar to the one the [[Social Victorians/People/Spencer Compton Cavendish#The Insignia of the Order of the Golden Fleece|Duke of Devonshire is wearing in the 1897 Lafayette portrait]].

The badges and collars that Knights of the Order actually wore vary quite a bit.

The 1842 Franz Xaver Winterhalter portrait (left) of Prince Consort Albert, Victoria's husband and father of the Prince of Wales, shows him wearing the Golden Fleece on a red ribbon around his neck and the star of the Garter on the front of his coat.&lt;ref&gt;Winterhalter, Franz Xaver. ''Prince Albert''. {{Cite web|url=https://www.rct.uk/collection/search#/16/collection/401412/prince-albert-1819-61|title=Explore the Royal Collection Online|website=www.rct.uk|access-date=2023-01-16}} https://www.rct.uk/collection/search#/16/collection/401412/prince-albert-1819-61.&lt;/ref&gt;

=== Royal Victorian Order ===
(GCVO, Knight or Dame Grand Cross; KCVO or DCVO, Knight or Dame Commander; CVO, Commander; LVO, Lieutenant; MVO, Member)

=== St. John ===
The Order of the Knights of St. John

=== Star of India ===
Most Exalted Order of the Star of India (GCSI, Knight Grand Commander; KCSI, Knight Commander; CSI, Companion)

=== Thistle ===
The Most Ancient and Most Noble Order of the Thistle

== Jewelry and Stones ==

=== Cabochon ===
This term describes both the treatment and shape of a precious or semiprecious stone. A cabochon treatment does not facet the stone but merely polishes it, removing &quot;the rough parts&quot; and the parts that are not the right stone.&lt;ref&gt;&quot;cabochon, n.&quot; ''OED Online'', Oxford University Press, December 2022, www.oed.com/view/Entry/25778. Accessed 7 February 2023.&lt;/ref&gt; A cabochon shape is often flat on one side and oval or round, forming a mound in the setting.

=== Jet ===

=== ''Orfèvrerie'' ===
Sometimes misspelled in the newspapers as ''orvfèvrerie''. ''Orfèvrerie'' is the artistic work of a goldsmith, silversmith, or jeweler.

=== Turquoises ===

== Military ==
Several men from the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 1897 fancy-dress ball at Devonshire House]] were dressed in military uniforms, some historical and some, possibly, not.

=== Baldric ===
According to the ''Oxford English Dictionary'', the primary sense of ''baldric'' is &quot;A belt or girdle, usually of leather and richly ornamented, worn pendent from one shoulder across the breast and under the opposite arm, and used to support the wearer's sword, bugle, etc.&quot;&lt;ref&gt;&quot;baldric, n.&quot; ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/14849. Accessed 17 May 2023.&lt;/ref&gt; This sense has been in existence since c. 1300.

=== Cuirass ===
According to the ''Oxford English Dictionary'', the primary sense of ''cuirass'' is &quot;A piece of armour for the body (originally of leather); ''spec.'' a piece reaching down to the waist, and consisting of a breast-plate and a back-plate, buckled or otherwise fastened together ....&quot;&lt;ref&gt;&quot;cuirass, n.&quot; ''OED Online'', Oxford University Press, March 2023, www.oed.com/view/Entry/45604. Accessed 17 May 2023.&lt;/ref&gt;

[[File:Knötel IV, 04.jpg|thumb|alt=An Old drawing in color of British soldiers on horses brandishing swords in 1815.|1890 illustration of the Household Cavalry (Life Guard, left; Horse Guard, right) at the Battle of Waterloo, 1815]]
=== Household Cavalry ===
The Royal Household contains the Household Cavalry, a corps of British Army units assigned to the monarch. It is made up of 2 regiments, the Life Guards and what is now called The Blues and Royals, which were formed around the time of &quot;the Restoration of the Monarchy in 1660.&quot;&lt;ref name=&quot;:3&quot;&gt;Joll, Christopher. &quot;Tales of the Household Cavalry, No. 1. Roles.&quot; The Household Cavalry Museum, https://householdcavalry.co.uk/app/uploads/sites/2/2021/06/Household-Cavalry-Museum-video-series-large-print-text-Tales-episode-01.pdf.&lt;/ref&gt;{{rp|1}} Regimental Historian Christopher Joll says, &quot;the original Life Guards were formed as a mounted bodyguard for the exiled King Charles II, The Blues were raised as Cromwellian cavalry and The Royals were established to defend Tangier.&quot;&lt;ref name=&quot;:3&quot; /&gt;{{rp|1–2}} The 1st and 2nd Life Guards were formed from &quot;the Troops of Horse and Horse Grenadier Guards ... in 1788.&quot;&lt;ref name=&quot;:3&quot; /&gt;{{rp|3}} The Life Guards were and are still official bodyguards of the queen or king, but through history they have been required to do quite a bit more than serve as bodyguards for the monarch.

The Household Cavalry fought in the Battle of Waterloo on Sunday, 18 June 1815 as heavy cavalry.&lt;ref name=&quot;:3&quot; /&gt;{{rp|3}} Besides arresting the Cato Steet conspirators in 1820 &quot;and guarding their subsequent execution,&quot; the Household Cavalry contributed to the &quot;the expedition to rescue General Gordon, who was trapped in Khartoum by The Mahdi and his army of insurgents&quot; in 1884.&lt;ref name=&quot;:3&quot; /&gt;{{rp|3}} In 1887 they &quot;were involved ... in the suppression of rioters in Trafalgar Square on Bloody Sunday.&quot;&lt;ref name=&quot;:3&quot; /&gt;{{rp|3}} 

==== Grenadier Guards ====
Three men — [[Social Victorians/People/Gordon-Lennox#Lord Algernon Gordon Lennox|Lord Algernon Gordon-Lennox]], [[Social Victorians/People/Stanley#Edward George Villiers Stanley, Lord Stanley|Lord Stanley]], and [[Social Victorians/People/Stanley#Hon. Ferdinand Charles Stanley|Hon. F. C. Stanley]] — attended the ball as officers of the Grenadier Guards, wearing &quot;scarlet tunics, ... full blue breeches, scarlet hose and shoes, lappet wigs&quot; as well as items associated with weapons and armor.&lt;ref name=&quot;:14&quot;&gt;“The Duchess of Devonshire’s Ball.” The ''Gentlewoman'' 10 July 1897 Saturday: 32–42 [of 76], Cols. 1a–3c [of 3]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0003340/18970710/155/0032.&lt;/ref&gt;{{rp|p. 34, Col. 2a}}

Founded in England in 1656 as Foot Guards, this infantry regiment &quot;was granted the 'Grenadier' designation by a Royal Proclamation&quot; at the end of the Napoleonic Wars.&lt;ref&gt;{{Cite journal|date=2023-04-22|title=Grenadier Guards|url=https://en.wikipedia.org/w/index.php?title=Grenadier_Guards&amp;oldid=1151238350|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Grenadier_Guards.&lt;/ref&gt; They were not called Grenadier Guards, then, before about 1815. In 1660, the Stuart Restoration, they were called Lord Wentworth's Regiment, because they were under the command of Thomas Wentworth, 5th Baron Wentworth.&lt;ref&gt;{{Cite journal|date=2022-07-24|title=Lord Wentworth's Regiment|url=https://en.wikipedia.org/w/index.php?title=Lord_Wentworth%27s_Regiment&amp;oldid=1100069077|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Lord_Wentworth%27s_Regiment.&lt;/ref&gt;

At the time of Lord Wentworth's Regiment, the style of the French cavalier had begun to influence wealthy British royalists. In the British military, a Cavalier was a wealthy follower of Charles I and Charles II — a commander, perhaps, or a field officer, but probably not a soldier.&lt;ref&gt;{{Cite journal|date=2023-04-22|title=Cavalier|url=https://en.wikipedia.org/w/index.php?title=Cavalier&amp;oldid=1151166569|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Cavalier.&lt;/ref&gt;

The Guards were busy as infantry in the 17th century, engaging in a number of armed conflicts for Great Britain, but they also served the sovereign. According to the Guards Museum,&lt;blockquote&gt;In 1678 the Guards were ordered to form Grenadier Companies, these men were the strongest and tallest of the regiment, they carried axes, hatches and grenades, they were the shock troops of their day. Instead of wearing tri-corn hats they wore a mitre shaped cap.&lt;ref&gt;{{Cite web|url=https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-2/|title=Service to the Crown|website=The Guards Museum|language=en-GB|access-date=2023-05-15}} https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-2/.&lt;/ref&gt;&lt;/blockquote&gt;The name comes from ''grenades'', then, and we are accustomed to seeing them in front of Buckingham Palace, with their tall mitre hats.

The Guard fought in the American Revolution, and in the 19th century, the Grenadier Guards fought in the Crimean War, Sudan and the Boer War. They have roles as front-line troops and as ceremonial for the sovereign, which makes them elite:&lt;blockquote&gt;Queen Victoria decreed that she did not want to see a single chevron soldier within her Guards. Other then [sic] the two senior Warrant Officers of the British Army, the senior Warrant Officers of the Foot Guards wear a large Sovereigns personal coat of arms badge on their upper arm. No other regiments of the British Army are allowed to do so; all the others wear a small coat of arms of their lower arms. Up until 1871 all officers in the Foot Guards had the privilege of having double rankings. An Ensign was ranked as an Ensign and Lieutenant, a Lieutenant as Lieutenant and Captain and a Captain as Captain and Lieutenant Colonel. This was because at the time officers purchased their own ranks and it cost more to purchase a commission in the Foot Guards than any other regiments in the British Army. For example if it cost an officer in the Foot Guards £1,000 for his first rank, in the rest of the Army it would be £500 so if he transferred to another regiment he would loose [sic] £500, hence the higher rank, if he was an Ensign in the Guards and he transferred to a Line Regiment he went in at the higher rank of Lieutenant.&lt;ref&gt;{{Cite web|url=https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-1/|title=Formation and role of the Regiments|website=The Guards Museum|language=en-GB|access-date=2023-05-15}} https://theguardsmuseum.com/about-the-guards/history-of-the-foot-guards/history-page-1/.&lt;/ref&gt;&lt;/blockquote&gt;

==== Life Guards ====
[[Social Victorians/People/Shrewsbury#Reginald Talbot's Costume|General the Hon. Reginald Talbot]], a member of the 1st Life Guards, attended the Duchess of Devonshire's ball dressed in the uniform of his regiment during the Battle of Waterloo.&lt;ref name=&quot;:14&quot; /&gt;{{rp|p. 36, Col. 3b}}

At the Battle of Waterloo the 1st Life Guards were part of the 1st Brigade — the Household Brigade — and were commanded by Major-General Lord Edward Somerset.&lt;ref name=&quot;:4&quot;&gt;{{Cite journal|date=2023-09-30|title=Battle of Waterloo|url=https://en.wikipedia.org/w/index.php?title=Battle_of_Waterloo&amp;oldid=1177893566|journal=Wikipedia|language=en}} https://en.wikipedia.org/wiki/Battle_of_Waterloo.&lt;/ref&gt; The 1st Life Guards were on &quot;the extreme right&quot; of a French countercharge and &quot;kept their cohesion and consequently suffered significantly fewer casualties.&quot;&lt;ref name=&quot;:4&quot; /&gt;

== Photography ==

== Footnotes ==
{{reflist}}</text>
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    <title>What is the impact of open-source AI?</title>
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      <text bytes="9944" sha1="bxhh0y87lxcmyjf7s4eqhlg9lz1t9cw" xml:space="preserve">{{Wikidebate}}
AI is a powerful tool that can be used to influence people's values, beliefs, and attitudes, but should it? Why or why not?

== Open-Source AI is likely to be a positive force in the world ==

=== Pro ===

* {{Argument for}} Open-Source AI can democratize access to powerful tools which can lead to scientific breakthroughs that can lead to positive impact, like curing diseases.
* {{Argument for}} Open-Source AI will enable entrepreneurs to develop products and services to make life easier, and save humans precious time doing mundane tasks.
* {{Argument for}}  ''&quot;Give us back our time!&quot; -'' People spend a majority of their lives at work. Having an AI tool that can expedite their task and enhances their productivity can give them back time to focus on their personal endeavors.
* {{Argument for}} It is important that AI used in impactful ways such as in medicine is {{w|Algorithmic transparency|transparent}}.
* {{Argument for}} It is important that AI used in impactful ways such as in medicine is designed for the public interest.
* {{Argument for}} People's privacy and data can be better be protected if AI is open source&lt;ref&gt;https://www.nature.com/articles/d41586-023-03803-y&lt;/ref&gt; and this may even be a requirement.
** {{Objection}} This especially is a requirement if you are working for most companies and would like to use AI LLMs to code, you need to give proprietary code.

=== Con ===
* {{Argument against}} Outsourcing tasks to AI can result in a narrowing of the scope of paid labor, significantly raising the threshold for hiring talent to perform paid work that otherwise cannot be completed effectively by AI
** {{Objection}}  As long as the financial benefits of automation and AI are sufficiently distributed in an authentically ethical way, then automation can help move all of humanity to a post scarcity state of abundance. AI &quot;taking jobs&quot; will be a good thing and not a problem, as long as the increased efficiently and productivity from jobs being &quot;taken&quot; by AI are felt sufficiently by all members of society. For example, an automation tax could be utilized to fund something similar to the Alaskan oil dividends distributed each year (but perhaps something significantly more substantial, depending on to what degree AI and technological automation eliminate jobs. It may be that an open source AI and related technological automation allows all humans to live lives of abundance and post-scarcity; that is, perhaps to live lives of abundance, 99% of all humans only need to &quot;work&quot; 1 to 4 hours per week, and the rest of their time can be spent on chosen pursuits, weather that be volunteering, recreation, spending time with family and friends, traveling, inventing, researching, writing, creating art, hobbies, and/or so on and so forth.
*** {{Objection}} There is no evidence or conclusive proof that automation can help move all of humanity to a post scarcity state of abundance, given the natural resource constraints. The four-hour week is arguably an unrealistic fantasy.
**** {{Objection}} - &quot;[...] that automation can help move all of humanity to a post scarcity state of abundance, given the natural resource constraints.&quot; is a theoretical possibility. There are already efforts to colonize Mars. With an internet search (&quot;solar system sustain a trillion&quot; without quotes) one can see that at some have stated that with space stations that this solar system could sustain a trillion humans. There is no proof that all humans cannot live lives of abundance either; and there is no proof that technological automation combined with ethical capitalism and just laws cannot help all humans to live lives of abundance and reduce the need for human labor to 32, 24, 16, 8, or 4 hours of labor/work per week. This is a theoretical possibility and not really disprovable nor provable until it either happens or never happens after an infinite amount of time.
***** {{Objection}} As for &quot;There are already efforts to colonize Mars&quot;: no such ''serious'' efforts are ongoing; rather, there is an effort to get into a potentially lucrative satellite Internet business.
****** {{Objection}} Although Elon Musk has made wrong predictions about feasibility/timing in the past, (like about how fast self driving cars will be functional and common) Elon Musk is the wealthiest human on Earth who has and can allocate billions of dollars in capital for research and development, who has developed several billion dollar companies, and Elon Musk has stated potentially credible plans about how humanity will colonize Mars.&lt;ref&gt;https://www.spacex.com/humanspaceflight/mars/&lt;/ref&gt;&lt;ref&gt;https://twitter.com/elonmusk/status/1622746027152273409?lang=en&lt;/ref&gt;&lt;ref&gt;https://twitter.com/elonmusk/status/1544636989647032321?lang=en&lt;/ref&gt;&lt;ref&gt;https://www.indiatoday.in/technology/news/story/elon-musk-says-humans-should-have-cities-on-mars-and-a-moon-base-2477516-2023-12-18&lt;/ref&gt;&lt;ref&gt;https://www.youtube.com/watch?v=V2AyDjcGRrk&lt;/ref&gt;
******* {{Objection}} The above statements are far from sufficient for the conclusion that Elon Musk genuinely/sincerely intends to colonize Mars (rather than expand rocket making business for various other ends), given the likely conclusion from [[Is colonization of Mars in this century realistic?]] and [https://www.procon.org/headlines/space-colonization-top-3-pros-and-cons/ Space Colonization - Pros &amp; Cons], procon.org. One item singled out: If he was serious about the project, he would be running a new version of Biosphere 2 project to ensure preparatory steps other than rocket making. Given Elon Musk's apparent intelligence and ability, it is unlikely he believes the things he is saying about Mars and space colonization.
***** {{Objection}} The above are arguably wild science-fiction fantasies with no basis in known reality, especially the idea that the Solar System can sustain a trillion (1,000,000,000,000) humans. See also [[Is colonization of Mars in this century realistic?]]. More is probably for a separate debate.
****** {{Objection}} You cannot predict distant future. Some stated that planes flying was impossible before that happened. A trillion humans in this solar system is something that in theory could happen in several hundred or several thousand years. To state that this distant theoretical (currently science fiction) possibility is impossible is not really falsifiable nor provable in a any real way. One could have said the same thing about The Space Station 1000 years ago. Regardless, this seems to becoming rather tangential to the original debate topic, &quot;What is the impact of open-source AI?&quot;. Perhaps another debate can be started, [[Can this Solar System potentially support a prosperous human population of over 500 billion humans who can all live lives of abundance and post-scarcity?]].&lt;ref&gt;https://slate.com/technology/2022/03/how-many-humans-solar-system-dyson-sphere.html&lt;/ref&gt;
******* {{Objection}} One can predict distant future, e.g. that Sun is going to expand and make the Earth uninhabitable.
******* {{Objection}} A statement of impossibility is in fact falsifiable/testable: it is refuted once someone makes the thing possible. By contrast, a statement of the form &quot;X will possibly happen in future&quot; cannot be directly falsified/refuted: no amount of X not happening is going to refute the possibility; and that casts doubt on its scientific character (what is not falsifiable, is not scientific).
******* {{Objection}} Inference of the form &quot;we did not anticipate planes, therefore anything is possible&quot; is invalid.
******* {{Objection}} To analyze the impact of AI in terms of extremely unlikely outcomes is to muddy the waters of the discussion and distract from realistic considerations.
******* {{Objection}} A de facto science-fiction article from the website of the grade that slate.com has can hardly be considered a serious argument.
***** {{Objection}} The concept of a purely theoretical possibility has almost no cognitive value; it is a theoretical, speculative possibility that we live in a simulation and that the simulation operator decides to halt it and solve many of our problems. There is almost no practical value in these kinds of speculations.
* {{Argument against}} Open-Source AI can democratize access to &quot;catastrophe weapons,&quot; meaning it can make it easier to develop chemical and biological weapons, plan kinetic attacks, and deploy cyber attacks.
** {{Objection}}  ''&quot;Regulations would stop it from getting to that point!&quot; -'' Global institutions would acknowledge the potential hazards posed by open-source AI and take proactive steps to safeguard against the development of such technology reaching critical, catastrophic capabilities. These measures might include rigorous monitoring, recurrent testing, and other security protocols. 
* {{Argument against}} Open-Source AI means that hostile foreign nations can access these tools freely, and use them to manipulate and control their populations, such as by generating fake news media to propagandize the public.
** {{Objection}} They can also do so with foreign proprietary AI. If they can't, they can develop it themselves.
** {{Objection}} This depends on which AI soft is made open source.
* {{Argument against}} '''AI not steerable&quot; -'''. Current generations of publicly available LLMs are seemingly unable to be censored. Simple prompt hacking routinely get even the most RLHFed models to provide dangerous information.
** {{Objection}} &quot;dangerous information&quot; can also be gained from properly searching the web and the information itself more or less isn't dangerous or isn't the dangerous factor.

== External links ==
* [https://www.kialo.com/is-keeping-ai-closed-source-safer-and-better-for-society-than-open-sourcing-ai-62470 Is keeping AI closed source safer and better for society than open sourcing AI?], largest structured {{w|arguments map}} relating to this subject on {{w|Kialo}}

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    <title>Understanding Emergence</title>
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      <text bytes="68225" sha1="ozdjboozzejid9boh3wv4z229h3tnji" xml:space="preserve">—Exploring the possible

[[File:Starling flock with nearby predator.jpg|thumb|Starlings flocking, a predator bird can be seen upper right]]

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== Introduction ==
[[w:Emergence|Emergence]], a phenomenon both awe-inspiring and enigmatic, lies at the heart of the intricate tapestry of the universe.&lt;ref&gt;[[w:ChatGPT|ChatGPT]] contributed to this text, responding to the prompt: “Write an extended article on the topic of emergence. Include examples from the Steven Strogatz book Sync along with other examples. Describe the concepts of weak emergence and strong emergence and provide examples of each. Discuss evolution in the context of emergence.” The text has since been extensively augmented and edited. &lt;/ref&gt; It is the phenomenon through which simple components interacting on a local level give rise to complex and novel behaviors at higher levels of organization. This course delves into the concept of emergence, drawing examples from nature and society. We will explore the nuances of weak and strong emergence and delve into the role of emergence in the context of evolution.

Remarkably, everything in the universe [[w:Emergence|emerged]] from the basic laws of physics. Quarks and electrons combine to form atoms which combine into molecules and form all materials. Air, water, rocks, and minerals emerge from various combinations of chemical elements.  Rain, wind, storms, lightening, and fire also emerged. Oceans formed, mountains rose, volcanos erupted, earthquakes shook, and eventually life emerged in the form of [[w:Microorganism|microorganisms]], plants, and animals.

Each of these transformations are remarkable, yet although each is really new, there is nothing really new.

While [[Thinking Scientifically/The role and limitations of scientific reduction|reductionism]] tells us what there is, emergence helps us understand what there can be; what can be constructed. 

== Objectives ==
The objectives of this course are to help students:

# Identify emergent phenomenon. 
# Explore the possibilities of emergence.
# Understand the limitations of emergence.
# Evaluate and understand claims of emergence. 
# Distinguish between a phase transition and emergence.
# Analyze the implications of a claim such as “B emerged from A”.
# Distinguish claims of weak emergence and strong emergence.
# Augment [[Thinking Scientifically/The role and limitations of scientific reduction|scientific reduction]].  

This course is part of the [[Wisdom/Curriculum|Applied Wisdom curriculum]] and of the [[Deductive Logic/Clear Thinking curriculum|Clear Thinking curriculum]].

If you wish to contact the instructor, please [[Special:EmailUser/Lbeaumont|click here to send me an email]] or leave a comment or question on the [[Talk:Understanding_Emergence|discussion page]].

== Characterizing Emergence ==
In everyday language the word ''emergent'' broadly refers to “the process of coming into being or becoming prominent”. However, in the context of philosophy or science, it refers more narrowly to  something that is “arising and existing only as a phenomenon of independent parts working together”. This course focuses on that second definition.&lt;ref&gt;The New Oxford American Dictionary, Second Edition, Oxford University Press, 2005. &lt;/ref&gt;

A property of a system is “[[w:Emergence|emergent]]” if it is not part of a detailed “fundamental” description of the system, but it becomes useful or even inevitable when we look at the system more broadly.&lt;ref&gt;{{cite book|title=The Big Picture: On the Origins of Life, Meaning, and the Universe Itself  |last=Carroll|first=Sean|date=May 16, 2017|publisher=Dutton|isbn=978-1101984253|pages=496|author-link=w:Sean_M._Carroll}}&lt;/ref&gt;

As a simple example, recognize that a painting emerges from the artist’s brushstrokes. At a fine-grained level of analysis, the painting is nothing more than a collection of brush strokes on a canvas. At a course-grained level of analysis, the way in which we typically view a painting, it is a visual representation of some scene, person, or imagined object. Examining the brush strokes ever more closely will not reveal the artistry and aesthetic value of the painting. 

[[File:Mona Lisa headcrop.jpg|thumb|The [[w:Mona_Lisa|Mona Lisa]] can be described as a collection of brush strokes, or as an enigmatic portrait.]]

Each emergent phenomenon can be accurately and consistently analyzed and described at two scales, ''fine grained'' and ''coarse grained''. The painting can be described at the fine-grained level by describing each individual brush stroke, or it can be described at the course-grained level by describing the scene that emerges from these brush strokes, and perhaps the mood that it evokes.

As another example, consider the air in the space surrounding you. The air can be described (at the coarse-grained or macroscopic level, perhaps by a meteorologist) as having a particular temperature, pressure, humidity, and density. It can also be described (at the fine-grained or microscopic level, perhaps by a physicist or chemist) as a collection of individual nitrogen, oxygen, and other atoms.

Note, in each case that the two different stories describing each system use very different vocabularies. When describing a painting we identify brush strokes at the microscopic level, and images at the macroscopic level. When describing air, we talk about molecules at the microscopic level, and temperature and pressure at the macroscopic level. Each story is correct and complete at its own level, but each of these stories must remain at their respective levels.

=== Disappearances ===
As our [[Thinking Scientifically#Understanding Evolves|understanding evolves]], several substances thought to exist have evaporated as the illusion was explained. Here are some examples.

The [[w:Phlogiston_theory|phlogiston theory]] is a superseded scientific theory that postulated the existence of a fire-like element called ''phlogiston'' contained within combustible bodies and released during combustion. 

Phlogiston theory stated that phlogisticated substances contain phlogiston and that they dephlogisticate when burned, releasing stored phlogiston which is absorbed by the air. Growing plants then absorb this phlogiston, which is why air does not spontaneously combust and also why plant matter burns as well as it does.

Work by  Antoine-Laurent de Lavoisier and others in the 1770s lead to the development of the modern [[w:Antoine_Lavoisier#Oxygen_theory_of_combustion|oxygen theory of combustion]].

Phlogiston was never found because it never existed, it was only an illusion. 

As another example, the [[w:Caloric_theory|caloric theory]] is an obsolete scientific theory that heat consists of a self-repellent fluid called caloric that flows from hotter bodies to colder bodies. Caloric was also thought of as a weightless gas that could pass in and out of pores in solids and liquids. The &quot;caloric theory&quot; was superseded by the mid-19th century in favor of the [[w:Mechanical_theory_of_heat|mechanical theory of heat]], but nevertheless persisted in some scientific literature—particularly in more popular treatments—until the end of the 19th century.

Caloric was never found because it never existed, it was only an illusion. 

[[w:Luminiferous_aether|Luminiferous aether]] (luminiferous meaning 'light-bearing') was the postulated medium for the propagation of light. It was invoked to explain the ability of the apparently wave-based light to propagate through empty space (a vacuum), something that waves should not be able to do.

The negative outcome of the [[w:Michelson–Morley_experiment|Michelson–Morley experiment]] (1887) suggested that the aether did not exist, a finding that was confirmed in subsequent experiments through the 1920s. This led to considerable theoretical work to explain the propagation of light without an aether. A major breakthrough was the [[w:Theory_of_relativity|theory of relativity]], which explains why the experiment failed to see aether, but was more broadly interpreted to suggest that it was not needed.

Luminiferous aether was never found because it never existed, it was only an illusion. 

''[[w:Élan_vital|Élan vital]]'' is a term coined by French philosopher [[w:Henri_Bergson|Henri Bergson]] in his 1907 book ''[[w:Creative_Evolution_(book)|Creative Evolution]]'', in which he addresses the question of [[w:Self-organization|self-organization]] and spontaneous [[w:Morphogenesis|morphogenesis]] of things in an increasingly complex manner. ''Élan vital'' was translated in the English edition as &quot;vital impetus” but is usually translated by his detractors as &quot;vital force&quot;. It is a hypothetical explanation for evolution and development of organisms, which Bergson linked closely with [[w:Consciousness|consciousness]] – the intuitive perception of experience and the flow of inner time.

The existence of ''Élan vital'', and the associated [[w:Vitalism|vitalism]] theories are superseded and discredited by the majority of modern scientists. 

''Élan vital'' was never found because it never existed, it was only an illusion. 

In each of these examples, although the initial hypothesis turned out to be incorrect, the investigations inspired by that hypothesis were valuable and contributed to our better understanding of the universe. These are example of how [[Thinking Scientifically#Understanding Evolves|understanding evolves]].  

Today the nature of [[w:Consciousness|consciousness]] and [[w:Free_will|free will]] are debated. Mechanisms explaining consciousness are not yet identified. The actual existence of free will is seriously debated. Perhaps these are emergent phenomena, or perhaps consciousness requires discovery of some new substance. Free will may be an  illusion. We await the answers as our understanding evolves. 

== Distinct from Phase Transition ==
Emergence is distinct from a [[w:Phase_transition|phase transition]]. A phase transition is a transformation that takes place over time. For example, when an ice cube melts, water transforms from the solid phase into the liquid phase. In contrast to a phase transition, emergence allows a single system to be described two very different ways, macroscopically and microscopically, at the same time. For example, a box of gas can be described macroscopically by temperature, pressure, humidity, and density. At the same time, it can be described microscopically by specifying the position and velocity of each molecule.

Several examples of each are listed in this table.
{| class=&quot;wikitable&quot;
|'''Phase Transition'''
|'''Emergence'''
|-
|Melting ice

Boiling water

Evaporation 

Sublimation
|Temperature

Hurricane 

Swarms

Fire

A painting 
|}

== Everything Emerged ==
We begin this course by examining, and eventually accepting the statement “everything in the universe emerged from the basic laws of physics”&lt;ref&gt;This is adapted from [[w:Sean_M._Carroll|Sean Carroll’s]] remark “Everything is an emergent phenomenon that does not appear in the most fundamental laws of physics, okay?” See: [https://www.preposterousuniverse.com/podcast/2023/09/04/ama-september-2023/ The Mindscape Podcast, AMA, September 2023], @ 1:44:01.4&lt;/ref&gt; . This is a brief statement of [[w:Naturalism_(philosophy)|naturalism philosophy]], often considered equivalent to materialism. According to naturalism, the causes of all phenomena are to be found within the universe and not transcendental factors beyond it. [[w:Supernatural|Supernatural]] claims are dismissed in the absence of [[w:Sagan_standard|extraordinary evidence]].
[[File:CMB Timeline300 no WMAP.jpg|thumb|Everything emerges as the [[w:Big_Bang|Big Bang]] continues to unfold.]]
The statement is true by definition, because the goal of [[w:Physics|physics]] is to understand how the universe behaves. The behavior of the universe is described by the laws of physics. Although these [[w:List_of_unsolved_problems_in_physics|laws are not now fully known]], our current understanding of the most fundamental laws of physics is the core theory of the [[w:Standard_Model|standard model particle physics]] plus [[w:General_relativity|general relativity]]. The standard model identifies [[w:Quarks|quarks]], [[w:Electrons|electrons]], [[w:Photons|photons]], [[w:Electromagnetism|electromagnetism]], and other [[w:Elementary_particle|elementary particles]] and fundamental forces of the universe. [[w:General_relativity|General relativity]] describes [[w:Gravity|gravity]], the fourth known force in the universe. That is [[Beyond Theism/What there is|what there is]]. That is all there is.

Our current understanding is that the universe expanded from an initial state of high density and temperature in a process known as the [[w:Big_Bang|big bang]]. Although this unfolding is described in detail, no one knows “what (if anything) banged”, or what, if anything, preceded the big bang. About one millionth of a second into the expansion, [[w:Quark|quarks]] and gluons combined to form baryons such as [[w:Proton|protons]] and neutrons. A few minutes into the expansion, neutrons combined with protons to form the universe's deuterium and helium nuclei in a process called [[w:Big_Bang_nucleosynthesis|Big Bang nucleosynthesis]]. Most protons remained uncombined as hydrogen nuclei.

Over a long period of time, the slightly denser regions of the (nearly) uniformly distributed matter gravitationally attracted nearby matter and thus grew even denser, forming gas clouds, stars, galaxies, and the other astronomical structures observable today. Eventually the [[w:Chemical_element|chemical elements]] formed, atoms combined into [[w:Molecule|molecules]] and our [[w:Sun|sun]], [[w:Solar_System|solar system]], and earth formed.

When we accept the premise that “everything emerged” what is special about the claim that some particular phenomenon emerged? Claiming that something emerged is the claim that the emergent phenomenon is ''qualitatively different'' from the components it emerged from. There ''is'' something new and different here. The image that emerges from a painting is qualitatively different from a collection of brush strokes. Although [[w:Water|water]] is composed of only hydrogen and oxygen, water is very different than [[w:Hydrogen|hydrogen]] or [[w:Oxygen|oxygen]]. Similarly, [[w:Salt|table salt]] is different from the [[w:Sodium|sodium]] or [[w:Chlorine|chlorine]] elements it is composed of.  

Claims of ''emergence'' are claims of ''novelty'' (also known as “autonomy”), referring to something new and different about the emergent object. 

This justifies the first part of the claim “''Although this is really new'', there is really nothing new”.

== Emerged ''From'' ==
Claiming that water emerged ''from'' hydrogen and oxygen is the claim that hydrogen and oxygen is the exhaustive list of all the constituent parts of water. The substance water is ''composed'' only of hydrogen and oxygen.

Similarly, the claim that [[w:fire|fire]] emerges ''from'' combustible materials and oxygen after ignition, is the claim that fire is composed of only these components. No [[w:Phlogiston_theory|phlogiston]] is to be found; none is needed. 

This justifies the second part of the claim “Although this is really new, ''there is really nothing new''”.

== Emerged ''How'' ==
Although the claim that water emerged from hydrogen and oxygen identifies the constituent parts, it does not explain ''how'' the transformation takes place. What ''mechanism'' works to transform hydrogen and oxygen into water? The [[w:Causality|causal]] factors must be identified and fully described. We now know that each water molecule contains one oxygen and two hydrogen atoms, connected chemically by [[w:Covalent_bond|covalent bonds]], where electrons are shared among the atoms.

This emergence can be reversed because hydrogen and oxygen can be made to emerge from water through electrolysis. Here electricity provides the energy to break the chemical bonds and release elemental hydrogen and oxygen. 

Furthermore, to support the claim of emergence, the mechanism described must be bottom-up rather than top-down. The mechanism must rely only on some organization or configuration of the laws of physics. No appeals to magic, supernatural phenomenon, or [[w:Teleology|teleology]] is permitted. 

== Emerged ''Why'' ==
Finally, it is helpful to be able to identify ''why'' the new phenomena emerged. The answer is often that the emergent phenomena represent a more advantageous (lower) energy state. At the same time, [[w:Entropy|entropy]] typically increases.

[[w:Thermodynamic_free_energy|Free energy]]—energy available to perform work—provides the power that enables emergence. This free energy is often in the form of a naturally occurring ''gradient''. 

For example, combining hydrogen with oxygen to form water releases energy. In fact, so much energy is released that hydrogen is often combined with oxygen and used as [[w:Liquid_rocket_propellant#Hydrogen|rocket fuel]]. [[w:Fuel_cell|Fuel cells]] typically work by combining hydrogen and oxygen to produce electricity. 

These ideas are explored more fully in the section [[Understanding Emergence#Powered by free energy|Powered by free energy]], below. 

== Four Claims ==
Claiming that “B emerged from A” makes the following four subsidiary claims:

# The emergent phenomenon is ''novel. Novelty'' (also known as “[https://plato.stanford.edu/entries/properties-emergent/ autonomy]”), describes what is new and different about the emergent object.
# ''composition'' (also known as “dependency”) if we claim that B emerged from A then we are claiming that all the constituent components in B are already in A.
# A ''transformation mechanism'' is identified, demonstrated, simulated, or suggested.  This [[w:Explanation|explains]] (or at least describes) ''how'' the transformation occurred. The ability to describe the transformation mechanism is the difference between science and magic.
# The ''energy source'' that powers the transformation is identified, explained, described, or suggested. This explains''why'' the transformation happened.

For example, the statement that “[[w:Abiogenesis|life emerges from a chemical soup]]” makes two ([[w:Ontology|ontological]]) claims that are almost contradictory. The first claim is that something new is created. This recognizes that [[w:life|life]] is qualitatively different from a chemical soup because it includes the [[w:Biological_process|biological processes]] of metabolism, reproduction, and growth along with the property of agency, if only as [[w:Chemotropism|chemotropism]]. The second claim is that nothing new is required. The life that emerges does not contain any constituent parts that are not originally in the chemical soup.  Life is just a bag of chemicals.

'''Although ''this'' is really new, there is really nothing new.'''

While the claim that life has emerged from a chemical soup is often made, and thorough analysis and decomposition of living organisms have failed to identify any substances in life that are beyond chemistry, the claim remains speculative until a mechanism can be verified. The claim could be [[w:Falsifiability|falsified]] by discovering something beyond chemistry, such as the ''[[w:Élan_vital|Élan vital]]'', that life requires.

Mechanisms for [[w:Abiogenesis|abiogenesis]] have been suggested, including the formation of a habitable planet, the prebiotic synthesis of organic molecules, molecular self-replication, self-assembly, autocatalysis, and the emergence of cell membranes. Many proposals have been made for different stages of the process. None of these are yet verified (for example by being duplicated in the lab) and none are yet falsified. 

One hypothesis for the energy source that propelled abiogenesis are the chemical and thermal gradients present around [[w:Hydrothermal_vent|hydrothermal vents.]] Other mechanisms have been proposed. Again, none of these are yet verified and none are yet falsified. 

== It only seems like Magic. ==
[[File:Origin of life stages.svg|thumb|400px|upright=2.5|Stages in the origin of life range from the well-understood, such as the [[w:planetary habitability|habitable Earth]] and the abiotic synthesis of simple molecules, to the largely unknown, like the derivation of the [[w:last universal common ancestor| last universal common ancestor]] (LUCA) with its complex molecular functionalities.]]

Emergence is about science, not magic.

This course emphasizes that emergence is not magic. Throughout this course we maintain a [[w:Naturalism_(philosophy)|naturalist]] worldview, the understanding that only natural laws and forces (as opposed to supernatural ones) operate in the universe. The only building blocks of the universe are those physicists [[Beyond Theism/What there is|have identified]]. None-the-less, remarkable structures emerge from these simple building blocks. 

Not all emergent phenomena are currently understood and can be explained in terms of causal mechanisms. While it may be tempting to replace the phrase “B emerged from A” with, “There was A, and then a [[w:Miracle|miracle]] happened and now B appeared”. That is not what we are saying or implying.

We recognize that there are [[w:Lists_of_unsolved_problems|many mysteries]], but no [[w:Supernatural|magic]]. 

The mechanisms that generate several of the phenomena studied here, including [[w:Phase_transition|phase transitions]], [[w:Temperature|temperature]], and [[w:Pressure|pressure]], are well understood. Mechanisms explaining other phenomena, including [[w:Abiogenesis|abiogenesis]], include transformation and processes that are largely unknown and can only be suggested or hypothesized at this time.

When an emergence claim is made, yet the subsidiary four claims cannot be justified in detail, then the claim is speculative and may suggest [[w:Hypothesis|hypothesis]] to be tested.

== Examples ==
Consider these various emergence claims:
*[[w:Temperature|Temperature]] emerges from the [[w:Kinetic_theory_of_gases|motion of gas particles]] and is an example of ''[[w:Statistical_mechanics|statistical dynamics]]''.
*Ice melting is a ''[[w:Phase_transition|phase transition]]''.
*[[w:Smelting|Smelting]] [[w:Iron_ore|iron ore]] to obtain iron is an example of ''chemical transformation''.
*When mountains, valleys, and volcanoes are formed by the movement of tectonic plates, that is an example of ''geological dynamics''.
*[[w:Tropical_cyclone|Hurricanes]] emerge from warm waters as ''flow of warm, moist, rapidly rising air, starts to rotate cyclonically'' as it interacts with the rotation of the earth.
*When an oak tree grows from an acorn that is an example of ''[[w:Developmental_biology|development]]''.
*When new species form, that is an example of ''[[w:Speciation|speciation]]'' and  ''[[w:Evolution|evolution]]''.
*[[w:Swarm_behaviour|Swarms]] emerge from ''collective motion'' of a large number of self-propelled entities.
*Formation of [[w:Culture|culture]] is an example of ''social behavior''.
*Building a chair from a felled tree is an example of ''[[w:Manufacturing|manufacturing]]''.
*The [[w:David_(Michelangelo)|statue of David]] emerged from a block of marble and magnificent ''[[w:Art|artistry]]''.
*Appearance of roads, bridges, computers, and the internet are examples of ''[[w:Engineering|engineering]]''.
*[[w:Superconductivity|Superconductivity]] emerges when certain materials are cooled below their characteristic critical temperature. 
*[[w:Wikipedia|Wikipedia]] emerged from volunteers and a ''[[w:Governance|governance system]]''. 

=== Assignment—Part 1 ===
For each of the emergence claims listed above, identify each of the four claims: 1) novelty, 2) composition (from), 3) mechanism (how), and 4) energy source (why).

=== Assignment—Part 2 ===
#This general [[/List of emergent phenomena/]] illustrates a range of emergent phenomena that occur in nature.
#Identify three examples from the list (or elsewhere) to study for this assignment.
#Construct each of the [[Understanding_Emergence#Four_Claims|four claims]] that define emergence, presented above.
#Are each of the [[Understanding_Emergence#Four_Claims|four claims]] viable? Is the phenomenon studied truly emergent? Why or why not?

== Levels ==
Philosopher [[w:Auguste_Comte|Auguste Comte]] described the [[w:Hierarchy_of_the_sciences|hierarchy of the sciences]]. This table is a modern adaptation of his ideas:

{| class=&quot;wikitable&quot;
!'''Discipline'''
!'''[[w:Holon_(philosophy)|Holons]]'''
!'''Catalogue'''
|-
|[[w:Sociology|Sociology]]
|People, interpersonal relationships, organizations, cultures, societies.
|[[w:Outline_of_sociology|Outline of sociology]]
|-
|[[w:Psychology|Psychology]]
|Neurons, brain, nervous system
|[[w:Connectome|Connectome]], [[w:DSM-5|DSM]], 
|-
|[[w:Biology|Biology]]
|Proteins, RNA, DNA, cells, tissues, organs, organisms.
|[[w:Taxonomy_(biology)#Alpha_and_beta_taxonomy|Biological taxonomy]], anatomical descriptions
|-
|[[w:Chemistry|Chemistry]]
|Atoms, molecules, compounds.
|The [[w:Periodic_table|periodic table]] of chemical elements.
|-
|[[w:Physics|Physics]]
|Electrons, quarks, photons, gravity, and other elementary particles and forces. 
|[[w:Standard_Model|Standard model]] of particle physics
|-
|[[w:Mathematics|Mathematics]]
|Numbers, [[w:Axiom|axioms]], operators, [[w:Theorem|theorems]], formulas, points, lines, planes, sets, and other structures.
|[[w:Mathematics_Subject_Classification|Mathematics subject classification]]
|}

Chemistry emerges from physics because atoms, molecules, and compounds can be described as collections of electrons, quarks, and forces. Similarly, biology emerges from chemistry because proteins and other biological substances can be described as chemicals. Similar emergent relationships occur at each level of the hierarchy. Mathematics holds a special place in this hierarchy because it is fundamental and foundational making it useful at all levels.

As another example, we can describe, study, analyze, and enjoy a book at many different levels, shown in the table below. 
{| class=&quot;wikitable&quot;
!'''Element'''
!'''Attributes'''
!'''Disciplines''' 
|-
|[[w:Culture|Culture]], [[w:Society|society]], [[w:Knowledge_sharing|shared knowledge]]
|Tradition, arts, crafts, all social institutions, forms of expression, and modes of social interaction…
|[[w:Cultural_studies|Cultural studies]], [[w:Social_studies|Social studies]], [[w:Information_science|Information science]]
|-
|[[w:Library|Library]], [[w:Bookselling|bookstore]]
|Number of volumes, scope, specialty, accessibility, organization, building design, location …
|[[w:Library_and_information_science|Library science]]
|-
|[[w:Genre|Genre]]
|Category, structural elements, stories, instances, exemplars …
|[[w:Genre_studies|Genre studies]]
|-
|[[w:Book |'''Book''']]
|Title, style, genre, creative contribution, action, character, dialogue, narration, pace, plot, point of view, setting, style, suspense, theme, tone, voice, accuracy, organization, intended audience, purpose, originality, clarity, vocabulary choice …
|[[w:Comparative_literature|Comparative literature]], [[w:Literary_criticism|literary criticism]], [[w:Creative_writing|creative writing]]
|-
|[[w:Chapter_(books)|Chapter]]
|Title, introduction, organization, unity, subheadings, conclusion, transition, length, references, illustrations, voice …
|[[w:Writing_style|Writing style]], [[w:Book_design|book design]]
|-
|[[w:Paragraph|Paragraph]]
|Topic sentence, unity, coherence, development, structure, transitions, length, clarity, consistency, concluding sentence, purpose, tone, target audience, focus …
|[[w:Writing_style|Writing style]]
|-
|[[w:Sentence_(linguistics)|Sentence]], clause, phrase
|Length, purpose, tense, voice, mood, modifiers, punctuation …
|[[w:Grammar|Grammar]]
|-
|[[w:Word|Words]]
|Word choice, part of speech, vocabulary, idioms …
|[[w:Spelling|Spelling]], [[w:Vocabulary|vocabulary]]
|-
|[[w:Letter_(alphabet)|Letters]], [[w:Symbol|symbols]]
|Choice of alphabet, upper or lower case, typefaces, point size, line length, line spacing,  page layout,  …
|[[w:Typography|Typography]], [[w:Graphic_design|graphic design]]
|}
A [[w:Printer_(publishing)|printer]], as differentiated from a reader or critic, may consider the type of ink, paper, trim size, cover materials, spine, and  binding style of the book.

A book emerges from letters, words, sentences, paragraphs, chapters, paper, and ink arranged by authors and printers. A book about typography spans several levels, as do books on grammar, writing style, and other literary elements. A printed book ''about'' typography is both an ''instance'' of typography, as well as a ''discussion'' of typography. It is [[w:Self-reference|self-referential]]. 

=== Assignment ===
#Study [https://archive.org/details/PrintEmergence this emergence diagram].
#Identify higher and lower levels and the associated holons for several items shown in the hierarchy of science diagram.

== Mechanisms ==
Claims 1 (novelty) and 2 (composition) are relatively easy to verify (or falsify). Claim 3, describing a mechanism, often more difficult. In this section we begin by exploring several mechanisms that are well-understood, and then identify other mechanisms that are not as well understood.

Here are examples of emergence caused by specific, typically well-understood, mechanisms. 

=== Phase Transitions ===
[[w:Phase_transition|Phase transitions]] are a common example of changes over time, that occur in equilibrium systems.&lt;ref&gt;[https://cmsr.rutgers.edu/images/people/lebowitz_joel/publications/ephenom.pdf Emergent Phenomena]; Entropy and phase transitions in macroscopic systems, by Joel L. Lebowitz, Physik Journal 6 (2007) Nr. 8/9&lt;/ref&gt;

In chemistry, thermodynamics, and other related fields, a [[w:Phase_transition|phase transition]] (or phase change) is the physical process of transition between one state of a medium and another. Commonly the term is used to refer to changes among the basic states of matter: solid, liquid, and gas, and in rare cases, plasma. 

During a phase transition of a given medium, certain properties of the medium change as a result of the change of external conditions, such as temperature or pressure. This can be a discontinuous change; for example, a liquid may become gas upon heating to its boiling point, resulting in an abrupt change in volume. The identification of the external conditions at which a transformation occurs defines the phase transition point.

[[w:Paradigm_shift|Paradigm shifts]] are significant changes in the way people perceive, think, or understand certain concepts or phenomena. These are phase transitions occurring in our understanding that are analogous to phase transitions that occur in materials.

Although phase transitions result in substantial qualitative changes—ice is different from water—this is not an example of emergence, as described above in the section [[Understanding Emergence#Distinct from Phase Transition|Distinct from Phase Transition.]]

==== Assignment: ====
#Study these [[/Examples of phase transitions/]].
#Study these [[/Examples of Paradigm Shifts/]].

=== The Kinetic Theory of Gases ===
[[File:Translational motion.gif|thumb|The temperature of the ideal gas is proportional to the average kinetic energy of its particles.]]

Although we have a clear intuitive sense of [[w:Temperature|temperature]] as a measure of our perceptions of hotness and coldness, we may not have a clear understanding of what temperature is and how it can be explained in terms of more fundamental constituents of nature.

Physicists now understand that temperature emerges from the motion of gas particles. 

We begin our detailed study of emergence mechanisms with the [[w:Kinetic_theory_of_gases|kinetic theory of gases]] because it provides a well-established and readily understood explanation for the macroscopic properties of gasses such as volume, pressure, and temperature, as well as the transport properties such as viscosity and thermal conductivity.  

=== Self-Organization, Slime Mold ===
[[w:Dictyostelium_discoideum|Dictyostelium discoideum]], commonly known as [[w:Slime_mold|slime molds]], represents a remarkable group of organisms. Initially, these creatures exist as solitary single-celled amoebas, nourishing themselves on the bacteria inhabiting decaying leaves within forest floors.&lt;ref&gt;[[w:ChatGPT|ChatGPT]] helped improve the text in this section.&lt;/ref&gt; However, when confronted with food scarcity, these numerous individual amoebas undergo a transformative process, coalescing into a singular entity.

[[File:Dictyostelium Aggregation.JPG|thumb| Dictyostelium exhibiting chemotaxis through aggregation]]

Several questions naturally arise. What orchestrates this aggregation of individual organisms? What acts as the ''pacemaker'', the central governing entity, dispatching chemical signals to summon this collective assembly? The key to this enigmatic phenomenon lies in the production of a chemical known as [[w:Cyclic_AMP|AMP]] by starving amoebas. When these individuals detect a concentration of AMP greater than their own output, they instinctively gravitate toward the source in a process known as [[w:Chemotaxis|chemotaxis]]. A few simple rules, followed by each single-celled amoeba, results in the emergence of the larger slime mold. No individual amoeba knows more that these simple rules, and each act as an autonomous agent. 

This process seemingly mirrors the function of a pacemaker, even though a conventional pacemaker is absent. This confounded biologists, who questioned, &quot;Where is the originator cell? Where is the pacemaker?&quot; It left them with a sense of dissatisfaction. In fact, this pacemaker hypothesis persisted as the prevailing model for another decade&lt;ref&gt;[https://muse.jhu.edu/article/403634/pdf The Force of the Pacemaker Concept in Theories of Aggregation in Cellular Slime Mold], Evelyn Fox Keller,1983, Perspectives in Biology and Medicine, Volume 26, Number 4, Summer 1983.&lt;/ref&gt; until a series of experiments compellingly demonstrated that slime mold cells were [[w:Self-organization|self-organizing]] from the [[w:Bottom–up_and_top–down_design|bottom up]].

Until this juncture, it had been widely assumed that the universe adhered to a top-down structure, as exemplified by religion and [[w:Teleology|teleology]]. However, as illustrated by this example, the natural world operates differently—there are no instances where change results from top-down, preconceived strategic designs or mandates issued by a solitary individual or authority figure. Nature adheres to a bottom-up paradigm.

Here are the fundamental principles governing slime molds:

* All low-level nodes are initially interchangeable.
* Each amoeba is indistinguishable from another.
* The emergent entity, the slime mold, emerges from these more basic constituents and possesses characteristics that are novel and irreducible compared to its individual components.
* The progression is from rudimentary rules at the lower level to increased complexity at higher levels.

These specific rules apply to interactions and network positioning:

* Produce AMP in response to food availability.
* When AMP reaches a certain threshold, move in its direction.
* Leave trails when experiencing &quot;hunger&quot; – depleted levels of AMP.
* Erase trails in the face of failure.
* Extend growth along successful trails.

A decentralized communication mechanism operates through chemical signals. While an individual amoeba constitutes a single organism, a threshold is reached wherein a slime mold emerges as the product of numerous interworking amoebas.

Slime molds emerge when many individual amoebas self-organize and coordinate to find food.  

Other examples of [[w:Swarm_intelligence|swarm intelligence]] in natural systems include [[w:Ant_colony|ant colonies]], [[w:Bee_colonies|bee colonies]], bird [[w:Flocking_(behavior)|flocking]], hawks [[w:Hunting|hunting]], animal [[w:Herding|herding]], [[w:Bacteria#Growth_and_reproduction|bacterial growth]], fish [[w:Shoaling_and_schooling|schooling]] and [[w:Microbial_intelligence|microbial intelligence]].

In 1972 Nobel prize winning physicist [[w:Philip_W._Anderson|Philip W. Anderson]] wrote an article called &quot;More is Different&quot;&lt;ref&gt; ''&quot;[http://robotics.cs.tamu.edu/dshell/cs689/papers/anderson72more_is_different.pdf More is Different]&quot; (PDF)    Bibcode1972Sci...177..393A doi10.1126/science.177.4047.393 PMID17796623 S2CID34548824''&lt;/ref&gt; in which he emphasized the limitations of reductionism and the existence of hierarchical levels of science, each of which requires its own fundamental principles for advancement.

=== Dynamic Equilibrium ===
Dynamic equilibrium refers to a balanced state between an open system and an environment that is feeding it a steady supply of free energy.&lt;ref&gt;{{cite book|title=The Romance of Reality: How the Universe Organizes Itself to Create Life, Consciousness, and Cosmic Complexity  |last=Azarian  |first=Bobby|date=June 28, 2022|publisher=BenBella Books|isbn=978-1637740446|pages=320}} @94 of 186.&lt;/ref&gt;

A car travelling along a straight line at a constant velocity is a simple example of a dynamic equilibrium. It is in ''equilibrium'' because all of the forces on it are balanced, and the acceleration is zero. However, the equilibrium is ''dynamic'' because the car is in motion.

The daily [[w:Earth's_rotation|rotation of the earth]] on its axis and the annual [[w:Earth's_orbit|revolution of the earth]] around the sun are both examples of dynamic equilibrium. Because each motion is constant, the system in equilibrium, because it is in motion the equilibrium is dynamic. 

The [[w:Water_cycle|water cycle]] is a more complex example of dynamic equilibrium. The quantity of water in the earth system remains constant as evaporation and precipitation continue to occur. 

[[w:Dynamic_equilibrium|Dynamic equilibrium]] is a prevalent concept in various natural processes, where opposing forces or reactions reach a balance that maintains stability over time.

==== Assignment ====
Study these [[/Examples of dynamic equilibrium/]].

=== Complexity and Chaos Theory ===
According to [[w:Karl_Popper|Karl Popper]], all problems are either clocks&lt;ref&gt;At the time of his writing clocks were entirely mechanical mechanisms consisting primarily of gears, springs, and levers. When mechanical clocks are disassembled the role of each mechanical component can be determined.  &lt;/ref&gt; or clouds. A clock is something you can take apart, analyze the parts, and understand how it works. A cloud is a dynamic system, you can't take it apart. The way to understand a cloud is to study it in a holistic way. [[w:Complexity|Complexity]] theory and [[w:Chaos_theory|chaos theory]] describe and analyze systems that are more similar to clouds than clocks. Clouds emerge in a way that is more complex than a clock. This section studies a few examples of systems that are more like clouds than clocks and concludes that [[w:Determinism|determinism]] and [[w:Predictability|predictability]] are very different things. Even if a chaotic system is unpredictable, it is still deterministic. 

==== The three-body problem ====
[[File:Three-body Problem Animation with COM.gif|thumb|320px|Approximate trajectories of three identical bodies located at the vertices of a scalene triangle and having zero initial velocities.]]

The [[w:Three-body_problem|three body problem]] is simply stated, yet there is no general closed form solution to the problem. 

Consider describing the motions of three objects, for example the sun, earth, and moon. The three-body problem is the problem of taking the initial positions and velocities (or momenta) of three point masses and solving for their subsequent motion according to Newton's laws of motion and Newton's law of universal gravitation.

Remarkably there is no general [[w:Closed-form_expression|closed-form solution]] to the three-body problem, meaning there is no general solution that can be expressed in terms of a finite number of standard mathematical operations. Moreover, the motion of three bodies is generally non-repeating, except in special cases.

This example illustrates that [[w:Determinism|determinism]] and [[w:Predictability|predictability]] are very different things. Even though the motions of the three bodies are unpredictable, their motions are still deterministic. The emerging motion is the result of their initial positions and momentum. The same complex motion will emerge every time whenever a system of three bodies begin with the same three initial positions and momenta.  The motions of the three bodies follow a specific path, even though we cannot predict in advance what that path will be.&lt;ref&gt;{{cite book|title=Determined: A Science of Life without Free Will  |last=Sapolsky|first=Robert M.|date=October 17, 2023|publisher=Penguin Press|isbn=978-0525560975|pages=528|author-link=w:Robert_Sapolsky}} @ 218 of 874.&lt;/ref&gt;

==== Lorenz Systems ====
[[File:A Trajectory Through Phase Space in a Lorenz Attractor.gif|frame|right|A sample solution in the Lorenz attractor when {{math|1=''ρ'' = 28}}, {{math|1=''σ'' = 10}}, and {{math|1=''β'' = {{sfrac|8|3}}}}]]

In the 1960’s MIT meteorologist [[w:Edward_Norton_Lorenz|Edward Lorenz]] was using computer simulations to model weather patterns. Serendipitously he noticed the results of the model were remarkably sensitive to very small changes in the initial conditions. As an example, the Lorenz attractor is a set of chaotic solutions of the [[w:Lorenz_system|Lorenz system]]. In popular media the &quot;butterfly effect&quot; stems from the real-world implications of the Lorenz attractor, namely that several different initial chaotic conditions evolve in phase space in a way that never repeats, so all chaos is unpredictable.

His work demonstrated that [[w:Chaos_theory|chaotic systems]] can be completely deterministic yet still be inherently unpredictable over long periods of time. Furthermore, the results are unpredictable not only in practice but in principle.

==== Prime Numbers ====
A [[w:Prime_number|prime number]] is a natural number greater than 1 that is not a product of two smaller natural numbers.

The first 25 prime numbers (all the prime numbers less than 100) are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

There are infinitely many prime numbers, as [[w:Euclid's_theorem|first proved]] by the ancient Greek mathematician Euclid.

The [[w:Largest_known_prime_number|largest known prime number]] (as of October 2023) is 2&lt;sup&gt;82,589,933&lt;/sup&gt; − 1, a number which has 24,862,048 digits when written in base 10.

Although the next prime number is determined, it is unknown and cannot be predicted.

==== Cellular automata ====
[[w:Cellular_automaton|Cellular automata]] also demonstrate deterministic systems that are unpredictable. 

[[w:Conway's_Game_of_Life|Conway’s game of life]] is a popular example of cellular automata. 

An [[w:Elementary_cellular_automaton|elementary cellular automaton]] is a one-dimensional cellular automaton where there are two possible states (labeled 0 and 1) and the rule to determine the state of a cell in the next generation depends only on the current state of the cell and its two immediate neighbors. More generalized [[w:Cellular_automaton|cellular automata]] having more dimensions are also defined.

Rule 22 is particularly interesting. The rule has a three-cell neighborhood, where a cell takes state “1” if there is exactly one neighbor, including the cell itself, in state 1.

Here are the possible results of one iteration applying rule 22:

[[File:Elementary cellular automata rule 22 possible transitions.jpg|800px|The transitions determined by elementary cellular automata rule 22.]]

[[File:WolframRule22.png|thumb|A pattern emerging from Wolfram Rule 22]]

In cellular automata simple rules can generate remarkably complex patterns. For example, rule 22 is considered chaotic because:&lt;ref&gt;[https://content.wolfram.com/uploads/sites/13/2019/06/28-2-1.pdf On Patterns and Dynamics of Rule 22 Cellular Automaton], Wolfram Alpha, Genaro J. Martinez, et al. &lt;/ref&gt;

#Future configuration of the automaton is completely determined from its initial state because of the deterministic rule and synchronous updating.
#Development of the automaton is sensitive to initial conditions (tiny perturbation might lead to dramatic events).
#The resulting global transition graph has dense periodic orbits (attractors).
#Configurations evolved can be characterized as random.

Although the results of iteration N can only be determined by running the rule N times, the result is always the same and depends only on the initial state. The system is deterministic yet unpredictable.

==== Summarizing Complex Systems ====
Various systems attain a level of complexity that makes them more like clouds than clocks. A [[w:Reductionism|reductionist]] approach to analyzing these systems often neglects their defining characteristics. However, although these complex, dynamic, nonlinear, and chaotic systems are unpredictable, they are deterministic. 

=== Downward causation ===
If I am sitting at my desk and decide I would like a bowl of ice cream, I get up from my desk, walk to the kitchen, open the freezer, take out a box of ice cream and indulge.
[[File:Wagon wheel, Corrigall Farm Museum - geograph.org.uk - 3773805.jpg|thumb|When the wheel turns, so must each spoke.]]
This is an example of ''[[w:Downward_causation|downward causation]]''. Thoughts emerging from my (higher level) brain eventually cause [[w:Motor_neuron|motor neurons]] to fire in ways that contract muscles and propel me toward the box of ice-cream.

Here are several more examples.

When a wheel is rolling the (high-level) wheel-ness causes the constituent wheel parts to do forward rolls.&lt;ref&gt;{{cite book|title=Determined: A Science of Life without Free Will  |last=Sapolsky|first=Robert M.|date=October 17, 2023|publisher=Penguin Press|isbn=978-0525560975|pages=528|author-link=w:Robert_Sapolsky}} @ 338 of 874.&lt;/ref&gt;

Changing climate conditions can cause various species to migrate toward new regions or become extinct. 

In another example the state of the economy can influence the purchasing behavior of individuals. People  are more likely to purchase goods when the economy is growing than during a recession. Note there is a [[w:Feedback|feedback loop]] here. Individual purchases also effect the overall state of the economy.

Various social norms can cause an individual to ''go along to get along''—conform to the group norm to have acceptance and security.

While these examples demonstrate that an emergent phenomenon, such as the mind, climate, economy, or group behavior, can influence the actions of an individual neuron, organism, or individual behavior, it is a mistake to believe that the higher-level system causes the building blocks themselves to acquire new skills. Neurons act as neurons even when getting a box of ice cream. Base elements do not behave in novel ways when they operate as part of the higher-order system.&lt;ref&gt;{{cite book|title=Determined: A Science of Life without Free Will  |last=Sapolsky|first=Robert M.|date=October 17, 2023|publisher=Penguin Press|isbn=978-0525560975|pages=528|author-link=w:Robert_Sapolsky}} @ 290 of 874&lt;/ref&gt;

=== Self-referential Loops ===
[[w:Self-reference|Self-referential loops]], explored in the book &quot;[[w:I_Am_a_Strange_Loop|I Am a Strange Loop]]&quot; by [[w:Douglas_Hofstadter|Douglas Hofstadter]], are fascinating phenomena that occur in nature and cognition.  [[w:Strange_loop|These loops]] involve systems that reference themselves in a cyclic manner, giving rise to intricate patterns and emergent properties.

Prominent examples are the emergence of consciousness and self-awareness. Because our [[w:Mental_model|mental model]] of the world includes a model of ourselves, we become aware of ourselves, and subjective experiences arise in our conciseness.

The [[w:Liar_paradox|liar paradox]], “this statement is false” demonstrates the enigma of self-references. 

==== Assignment: ====
Study these examples of [[/Self-referential loops/]].

=== Complex Adaptive Systems ===
A [[w:Complex_adaptive_system|complex adaptive system]] is a system that is complex in that it is a [[w:Dynamic_network_analysis|dynamic network of interactions]], but the behavior of the ensemble may not be predictable according to the behavior of the components. It is ''[[w:Adaptive_system|adaptive]]'' in that the individual and collective behavior mutate and self-organize corresponding to the change-initiating micro-event or collection of events.

Typical examples of complex adaptive systems include: climate; cities; firms; markets; governments; industries; ecosystems; social networks; power grids; animal swarms; traffic flows; social insect (e.g. ant) colonies; the brain and the immune system; and the cell and the developing embryo. Human social group-based endeavors, such as political parties, communities, geopolitical organizations, war, and terrorist networks are also considered as complex adaptive systems.

=== Problems create knowledge ===
In the grand narrative of human advancement, the intricate relationship between problems and knowledge creation has been a driving force that propels us toward greater understanding and progress.&lt;ref&gt;[[w:ChatGPT|ChatGPT]] created this essay responding to the prompt: “Write an essay entitled ‘Problems create knowledge’. Include insights from Karl Popper.”&lt;/ref&gt; The foundation of this concept, which finds resonance in the philosophical insights of [[w:Karl_Popper|Karl Popper]]&lt;ref&gt;Thornton, Stephen, &quot;Karl Popper&quot;, The Stanford Encyclopedia of Philosophy (Fall 2023 Edition), Edward N. Zalta &amp; Uri Nodelman (eds.), forthcoming URL = &lt;&lt;nowiki&gt;https://plato.stanford.edu/archives/fall2023/entries/popper/&lt;/nowiki&gt;&gt;.&lt;/ref&gt;, underscores how challenges stimulate our intellectual growth, guide our explorations, and shape the contours of our collective wisdom.

Encountering problems and thereby creating knowledge is an important mechanism that often causes emergence. 

==== Assignment ====
Read the essay [[/Problems Create Knowledge/]].

=== Evolution and Emergence: ===
[[w:Evolution|Evolution]], one of the most remarkable processes in nature, is deeply intertwined with emergence. Through the mechanism of natural selection, complex and adaptive behaviors, traits, and structures emerge over time. The evolution of the eye, for instance, involves the gradual emergence of light-sensitive cells, followed by the refinement of these structures over millions of years to create sophisticated visual organs.

The evolutionary emergence of complex ecosystems showcases how the interaction between species leads to intricate relationships, with each species influencing and being influenced by others. This interplay gives rise to emergent properties such as ecological stability, biodiversity, and intricate food webs.

[[w:Developmental_biology|Development]]—the growth and maturing of an organism or system over time—is distinct from [[w:Evolution|evolution]]—the change in heritable characteristics of organisms across generations. 

In short, evolution is powered by free energy, and constrained by reality.

==== Assignment ====

The book ''The Emergence of Everything''&lt;ref&gt;{{cite book|title=The Emergence of Everything: How the World Became Complex  |last=Morowitz|first=Harold J.|date=November 7, 2002|publisher=Oxford University Press|isbn=978-0195135138|pages=224|author-link=w:Harold_J._Morowitz}}&lt;/ref&gt;, by Harold J. Morowitz, describes the emergence of everything in the universe in a series of 28 steps beginning with “the first emergence” (the origins of the big bang) and ending with “the spirit”. Interested students may wish to read that book and identify the [[Understanding Emergence#Four Claims|four claims]] for each claimed emergence. 

== Powered by free energy ==
[[w:Thermodynamic_free_energy|Free energy]]—energy available to perform work—provides the power that enables emergence. This free energy is often in the form of a naturally occurring ''gradient''.

A [[w:Gradient|gradient]] is the difference between two interacting systems that creates [[w:Instability|instability]], whether it be a difference in temperature, pressure, chemical concentration, electrical charge, or some other characteristic. A hillside is a common example of a gravitational gradient. 

When such a difference exists, there will be spontaneous flow from one system to the other until that difference, or the gradient, is eliminated, and a [[w:Stable_equilibrium|stable equilibrium]] is achieved. This happens spontaneously to dissipate the free energy contained in the gradient.&lt;ref&gt;{{cite book|title=The Romance of Reality: How the Universe Organizes Itself to Create Life, Consciousness, and Cosmic Complexity  |last=Azarian  |first=Bobby|date=June 28, 2022|publisher=BenBella Books|isbn=978-1637740446|pages=320}} @ 22 of 186&lt;/ref&gt; As an example, materials tend to move downhill in response to the gravitational gradient.

Identifying the source of free energy helps to answer ''why'' this emerged. 

=== Assignment ===
Complete the Wikiversity course on [[Gradients in Nature]].

== Constrained by reality ==
Any system, including emergent systems, must become stable to persist. This requires emergent systems [[w:Adaptation|adapt]] to their environment. We often learn in everyday life by [[w:Trial_and_error|trial and error]]. In biological evolution this is called the [[w:Survival_of_the_fittest|survival of the fittest]]. In the context of adaptation, it is called variation and selection. In scientific research this is called conjecture and refutation or simply [[w:Statistical_hypothesis_testing|hypothesis testing]]. In Artificial Intelligence it is called generate and test.

In each case the viable possibilities are constrained by reality. [[Facing Facts/Reality is the Ultimate Reference Standard|Reality serves as the ultimate reference standard]] against which all ideas, theories, and perspectives are measured and validated. It is the foundation that guides our exploration of the universe, shapes our perceptions, and directs our pursuits of what is real and what us true.

Although everything in existence has emerged, it is not true that anything whatsoever can emerge. 

=== Assignment ===

* Study the essay [[Facing Facts/Reality is the Ultimate Reference Standard|Reality is the Ultimate Reference Standard]].
* Study the essay [[Exploring Worldviews/Aligning worldviews|Aligning worldviews]].

== Weak and Strong Emergence ==
The term ‘emergence’ often causes confusion in science and philosophy, as it is used to express at least two quite different concepts. We can label these concepts ''strong emergence'' and ''weak emergence''. Both of these concepts are important, but it is vital to keep them separate.&lt;ref&gt;[https://consc.net/papers/emergence.pdf. Strong and Weak Emergence], David J. Chalmers, Research School of Social Sciences Australian National University.&lt;/ref&gt;

=== Weak Emergence: ===
[[w:Emergence#Strong_and_weak_emergence|Weak emergence]] occurs when the emergent properties of a system can be understood and explained by analyzing its constituent parts and their interactions. The behavior of a system at a higher level emerges from the interactions of its components, and while surprising, it is not fundamentally irreducible. The synchronization of fireflies and the flocking behavior of birds are examples of weak emergence.

==== Assignment: ====

# Study these [[/Examples of Weak Emergence/]].
# Choose one or more examples to study for this assignment.
# Identify and describe each of the [[Understanding Emergence#Four Claims|four claims]] that pertain to the example phenomena chosen.

=== Strong Emergence: ===
Strong emergence is a hypothesized kind of emergence that creates the existence of something ontologically new in nature.&lt;ref&gt;{{cite book|title=The Romance of Reality: How the Universe Organizes Itself to Create Life, Consciousness, and Cosmic Complexity  |last=Azarian  |first=Bobby|date=June 28, 2022|publisher=BenBella Books|isbn=978-1637740446|pages=320}} @99 of 186.&lt;/ref&gt; Most researchers who have studied these claims reject the existence of strong emergence. 

[[w:Emergence#Strong_and_weak_emergence|Strong emergence]] suggests that emergent properties cannot be fully explained by the interactions of constituent parts. In other words, the whole is more than the sum of its parts, and new properties arise that cannot be predicted from a reductionist perspective. Consciousness, subjective experience, and the emergence of novel biological functions are often cited as examples of strong emergence. “In strong emergence, the behavior of a system with many parts is not reducible to the aggregate behavior of all those parts, even in principle.”&lt;ref name=&quot;:0&quot;&gt;{{cite book|title=The Big Picture: On the Origins of Life, Meaning, and the Universe Itself  |last=Carroll|first=Sean|date=May 16, 2017|publisher=Dutton|isbn=978-1101984253|pages=496|author-link=w:Sean_M._Carroll}} Chapter 13.&lt;/ref&gt;

For example, when considering behavior of a particular atom that is part of a human body, an advocate of strong emergence might claim: &lt;blockquote&gt;“That atom is part of you, a person, and you can’t predict the behavior of that atom without understanding something about the bigger person-system. Knowing about the atom and its surroundings is not enough.”&lt;ref name=&quot;:0&quot; /&gt;&lt;/blockquote&gt;Sean Carroll remarks “If it’s how the world actually does work, then our purported microscopic theory of the atom is simply wrong. ”&lt;ref name=&quot;:0&quot; /&gt;

There is no direct evidence of this happening. Strong emergence claims to explain consciousness, or establish the existence of free-will, souls, or an afterlife. Claims of strong emergence are generally dismissed as magical explanations for mysteries that are not yet understood.  

==== Assignment: ====

# Study these [[/Examples of Strong Emergence/]].
# Choose one or more of the examples to study.
# Decide if you believe 1) the phenomena described is an illusion, or 2) the emergent phenomena will eventually be explained in terms of presently known and understood constituent parts, or 3) the emergent phenomena is the result of yet to be discovered constituent parts, or newly discovered behavior of the constituent parts.

== What there is ==
Our studies throughout this course allow us to provide a more nuanced description of what there is, what is real in our universe. Below is a table, adapted from ''The Big Picture'', chapter 13. The fundamental building blocks of the universe are listed in the left-most column. These are discovered and described using techniques of [[Thinking Scientifically/The role and limitations of scientific reduction|scientific reduction]], drilling down and continuously decomposing objects into the constituent parts.  We then see [[Understanding Emergence#Levels|higher levels]] that emerge from the fundamental building blocks where useful analysis can also take place. Chemists study atoms, molecules, compounds, chemical bonds, chemical reactions, and energy transfers. Biologist study proteins, cells, tissues, organs, and organisms. Continuing toward the right, we include the subjective realm. There are certainly real concepts of morality, aesthetics, and aspirations, however the mechanisms and arrangements of the constituent parts are not yet well understood. All of that is real, some is fundamental, and some is emergent. The right-most column lists previously hypothesized substances that are now known to be illusions. Phlogiston was a good theory while it lasted, but as our understanding evolved, it is now known to be a superseded theory based on an illusion. 

The existence of [[w:Free_will|free will]], the [[w:Soul|soul]], [[w:Afterlife|afterlife]], and an almighty [[w:God|god]] are still debated. Perhaps these will also be generally accepted as illusions or proved conclusively to exist. The investigations, and [[Practicing Dialogue|dialogue]] continue. 

{| class=&quot;wikitable&quot; 
|-
!! style=&quot;background:SkyBlue; color:black;&quot;  | '''Fundamental'''
!!style=&quot;background:Wheat; color:black;&quot;colspan=&quot;3&quot; |'''Emergent / Apparent'''
|-
|Underlying Physical Reality,

The [[w:Standard_Model|Standard Model]] of Particle Physics,

[[w:Theory_of_relativity|Relativity]]
|Psychology,

Biology,

Chemistry,  

Physics
|Morality,

Aesthetics,

Asperations

|Phlogiston,

Aether,

Caloric,

''Élan vital,''

Unicorns
|-
!!  style=&quot;background:SkyBlue; color:black;&quot;colspan=&quot;3&quot; |  '''Real'''
!!style=&quot;background:Wheat; color:black; |'''Illusions'''
|-
| style=text-align:center colspan=&quot;2&quot; |←←← Factual / Objective →→→
| style=text-align:center| Constructed / Subjective
| style=text-align:center|Obsolete, Superseded, Fantasy, Myth
|-
| style=text-align:center colspan=&quot;2&quot; |←←←← [[Thinking Scientifically/The role and limitations of scientific reduction|Reduction]] →→→→
| style=text-align:center|←←←Emergent →→→
| style=text-align:center|Dreaming, Storytelling,[[Thinking Scientifically#Understanding Evolves| Understanding evolves]]
|}

=== Assignment ===

# Read the essay [[Beyond Theism/What there is|What there is]].
# Read the essay [[Thinking Scientifically/The role and limitations of scientific reduction|The role and limitations of scientific reduction]].
# Carefully examine the table above. Add or move entries in the table to align with your best understanding of what there is.

== Summary and Conclusions ==
Key conclusions studied in this course are summarized here. 

* Remarkably, everything in the universe [[w:Emergence|emerged]] from the basic laws of physics.
* A property of a system is “emergent” if it is not part of a detailed “fundamental” description of the system, but it becomes useful or even inevitable when we look at the system more broadly.
* While reductionism tells us what there is, emergence helps us understand what there can be; what can be constructed.
* '''Although ''this'' is really new, there is really nothing new.'''
* Several hypothesized substances, such as Phlogiston, and caloric were never found because they never existed, they were only an illusion.
* Claiming the “B emerged from A” makes four subsidiary claims: 1) B is ''novel'', 2) B is ''composed'' only of A, 3) the ''transformation mechanism'' is identified, and 4) the ''energy source'' powering the transformation is identified.
* Emergence is about science, not magic.
* Many mechanisms may explain the emergence. These include phase transitions, the kinetic theory of gasses, self-organization, dynamic equilibrium, complexity and chaos theory, downward causation, self-referential loops, and complex adaptive systems.
* Although complex and chaotic systems are unpredictable, they are deterministic.
* In downward causation, it is a mistake to believe that the higher-level system causes the building blocks themselves to acquire new skills.
* Problems create knowledge.
* Both development and evolution result in change over time. Development—the growth and maturing of an organism or system over time—is distinct from evolution—the change in heritable characteristics of organizations across generations.
* In short, emergence is powered by free energy, and constrained by reality.
* [[w:Thermodynamic_free_energy|Free energy]]—energy available to perform work—provides the power that enables emergence. This free energy is often in the form of a naturally occurring ''gradient''.
* In each case the viable possibilities are constrained by reality. [[Facing Facts/Reality is the Ultimate Reference Standard|Reality serves as the ultimate reference standard]]&lt;nowiki/&gt;against which all ideas, theories, and perspectives are measured and validated. It is the foundation that guides our exploration of the universe, shapes our perceptions, and directs our pursuits of what is real and what us true.
* Although everything in existence has emerged, it is not true that anything whatsoever can emerge.

In conclusion, emergence is a captivating lens through which we can perceive the interconnectedness of the universe. From the synchronous fireflies to the intricate web of neural connections in our brains, the world is teeming with examples that highlight the emergence of complexity from simplicity. Whether weak or strong, emergence reveals the elegance with which nature orchestrates a harmonious dance of elements. In the grand narrative of evolution, emergence serves as the catalyst for innovation, adaptation, and the breathtaking diversity that defines life's journey.

While [[Thinking Scientifically/The role and limitations of scientific reduction|reductionism]] is successful at understanding much of our world, it has its limits. Chaos, complexity,  self-organization, self-referential loops, and the harnessing of free energy demonstrate that many interesting and real phenomena can only be understood by analyzing systems in addition to identifying their components.

== Recommended Reading ==
Students who are interested in learning more about emergence may wish to read these books:

*{{cite book |last=Strogatz |first= Steven H. |author-link=w:Steven_Strogatz |date=March 5, 2003 |title=Sync: The Emerging Science of Spontaneous Order  |publisher=Hachette Books |pages=338 |isbn=978-0786868445}}
*{{cite book |last=Morowitz |first=Harold J. |author-link=w:Harold_J._Morowitz |date=November 7, 2002 |title=The Emergence of Everything: How the World Became Complex  |publisher=Oxford University Press |pages=224 |isbn=978-0195135138}}
*{{cite book |last=Azarian  |first=Bobby |date=June 28, 2022 |title=The Romance of Reality: How the Universe Organizes Itself to Create Life, Consciousness, and Cosmic Complexity  |publisher=BenBella Books |pages=320 |isbn=978-1637740446}}
*{{cite book |last=Sapolsky |first=Robert M. |author-link=w:Robert_Sapolsky |date=October 17, 2023 |title=Determined: A Science of Life without Free Will  |publisher=Penguin Press |pages=528 |isbn=978-0525560975}}
*{{cite book |last=Hoel |first=Erik |author-link=w:Erik_Hoel |date=July 25, 2023 |title=The World Behind the World: Consciousness, Free Will, and the Limits of Science  |publisher=Avid Reader Press |pages=256 |isbn=978-1982159382}}
*{{cite book |last=Hofstadter |first=Douglas R  |author-link=w:Douglas_Hofstadter |date=February 5, 1999 |title=Gödel, Escher, Bach: An Eternal Golden Braid  |publisher=Basic Books |pages=824 |isbn=978-0465026562}}
*{{cite book |last=Hofstadter |first=Douglas R  |author-link=w:Douglas_Hofstadter |date=July 8, 2008 |title=I Am a Strange Loop  |publisher=Basic Books |pages=432 |isbn=978-0465030798}}
*{{cite book |last=Carroll |first=Sean |author-link=w:Sean_M._Carroll |date=May 16, 2017 |title=The Big Picture: On the Origins of Life, Meaning, and the Universe Itself  |publisher=Dutton |pages=496 |isbn=978-1101984253}}
*{{cite book |last=West  |first=Geoffrey |author-link=w:Geoffrey_West |date=May 15, 2018 |title=Scale: The Universal Laws of Life, Growth, and Death in Organisms, Cities, and Companies  |publisher=Penguin Books |pages=496 |isbn=978-0143110903}}
*{{cite book |last=Pearl |first=Judea |author-link=w:Judea_Pearl |date=August 25, 2020 |title=Book of Why   |publisher=Basic Books |pages=432 |isbn=978-1541698963}}
*{{cite book |last=Zemansky  |first=Mark Waldo |author-link=w:Mark_Zemansky |date=September 1, 1981 |title=Heat and thermodynamics |publisher=McGraw-Hill|pages=560 |isbn=978-0070666474}}

I have not yet read the following books, but they seem interesting and relevant. They are listed here to invite further research.

*Vital Dust: The Origin and Evolution of Life on Earth, by Christian De Duve

== References ==
&lt;references/&gt;

{{CourseCat}}
[[Category:Applied Wisdom]]

[[Category:Philosophy]]

[[Category:Clear Thinking]]

[[Category:Courses]]

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  <page>
    <title>Large language models</title>
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      <text bytes="19662" sha1="s6nxftu7ik5z310wfwoznqlfkwzlic5" xml:space="preserve">Large language models (LLM's) are software programs that are also known as a form of &quot;artificial intelligence&quot; (AI); LLM's are specifically an aspect of generative AI. This wiki area is for learning, teaching, and research related to LLM's. 
{{RightTOC}}
[[Image:Multiple attention heads.png|right|280px|thumb|An illustration of multiple attention heads, each having its own criteria of relevance of other tokens for one of the tokens within the scope of a context window. (For the purpose of illustration, the context window consists of only one sentence.]]

==Discourse and ideas==
Here is discourse and ideas related to large language models. Perhaps once significantly developed/refined, some of these can have their own sub-page or become a unique learning resource. 
===Learning wikis as training data===
Unless laws change, Creative Commons content appears to be valid training data for LLM's. As LLM's progress and advance, more and more data can be utilized to training increasingly complex models. Learning wikis devoted to learning, teaching, and resource, that allow for original research and original content creation (related to learning, teaching, and research), can potentially be extremely valuable (in terms of educational value) for large language models. Perhaps in the future (if this does not already exist), large language models will be able to continuously be trained on, retain, and learn from new data and information. Perhaps in the future, an open source large language model could only be trained on Creative Commons data, and therefore, all generated content would also be licensed under Creative Commons. 

==Discussion questions==
Here are some learning and teaching oriented discussion questions related to large language models. Humans can use language and mental effort to explore these ideas collaboratively, or some of these could be used as prompts to see how an LLM might respond. 
* Would a large language model that is only trained on Creative Commons licensed data only be capable of generating responses to prompts that can also be rightly and correctly licensed under a Creative Commons license? 
* How might large language models affect learning and research. Will LLM's eventually seen like calculators are in math and sciences now? But for everything (all subjects/topics, including math, physics, ethics, biology, psychology, chemistry, engineering, art)? 
* What are some ethical considerations related to large language models that should be considered? 
* What are some pros and cons to open source large language models? Will open source LLM's likely become more advanced the propriety LLM's eventually? What do you think? 
* How can large language models help to advance and accelerate technological automation in ways that will benefit all of humanity? 
* In what ways can large language models help programmers to code?
* Can music be thought of a language within the realm of large language models? 
* What is differentiable computing and how does differentiable computing relate to large language models? 
* How can teachers utilize large language models to help accelerate student learning and to help students learn more efficiently? 

== Educational prompt ideas==
These are original prompt ideas regarding ways to learn about large language models, and also to explore using LLM's for learning, teaching, and research. Input these into your preferred LLM (without quotes) to see what results are generated. LLM's might produce interesting or useful answers in response to these prompts. Some of these prompts may be interesting or useful for discussions among and between humans. 

* &quot;Describe to me how large language models can be utilized for learning, teaching, and research. Do this in an about 200 word two paragraph mini essay. Explain it to me like I am a freshman in community college.&quot; 
* &quot;Give me a list of 12 ways that large language models can be utilized for learning, teaching, and research.&quot;
* &quot;How can LLM's be utilized to accelerate the pace of research and scientific discovery?&quot;
* &quot;What are some ethical considerations related to large language models that should be considered?&quot;
* &quot;What are some pros and cons to open source large language models? Will open source LLM's likely become more advanced the propriety LLM's eventually? What do you think?&quot;
* &quot;What are some project ideas to integrate large language models in with humanoid robots, and/or other sorts of robots? Please give me 15 project ideas that can be relatively simple or extremely complex.&quot;
* &quot;Please search the Internet if possible. In what ways have university professors and academic researchers been using large language models in the last year? Please respond in list form.&quot;
* &quot;In what ways can large language models help programmers to code? Please provide me 8 examples and respond in list form.&quot;
* &quot;Can music be thought of a language within the realm of large language models?&quot;
* &quot;What is differentiable computing and how does differentiable computing relate to large language models?&quot;
* &quot;How can one fine tune an open source large language model?&quot;
* &quot;What are some popular state of the art open source large language models. Please search the internet as helpful and respond to me in list form.&quot;
* &quot;Please give me a list of important terminology that I should be aware of when working with and training open source large language models. Please be comprehensive. Please respond in list form. And please search the internet as helpful.&quot;
* &quot;What sort of hardware should I utilize to run the most competent open source large language models that I want to utilize for learning, teaching, and research? Please search the internet as helpful.&quot;
* &quot;How can teachers utilize large language models to help accelerate student learning and to help students learn more efficiently? Please respond in list form.&quot;
* &quot;How can researchers utilize large language models to create theories, hypothesis, and to formulate potential research studies? Please respond in short paragraphs, but in list form.&quot;

==Readings and learning media==
===External===
* [https://stpp.fordschool.umich.edu/tags/large-language-models Large Language Models] - Articles
* [https://hai.stanford.edu/news/how-large-language-models-will-transform-science-society-and-ai How Large Language Models Will Transform Science, Society, and AI]
* [https://insights.sei.cmu.edu/blog/harnessing-the-power-of-large-language-models-for-economic-and-social-good-foundations/ Harnessing the Power of Large Language Models For Economic and Social Good: Foundations]
* [https://courses.grainger.illinois.edu/CS447/sp2023/Slides/Lecture27.pdf Lecture 27: Intro to Large Language Models]

==== Introduction to Hugging Face NLP ====
Introductory course about natural language processing (NLP) using libraries from the Hugging Face ecosystem &amp;ndash; Transformers, Datasets, Tokenizers, and Accelerate.

: [https://huggingface.co/learn/nlp-course/chapter0/1 '''NLP Course''']
:: [https://huggingface.co/learn/nlp-course/chapter1/1 transformer models]
::: [https://huggingface.co/learn/nlp-course/chapter1/2 NLP], [https://huggingface.co/learn/nlp-course/chapter1/3 What], [https://huggingface.co/learn/nlp-course/chapter1/4 How], [https://huggingface.co/learn/nlp-course/chapter1/5 Encoder], [https://huggingface.co/learn/nlp-course/chapter1/6 Decoder], [https://huggingface.co/learn/nlp-course/chapter1/7 Sequence-to-sequence], [https://huggingface.co/learn/nlp-course/chapter1/8 Bias and limitations], 
:: [https://huggingface.co/learn/nlp-course/chapter2/1 using transformers]:
::: [https://huggingface.co/learn/nlp-course/chapter2/2 pipeline], [https://huggingface.co/learn/nlp-course/chapter2/3 models], [https://huggingface.co/learn/nlp-course/chapter2/4 tokenizer], [https://huggingface.co/learn/nlp-course/chapter2/5 batching], decoding, padding, attention mask
:: [https://huggingface.co/learn/nlp-course/chapter3/1 fine-tuning a pretrained model]:
::: [https://huggingface.co/learn/nlp-course/chapter3/2 Preprocessing]&lt;small&gt;: tokenization, padding&lt;/small&gt;, [https://huggingface.co/learn/nlp-course/chapter3/3 Fine-tuning], [https://huggingface.co/learn/nlp-course/chapter3/4 Full training], map, [https://huggingface.co/docs/datasets/index dataset], dynamic padding, batch, collate function, train, predict, evaluate, [https://github.com/huggingface/accelerate accelerate]
:: [https://huggingface.co/learn/nlp-course/chapter4/1 sharing models and tokenizers]:
::: hub, model card
:: [https://huggingface.co/learn/nlp-course/chapter5/1 the datasets library]:
::: batch, DataFrame, validation, splitting, embedding, FAISS
:: [https://huggingface.co/learn/nlp-course/chapter6/1 the tokenizers library]:
::: [https://huggingface.co/learn/nlp-course/chapter6/2 training tokenizer], grouping, QnA, [https://huggingface.co/docs/tokenizers/api/normalizers normalizers], pre-tokenization, [https://huggingface.co/docs/tokenizers/api/models models],[https://huggingface.co/docs/tokenizers/api/trainers trainers]: [https://huggingface.co/learn/nlp-course/en/chapter6/5 Byte-Pair Encoding (BPE)], WordPiece, Unigram, [https://huggingface.co/docs/tokenizers/api/post-processors post processors], [https://huggingface.co/docs/tokenizers/components#decoders decoders]
:: [https://huggingface.co/learn/nlp-course/chapter7/1 main nlp tasks]:
::: [https://huggingface.co/learn/nlp-course/chapter7/2 token classification], metrics, perplexity, [https://huggingface.co/learn/nlp-course/chapter7/4 translation], [https://huggingface.co/learn/nlp-course/chapter7/5 summarization], [https://huggingface.co/learn/nlp-course/chapter7/6 training CLM], [https://huggingface.co/learn/nlp-course/chapter7/7 QnA],  
:: [https://huggingface.co/learn/nlp-course/chapter8/1 how to ask for help]
:: [https://huggingface.co/learn/nlp-course/chapter9/1 building and sharing demos]

==== Hugging Face docs ====
: https://huggingface.co/docs
:: Core libraries
::: [https://huggingface.co/docs/transformers Transformers] &amp;ndash; State-of-the-art ML for Pytorch, TensorFlow, and JAX.
:::: [https://huggingface.co/docs/transformers/main_classes/pipelines#transformers.pipeline pipeline] &amp;ndash; simple interface for inference with models.
:::: [https://huggingface.co/docs/transformers/model_doc/auto#auto-classes Auto classes]: AutoConfig, AutoModel, and AutoTokenizer. The from_pretrained method.
:::: [https://huggingface.co/docs/transformers//main_classes/trainer#transformers.Trainer Trainer] and [https://huggingface.co/docs/transformers/main_classes/trainer#transformers.TrainingArguments TrainingArguments] 
::: [https://huggingface.co/docs/datasets Datasets] &amp;ndash; Access and share datasets for computer vision, audio, and NLP tasks.
:::: [https://huggingface.co/docs/datasets/tutorial Tutorials]
:::: [https://huggingface.co/docs/datasets/how_to How-to guides]
:::: [https://huggingface.co/docs/datasets/about_arrow Conceptual guides]
:::: [https://huggingface.co/docs/datasets/package_reference/main_classes Reference]
::: [https://huggingface.co/docs/accelerate Accelerate] &amp;ndash; Easily train and use PyTorch models with multi-GPU, TPU, mixed-precision.
::: [https://huggingface.co/docs/tokenizers Tokenizers] &amp;ndash; Fast tokenizers, optimized for both research and production.
:::: Components: Normalizers, Pre-tokenizers, Models, Post-Processors, Decoders ...
:: [https://huggingface.co/docs/hub Hub] &amp;ndash; Host Git-based models, datasets and Spaces on the Hugging Face Hub.
:: [https://huggingface.co/docs/diffusers Diffusers] &amp;ndash; State-of-the-art diffusion models for image and audio generation in PyTorch.
:: [https://huggingface.co/docs/huggingface_hub Hub Python Library] &amp;ndash; Client library for the HF Hub: manage repositories from your Python runtime.
:: [https://huggingface.co/docs/huggingface.js Huggingface.js] &amp;ndash; A collection of JS libraries to interact with Hugging Face, with TS types included.
:: [https://huggingface.co/docs/transformers.js Transformers.js] &amp;ndash; Community library to run pretrained models from Transformers in your browser.
:: [https://huggingface.co/docs/api-inference Inference API (serverless)] &amp;ndash; Experiment with over 200k models easily using the serverless tier of Inference Endpoints.
:: [https://huggingface.co/docs/inference-endpoints Inference Endpoints (dedicated)] &amp;ndash; Easily deploy models to production on dedicated, fully managed infrastructure.
:: [https://huggingface.co/docs/peft PEFT] &amp;ndash; Parameter efficient fine-tuning methods for large models
::: Soft prompting, LoRA, IA3
:: [https://huggingface.co/docs/optimum Optimum] &amp;ndash; Fast training and inference of HF Transformers with easy to use hardware optimization tools.
:: [https://huggingface.co/docs/optimum-neuron AWS Trainium &amp;amp; Inferentia] &amp;ndash; Train and Deploy Transformers &amp;amp; Diffusers with AWS Trainium and AWS Inferentia via Optimum
:: [https://huggingface.co/docs/evaluate Evaluate] &amp;ndash; Evaluate and report model performance easier and more standardized.
::: types: metrics, comparisons, measurements
:: [https://huggingface.co/tasks Tasks]
::: extraction, question answering, classification, generation ...
:: [https://huggingface.co/docs/dataset-viewer Dataset viewer] &amp;ndash; API to access the contents, metadata and basic statistics of all Hugging Face Hub datasets.
::: Splits and subsets, [https://github.com/huggingface/dataset-viewer dataset-viewer]
:: [https://huggingface.co/docs/trl TRL] &amp;ndash; Transformer Reinforcement Learning
:::  reward modeling, fine-tuning, optimizations, 
:: [https://huggingface.co/docs/sagemaker Amazon SageMaker] &amp;ndash; Train and Deploy Transformer models with Amazon SageMaker and Hugging Face Deep Learning Containers (DLC).
:: [https://huggingface.co/docs/timm timm] &amp;ndash; Pytorch Image Models.
::: State-of-the-art computer vision models, layers, optimizers, training/evaluation, and utilities.
:: [https://huggingface.co/docs/safetensors Safetensors] &amp;ndash; Simple, safe way to store and distribute neural networks weights.
:: [https://huggingface.co/docs/text-generation-inference Text Generation Inference (TGI)] &amp;ndash; Toolkit to serve Large Language Models.
:: [https://huggingface.co/docs/autotrain AutoTrain] &amp;ndash; AutoTrain API and UI.
::: [https://huggingface.co/autotrain autotrain]
:: [https://huggingface.co/docs/text-embeddings-inference Text Embeddings Inference] &amp;ndash; Toolkit to serve Text Embedding Models.
:: [https://huggingface.co/docs/competitions Competitions] &amp;ndash; Create your own competitions on Hugging Face.
:: [https://huggingface.co/docs/bitsandbytes Bitsandbytes] &amp;ndash; Toolkit to optimize and quantize models.
:: [https://huggingface.co/docs/optimum-tpu Google TPUs] &amp;ndash; Deploy models on [https://cloud.google.com/tpu/docs Google TPUs] via Optimum.
:: [https://huggingface.co/docs/chat-ui Chat UI] &amp;ndash; Open source chat frontend, powers the [https://huggingface.co/chat HuggingChat] app.
:: [https://huggingface.co/docs/leaderboards Leaderboards] &amp;ndash; Create your own Leaderboards on Hugging Face.
:: [https://huggingface.co/docs/hugs Hugging Face Generative AI Services (HUGS)] &amp;ndash; optimized, zero-configuration inference microservices designed to simplify and accelerate the development of AI applications with open models.

===Videos===
* [https://www.youtube.com/watch?v=5sLYAQS9sWQ How Large Language Models Work]
* [https://www.youtube.com/watch?v=JhCl-GeT4jw Large Language Models and The End of Programming - CS50 Tech Talk with Dr. Matt Welsh]
* [https://www.youtube.com/watch?v=yBI1nPep72Q LMStudio Tutorial Run ANY Open-Source Model LOCALLY]
* [https://www.youtube.com/watch?v=UU1WVnMk4E8 Create a Large Language Model from Scratch with Python – Tutorial]
* [https://www.youtube.com/watch?v=eC6Hd1hFvos Fine-tuning Large Language Models (LLMs) | w/ Example Code]

===Data sets===
* [https://huggingface.co/blog/Pclanglais/two-trillion-tokens-open Releasing the largest multilingual open pretraining dataset]
:: [https://huggingface.co/datasets/PleIAs/common_corpus Common Corpus]
:: [https://huggingface.co/datasets/PleIAs/common_corpus/tree/main Files and versions]

===Wikipedia===
{{colbegin|5}}
* [[w:Large language model|Large language model]]
* [[w:Prompt engineering|Prompt engineering]]
* [[w:GPT-4|GPT-4]]
* [[w:Category:Large language models|Category:Large language models]]
* [[w:LLaMA|LLaMA]]
* [[w:Mistral AI|Mistral AI]]
* [[w:Foundation model|Foundation model]]
* [[w:Natural-language understanding|Natural-language understanding]]
* [[w:Ethics of artificial intelligence|Ethics of artificial intelligence]]
* [[w:Artificial general intelligence|Artificial general intelligence]]
* [[w:Intelligence amplification|Intelligence amplification]]
* [[w:Outline of artificial intelligence|Outline of artificial intelligence]]
* [[w:Synthetic intelligence|Synthetic intelligence]]
* [[w:Weak artificial intelligence|Weak artificial intelligence]]
* [[w:History of artificial intelligence|History of artificial intelligence]]
* [[w:Timeline of artificial intelligence|Timeline of artificial intelligence]]
* [[w:Progress in artificial intelligence|Progress in artificial intelligence]]
* [[w:History of natural language processing|History of natural language processing]]
* [[w:Hardware for artificial intelligence|Hardware for artificial intelligence]]
* [[w:AI safety|AI safety]]
* [[w:Neural scaling law|Neural scaling law]]
* [[w:Philosophy of artificial intelligence|Philosophy of artificial intelligence]]
* [[w:Philosophy of mind|Philosophy of mind]]
* [[w:Computational theory of mind|Computational theory of mind]]
* [[w:Regulation of artificial intelligence|Regulation of artificial intelligence]]
* [[w:LangChain|LangChain]]
* [[w:Generative pre-trained transformer|Generative pre-trained transformer]]
* [[w:GitHub Copilot|GitHub Copilot]]
* [[w:ChatGPT|ChatGPT]]
* [[w:Generative artificial intelligence|Generative artificial intelligence]]
* [[w:Category:Generative artificial intelligence|Category:Generative artificial intelligence]]
* [[w:Music and artificial intelligence|Music and artificial intelligence]]
* [[w:Workplace impact of artificial intelligence|Workplace impact of artificial intelligence]]
* [[w:Applications of artificial intelligence|Applications of artificial intelligence]]
* [[w:Artificial intelligence in Wikimedia projects|Artificial intelligence in Wikimedia projects]]
* [[w:Wikipedia:Artificial intelligence|Wikipedia:Artificial intelligence]]
* [[w:Artificial intelligence in healthcare|Artificial intelligence in healthcare]]
* [[w:Automated reasoning|Automated reasoning]]
* [[w:Machine learning in physics|Machine learning in physics]]
* [[w:Quantum neural network|Quantum neural network]]
* [[w:ChatGPT in education|ChatGPT in education]]
* [[w:Artificial intelligence content detection|Artificial intelligence content detection]]
* [[w:Turing test|Turing test]]
* [[w:List of datasets for machine-learning research|List of datasets for machine-learning research]]
* [[w:Fine-tuning (deep learning)|Fine-tuning (deep learning)]]
* [[w:Attention (machine learning)|Attention (machine learning)]]
* [[w:Mixture of experts|Mixture of experts]]
* [[w:Gemini (language model)|Gemini (language model)]]
* [[w:Auto-GPT|Auto-GPT]]
* [[w:VideoPoet|VideoPoet]]
{{colend}}

==See also==
* [[Artificial intelligence]]
* [[Artificial Intelligence &amp; Machine Learning]]
* [[Artificial Intelligence and Robotics Laboratory]]
* [[Computer science]]
* [[Artificial Consciousness]]
* [[Supersymmetric Artificial Neural Network]]
* [[History of artificial intelligence]]
[[Category: Computer science]]
[[Category: Machine learning]]
[[Category: Artificial intelligence]]</text>
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    <title>User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells</title>
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      <text bytes="208373" sha1="28o80f9w9tz3lxpw6aa6zk65lecfdui" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|June 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a regular 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the &lt;s&gt;120-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 5, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 11 [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] cells. Eleven 11-cells (plural) are the 137-cell, the &lt;s&gt;600-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 3, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 137 [[W:Rhombic triacontahedron|triacontahedron]] cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=''Regularity of Graphs, Complexes and Designs''}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=''A Symmetrical Arrangement of Eleven Hemi-Icosahedra''}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]]
The 4-dimensional regular polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:Simplex|simplex]] (called the [[5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:Orthoplex|orthoplex]] (called the [[16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular convex 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the [[w:Tesseract|16-point (8-cell) tesseract]], the 4-cubic, which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found within any of the other regular convex 4-polytopes, except in the largest and most complex one, the [[120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). However, we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The pentad 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the hexad 16-cell is their very starting point, and the most frequently used tool in the box.

== 5-cells and hemi-icosahedra in the 11-cell ==
[[File:Symmetrical_5-set_Venn_diagram.svg|thumb|The 5-point (10-face) regular 5-cell (the regular 4-simplex). Grünbaum's rotationally symmetrical 5-set Venn diagram{{Sfn|Grünbaum|1975|loc=''Rotationally symmetrical 5-set Venn diagram'', Fig 1 (e)|ps=; partitions the individual elements of the 5-cell.}} is an illustration of the 5-cell labeling each of its &lt;math&gt;2^5&lt;/math&gt; elements.{{Sfn|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled; image includes the Python code to render it, optimising for maximum area of the smallest regions.}}]]
[[File:Hemi-icosahedron.png|thumb|The 6-point (10 face) [[W:hemi-icosahedron|hemi-icosahedron]], an abstraction of the regular icosahedron, has half as many faces, edges and vertices. Each element of the abstract polyhedron represents two real elements found in different places in the 11-cell's realization.{{Sfn|Ruen: hemi-icosahedron|2007}}]]
The most apparent relationship between the pentad 5-cell and the hexad hemi-icosahedron is that they both have 10 triangular faces. When we see a facet congruence between a 4-polytope and a 3-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 6-point 5-simplex. It is also the same size as the 6-point 3-orthoplex; thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no chord congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals, pentad and hexad, who are famously stand-offish with each other. Interesting!

The cell of the 11-cell is an abstract hexad hemi-icosahedron containing pentads in its interior, most handsomely illustrated by Séquin.{{Sfn|Séquin &amp; Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing rings of polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The 11-cell and 5-cell of radius {{radic|2}} both have edge length {{radic|5}}, so their triangular faces are congruent. The 11-cell's 11 hemi-icosahedral cells have 10 triangle faces each, and a 5-cell is behind each face. Only 3 of the 5-cell's 5 vertices are visible on the surface of the hemi-icosahedron as the exposed face, but its other two vertices are visible on the opposite side of the hemi-icosahedron as an exposed edge.{{Sfn|Coxeter|1984|p=110|loc=§6. The Petrie polygon [of the 11-cell]|ps=; &quot;We may reasonably call this edge and face ''opposites''. It is easy to find the face opposite to a given edge by looking at the faces to which a given edge belongs. ... Conversely, given a face, we can find the opposite edge by seeing which vertices belong to neither of the hemi-icosahedra which share that face. The ten edges opposite to the ten faces of one hemi-icosahedron are the edges of the complementary &lt;math&gt;a_4&lt;/math&gt; [4-simplex], that is, the joins of all pairs of the five vertices [of the 11-cell] not belonging to the given hemi-icosahedron.&quot;}} The other parts of the 5-cell (its other 6 edges, its other 9 faces and its 5 tetrahedral cells) lie outside the hyperplane occupied by the hemi-icosahedron.

There are 11 disjoint 5-cells in each 11-cell, which is comprised of 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. The real 11-cell is harder to see than the abstract 11-cell that represents it, because the real hemi-icosahedron that the abstract hemi-icosahedron represents is hard to see, but seeing these objects will be easier once we have identified the real hemi-icosahedron and exactly where the 11-cell's real elements reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each hemi-icosahedron's 10 faces come from 10 disjoint 5-cells, and there is just one 5-cell with which it does not share a face.{{Efn|As Coxeter observes (in the previous citation), that unrepresented 5-point (5-cell) is the other 5 vertices of the 11-point (11-cell) that are not vertices of this 6-point (hemi-icosahedron), the hemi-icosahedron's disjoint complement.}} Each 5-cell has 10 faces from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share a face.

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in two different planes which are completely orthogonal to each other. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must be sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness: 8 concentric hulls|2022|loc=Hull #8 (lower right)|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell of the 11-cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Compounds in the 120-cell|§Compounds in the 120-cell]]'' that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software which uses Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s to render the polyhedra which are found in the interior of ''n''-dimensional polytopes.{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the tetrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is a real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (Hull #8) is a concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this {12} irregular great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing {6} irregular great hexagons (truncated triangles) of alternating 120-cell edges and 5-cell edges. The great dodecagon is a great circle of the hemi-icosahedral cell, and also a great circle of the 120-cell, because unlike the cells of most 4-polytopes, the 11-cell's cells are embedded in 4-space at the exact center of the 4-polytope, making them inscribed 4-polytopes themselves.{{Efn|name=Embedding point}} There are 25 great dodecagons and 120 5-cell edges in Moxness's 60-point (Hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (Hull #8) polyhedron altogether.

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point (Hull #8) is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] similar to the one from the catalog shown here,{{Sfn|Piesk: Rhombicosidodecahedron|2018}} but a slightly shallower truncation with smaller red pentagons (120-cell faces) and narrower rhombs. Rhombicosidodecahedra are made by truncating the [[W:Rhombic triacontahedron|rhombic triacontahedron]], which is the 30-sided polyhedron with only one kind of face, the dual of the 30-point icosidodecahedron. The 120-cell contains 120 of these 11-cell cells.]]
Moxness's 60-point (Hull #8) is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (Hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|The 5-point (regular 5-cell) can be another abstraction of Moxness's 60-point (Hull #8), 12-vertices-into-1, but it is not strictly the result of a contraction operation as described by Boole Stott, since she did not apply her expansion and contraction operations to uniform polytopes with more than one edge length.}} In fact Moxness was not the first to find a rhombicosidodecahedron in the 120-cell; Alicia Boole Stott was. It is the 6th section of {5,3,3} beginning with a cell, which Boole Stott identified as the ''e&lt;sub&gt;2&lt;/sub&gt; expansion'' of the icosahedron (or equivalently of its dual polyhedron the dodecahedron).{{Sfn|Boole Stott|1910|loc=§Examples of the e&lt;sub&gt;2&lt;/sub&gt; expansion|p=7}} But that rhombicosidodecahedron identified by Boole Stott is not Moxness's Hull #8: rather, it is the semi-regular Archimedean solid, with a single edge length and square faces. Moxness's Hull #8 with its two distinct edge lengths and rectangular faces is Coxeter's 8th section of {5,3,3} beginning with a cell, which is missing from the sections illustrated by Boole Stott.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than tetrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. The 120-cell contains 120 of Moxness's 60-point (11-cell cells). They are non-disjoint, with 12 60-points sharing each vertex.

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (Hull #8) is the real hemi-icosahedron cell, with 120 5-cell edges in it as chords lurking under its surface. The rhombicosadodecahedron has enough of those 5-cell edges in its interior to make 20 disjoint tetrahedral cells which belong to 20 distinct 5-cells.

The 600-point (120-cell) may be constructed as a semi-regular honeycomb of two kinds of cells, the 120 non-disjoint 60-point (Moxness's rhombicosadodecahedra) and their neighbors the 120 disjoint 5-point (5-cells), cell-bonded together by shared tetrahedral cells of edge-length {{radic|5}}. Each rhombicosadodecahedron is bonded to 20 islanded 5-cells, sharing with each a 5-cell tetrahedron (4 of its interior 5-cell faces, and 6 of its interior 5-cell edges). The rhombicosadodecahedron is also bonded directly to 10 other rhombicosadodecahedra (since each 11-cell has 11 rhombicosadodecahedron cells), again not by sharing one of its exterior faces, but this time by sharing an interior face, a {{radic|6}} triangle which lies in a central plane.{{Efn|The {{radic|6}} chord is another hidden chord lurking beneath the 60-point rhombicosadodecahedron's surface, like the {{radic|5}} 5-cell edge. Both these hidden chords lie on the same great circle with the 120-cell edges (the pentagon face edges). Three {{radic|6}} chords make a triangle which lies in a central plane. This {{radic|6}} great triangle must not be confused with the {{radic|5}} face triangle of the 5-cell, which does not lie in a central plane.}} The 120 rhombicosadodecahedra are also bonded directly together at ''all'' their exterior faces: at their pentagon faces (which are 120-cell faces of dodecahedral cells); at their small triangular faces (which are 600-cell faces of small tetrahedral cells, 10 of which are inscribed in each dodecahedral cell); and at their rectangular faces (which are faces of 10-point [[w:Pentagonal_prism|pentagonal prism]] cells, where the pentagonal prism is a column inscribed in a dodecahedral cell, joining its opposite pentagon faces). The 120 rhombicosadodecahedra are bonded directly to each other at their own exterior faces; no dodecahedra, small tetrahedra, or pentagonal prisms lie between them, and those objects are not cells of this honeycomb; we mention them here only to help locate the rhombicosadodecahedra within the 120-cell. The only objects which fill holes between the rhombicosadodecahedra are the 120 islanded 5-cells.{{Efn|This honeycomb of the 120-cell resembles the [[600-cell#Icosahedra|honeycomb of the 600-cell consisting of 24 face-bonded regular icosahedra]] with 24 clusters of 5 tetrahedra filling the holes between them. In the 120-cell there are 5 times as many of everything as in the 600-cell, so there are 120 (rhombicosadodecahedron) icosahedral cells with 600 tetrahedra filling the holes between them.}}

It is possible for rhombicosadodecahedra to meet each other at more than one kind of exterior face because they are not meeting there as cells of the same 11-cell; they belong to different 11-cells. The 11 rhombicosadodecahedra in the same 11-cell do not meet at their own exterior faces, they meet at their interior {{radic|6}} triangle faces. As a cellular 4-polytope, the 11-cell is inside-out. Its cells meet each other at interior planes, sharing interior {{radic|6}} triangle faces, instead of in the usual way that 4-polytopes' cells share an exterior face. It is a deeper kind of intertwining to be sure, but the 11 rhombicosadodecahedra in the same 11-cell are still volumetrically disjoint cells, and the 11-cell is a perfectly real 4-polytope with real cells. It is a semi-regular 4-polytope, because it has two different kinds of cells: 11 rhombicosadodecahedra and 11 5-cells. The 11 5-cells are islanded (not bound to each other), and the 11 rhombicosadodecahedra are bound both to 5-cells (at {{radic|5}} 5-cell faces) and to each other (at {{radic|6}} central triangle faces). In the section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#What's in the box|§What's in the box]]'', below, we will look closely at the relation between the {{radic|5}} chord and the {{radic|6}} chord, which is the relation between the pentad and the hexad, and at the general phenomenon of hemi-polytopes which meet each other at a central plane. Then in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Cell rings of the 11-cells|§Cell rings of the 11-cells]]'' we will look at the two-cell cellular structure of the 11-cell in more detail.

Considered as a single complex of overlapping cells rather than as distinct honeycombs of volumetrically disjoint cells, the 120-cell contains 60-point (rhombicosadodecahedral cells), islanded 5-point (5-cells), 20-point (dodecahedral cells), and 10-point ([[w:Pentagonal_prism|pentagonal prism]] cells). The 60-points meet each other in pairs at their pentagon faces, meet regular dodecahedra at those pentagon faces also, meet each other in quads at their 600-cell faces where they share an interior 4-point (600-cell tetrahedral cell) and its four faces (just as they share a larger interior 5-cell tetrahedron and its faces), and meet each other and the 10-point (pentagonal prisms) at their rectangle faces. We might try to imagine such a complex with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with parts truncated from 30 neighboring truncated dodecahedra pentagonal prisms). There are 120 of each of these 3 kinds of dodecahedron. 

Moxness's 60-point (rhombicosadodecahedron cell) is a central object of the 120-cell, and contains many wonders. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra by expansion and contraction operations, as Alicia Boole Stott discovered before 1910,{{Sfn|Polo-Blanco: ''Theory and history of geometric models of Alicia Boole Stott''|2007|loc=§5.3.2 1910 paper on semi-regular polytopes|pp=152-158|ps=; summarizes Boole Stott's method and results from {{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}}, including her tables of expansion-contraction dimensional analogies and a few of her illustrations.}} and those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, rediscovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter &amp; Petrie|1938|p=4|loc=''[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]''}}|name=Concentric polyhedra}} Before we move on from the 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice one more thing about the hidden {{radic|5}} chords lurking below its surface. The 12 little pentagon faces are spanned by sets of 6 {{radic|5}} chords (5-cell edges). The 6 chords of each set are disjoint (they don't touch or form 5-cell faces), but they are symmetrically arranged in 3 parallel pairs, {{radic|3}} apart, which lie in 3 orthogonal central planes. It turns out they are the 6 reflex edges of a 12-point non-convex polyhedron called the [[w:Jessen's_icosahedron|Jessen's icosahedron]], inscribed in the 60-point cell. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again and work our way outwards, before eventually in ''[[#The perfection of Fuller's cyclic design|§The perfection of Fuller's cyclic design]]'' we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cell)s ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks, showing its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Pentahemidemicube.png|120px]]
|[[File:Tetrahemihexahedron_rotation.gif|120px]]
|[[File:Pentahemicosahedron.png|120px]]
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts needed to build the 120-cell (the picture on the cover of the box), and everything else that's invisible in that picture which fits inside the 120-cell: all the astonishing regular 4-polytopes. The pentad building block is half a pair of orthogonal 5-point (regular 5-cells), the hexad building block is half an 8-point (16-cell), and the heptad building block is a quarter of an 11-point (11-cell). Everything can be built from these blocks by embedding them at the center of the 3-sphere, inscribing them as 4-polytopes to make compounds as with the 16-cell building block, rather than by assembling them face-to-face as 3-polytopes to make honeycombs in which they are cells or vertex figures.{{Efn|name=Embedding point}}

The pentad building block is the 5-point '''pentahemidemicube''', an abstract polyhedron with 10 equilateral triangle faces, 10 {{radic|5}} edges, and 5 vertices.{{Sfn|Christie: Pentahemidemicube|2024|loc=File:Pentahemidemicube.png|ps=; &quot;The 5-point ''pentahemidemicube'' is an abstract polyhedron with 10 equilateral triangle faces, 10 edges, and 5 vertices. It has a real presentation as the regular 5-cell.&quot;}} It has a real presentation as the 5-point (regular 5-cell 4-polytope), a regular tetrahedron expanded by addition of a fifth vertex equidistant from all the others in a fourth dimension. The pentad block is a tetrahedral pyramid (a 5-cell) inscribed in a cube: a tetrahedron inscribed in a cube (a demicube) expanded by addition of a fifth vertex, one of the cube's other 4 vertices. Each pentad (5-cell) building block lies in the 120-cell orthogonal to three other pentad (5-cell) building blocks. They are four completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges. Each regular 5-cell in the 120-cell is a pentad (pentahemidemicube) building block, in one of 30 orthogonal sets of 4. There are 120 pentad building blocks in the box.

Every vertex of the 16-cell, and therefore every vertex of the 8-cell, 24-cell, 600-cell and 120-cell, is surrounded by a 6-point (octahedral vertex figure). The hexad building block is that 6-point octahedron embedded at the center of the 4-polytope instead of at a vertex, that is, inscribed in the 3-sphere as a 4-polytope, the 6-point '''[[w:Tetrahemihexahedron|tetrahemihexahedron]]''' or '''hemi-cuboctahedron'''.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; &quot;The tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U&lt;sub&gt;4&lt;/sub&gt;. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices.&quot;}} The hexads have 4 red triangle 5-cell faces, bounded by 12 {{radic|5}} edges. Their 3 yellow square faces lie in central planes, because the building block is an abstract 4-polytope. The hexad is an irregular abstract polyhedron that also has 6 {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

Two orthogonal 5-point (pentad) building blocks intersect in a 4-point (inscribed tetrahedron) to form a 6-point (hexad) building block. Two orthogonal 6-point (hexad) building blocks intersect in a 4-point (central square) to form an 8-point (16-cell) building block, the most usable building block, from which everything larger can be built by compounding.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 120 pentads to pair with a completely orthogonal hexad, and although there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed) they only result in 120 pairs two different ways, so there are 120 11-cells. The 11-cell is 6 pentads' and 5 hexads' smallest joint compound, and the 120-cell is their largest, 120 pentads and 100 hexads.

The heptad{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''|pages=22–31}} building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|4}} edges and 9 {{radic|5}} edges), and 7 vertices.{{Sfn|Christie: Pentahemicosahedron|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;}} It is a compound of a 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]) and a 6-point (hexad [[W:hemi-icosahedron|hemi-icosahedron]]) intersecting in a 4-point (tetrahedron), that is, of three orthogonal 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]&lt;nowiki/&gt;s) intersecting in their common 4-point (tetrahedron). It is also the abstract cube contracted by removal of one vertex, which reduces one of the cube's two inscribed tetrahedra (demicubes) to a single remaining triangular face. As with all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Four &lt;small&gt;&lt;math&gt;(11-7)&lt;/math&gt;&lt;/small&gt; orthogonal 7-point (pentahemicosahedron) blocks can be joined at a common Petrie polygon (a skew hexagon which contains their orthogonal 6-edge paths) to construct a real 11-point ([[W:11-cell|11-cell]] semi-regular 4-polytope), which magically contains six &lt;small&gt;&lt;math&gt;(11-5)&lt;/math&gt;&lt;/small&gt; 5-point (1-pentad regular decahedra) as cells and five &lt;small&gt;&lt;math&gt;(11-6)&lt;/math&gt;&lt;/small&gt; 12-point (2-hexad semi-regular cuboctahedra) as cells. There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like [[W:Euclid's postulates|Euclid's postulates]], and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as honeycombs of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles. It possesses the complete symmetry of the polytope without any redundancy, because it contains one of each of the polytope's characteristic root elements.{{Efn|A [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be partitioned radially by its planes of symmetry (Coxeter's &quot;reflecting circles&quot;) into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated rotationally to generate its regular 4-polytope because it is the complete [[W:gene|gene]] for it, containing all of its elements and capturing all of its symmetry without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring that triggers the reflections.|name=Characteristic orthoscheme}} It is the gene for the polytope, which can be replicated to construct the polytope.

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its &lt;math&gt;A_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the regular compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;small&gt;&lt;math&gt;120\times 120 = 14400&lt;/math&gt;&lt;/small&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[16--cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its &lt;math&gt;B_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 384. The 120-cell is the convex hull of the regular compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 24-point (24-cell) is subdivided into 1152 instances of its [[24-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;F_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 1152. The 120-cell is the convex hull of the regular compound of 25 disjoint 24-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;25\times 1152 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has three equivalent origins in distinct root systems, though most are obviously constructed from one root, and not so obviously from the others. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything larger than the pentad can be constructed by compounding hexads, and we shall see that everything as large as a 24-cell has a construction from heptads which are a product of a pentad and a hexad. Surely every uniform polytope can be constructed by some function of expansions and contractions from any of three root systems, however indirect the recipe. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes, related by an equals sign between them, then give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics by [[W:Noether's theorem|Noether's theorem]].

=== Pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 octahedral cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell contains the 600 octahedral cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It also contains the 600 cubical cells of its 75 8-cells. It is a 600-cell in three ways, although the octahedra and cubes are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell and the 8-cell, respectively.|name=600 cells three ways}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (1-pentad 5-cell), out of the 8-point (2-hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (1-pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope that contains 5-point (1-pentad 5-cell) building blocks beside the largest one, it just isn't a ''regular'' 4-polytope. It is semi-regular with two kinds of cells, the 4-point (regular tetrahedron) and the 12-point (truncated cuboctahedron), which have just one kind of regular face between them, a 5-cell face. That semi-regular 4-polytope is the 11-cell.

We said earlier that the 11-cell had 11 disjoint 5-cells and 11 hemi-icosahedral cells in it, and that its construction from them might be hard to see, but now we are going to account numerically, at least, for how it can have 6 pentad cells plus 5 hexad cells magically adding to 11 of the same kind of elevenad cell. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, 11. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (5-cell regular decahedron) cells, and the 12 6-point (hexad) building blocks into 100 12-point (regular icosahedron) cells, and then magically adding one whole 5-point (5-cell regular decahedron) cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 12-point (regular icosahedron) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such (cell bundles of 6 + 5 = 11 cells), with each 11-cell bundle containing 11 of both kinds of cell. Very, very strange!{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden'': §II To Add is to Multiply|p=38|ps=; the sequence of regular polygons with harmonious proportions beginning {5}, {7}, {9}, {11} as a family of natural numbers; &quot;The numbers in this optimal family have remarkable properties [that allow] similar figures to be arranged easily or allow a figure to be dissected into a set of similar figures. This is the origin of the Greek idea of geometric progression — that ''multiplication can be accomplished instead by addition''.... For instance, it is well known that the square of the golden ratio is one more than itself (written above as 𝜙&lt;sup&gt;2&lt;/sup&gt; = 𝜙 + 1) and one less than 𝜙 is its reciprocal (1/𝜙 = 𝜙 - 1). The first relation equates multiplication with addition, while the second accomplishes division through subtraction. [These are pentagon {5} bisectional golden ratios, but] the heptagonal [{7} tri-sectional] ratios also behave in surprising ways — adding to multiply, subtracting to divide.... There are analogous constructions for [nonagon {9}] quadrisectional and [hendecagon {11}] pentasectional systems.&quot;}}

The 120-cell has three interchangeable cyclic expansion-contraction cycles, the pentad, hexad and heptad cyclic designs, rooted in the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems, respectively. Each of them constructs the whole sequence of regular 4-polytopes from a single building block. The key to all three cycles is that 5 + 6 = 11, and there is an 11-point quasi-regular 4-polytope in the sequence of convex 4-polytopes. It lies between the 8-point (16-cell) and the 16-point (8-cell), and is itself a &lt;math&gt;B_4&lt;/math&gt; polytope. Its role is to be a unified single building block common to all three cyclic expansion-contraction cycles. It is involved with the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems and expresses in one building block their relations to each other.

We have already seen how the hexad building cycle works. It is plain compounding of 4-orthoplexes at the beginning, increasing to densely intricate non-disjoint symmetry toward the end, making it difficult to see why it works once past the 24-cell. This is how the build works, in terms of sixes. The 16-cell is a 2-hexad (3 pairs of completely orthogonal great square planes). The 24-cell is 3 16-cells and a 4-hexad, 4 different ways: 16 non-disjoint hexads (great hexagon planes). The 120-point (600-cell) is 25 24-cells and a 20-hexad, 10 different ways: 200 non-disjoint hexads (great hexagon planes). The 600-point (120-cell) is 225 (9 times 25) 24-cells. ....

The pentad building cycle doesn't do plain compounding to regular 4-polytopes until the last one, the 120-cell. Fives don't even appear as pentagons until after the 24-cell. We have seen how there are 11 5-cells in the quasi-regular 11-cell, which occurs before the 8-cell, and also how everything can be compounded from irregular 5-cells (4-orthoschemes specialized for each 4-polytope). It is still not evident how a pentad cycle can work to build all the regular 4-polytopes from the regular 4-simplex. This is how is how the build works, in terms of fives. ....

This is how the build works, in terms of elevens. The 11-cell is 4 orthogonal heptads (intersecting in their skew hexagon Petrie polygon). The 24-cell is also 4 heptads (intersecting in 4 orthogonal digon axes, so each heptad contributes 6 unique points to a single fibration of 24 points). Heptads are quarter elevenads, so the 11-cell and 24-cell are both 1 elevenad. ....

See the tabular sequence of convex 4-polytopes in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Build with the blocks|§Build with the blocks]]'', below: table rows ''Inscribed, Pentad, Hexad, Heptad''.

=== 11 of them ===
There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of eleven 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum eleven of them. There is only a single Hopf fibration of the 11 great circles of an 11-cell. 

The 5-cell and the 11-cell are limit cases. The 5-cells are completely disjoint, but the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 10 other instances of itself.

== Cell rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer &amp; Segerman|2013|loc=''Puzzling the 120-cell''}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his section of the Wikipedia article on the 120-cell entitled ''[[120-cell#Visualization|§Visualization]]''.{{Sfn|Ruen &amp; Goucher et al. eds. 120-cell|2024|loc=Goucher's ''[[120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious place to look for the 11-cell's Hopf map is in Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other place it could be hiding. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The quasi-regular convex 11-cell|§The quasi-regular convex 11-cell]]'', and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen: triacontagon|2011|loc=''Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them except his illustrations. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Four orthogonal projections of the 120-cell to a central plane, selected from the table of 3 regular and 11 compound 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|100 icosahedra&lt;br&gt;120 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|5 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 icosahedra&lt;br&gt;6 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|120
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|4
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|50
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|7.2°
|align=center|36°
|align=center|6°
|align=center|7.2°
|}

....

Here we summarize our findings on the 11-cells so far, including in advance some results to be obtained in the next section two sections.

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section....

== The 15 major chords ring everything ==
The natural numbers are each a distinct flavor.

Each integer has its own distinct taste. The first six integers {0..5} are the six strongest, most familiar tastes in nature, instantly distinguished and utterly unique.

The seventh flavor {6} is the first ratio of two of the others. Thereafter follows in strange sequence all the ratios of all the flavors, each marriage of every two flavors, most of which are fractions &lt;math&gt;\{k/d\}&lt;/math&gt;, but some of which are integers &lt;math&gt;\{k\}&lt;/math&gt;.

Some of these ratios are successful flavors in their own right, as distinct and inimitable as the marriage of chocolate and oranges, but the marriage of many flavor pairs is mud. It is a rule of thumb among flavor specialists that nearly any taste can be successfully alloyed with either onions or chocolate, but the direct marriage of these two flavors is notably unsuccessful. Some flavors make famous pairings or infamous ones, and subtle new pairings are always to be discovered that work surprisingly well for some purpose, but the majority of flavor marriages are undistinguished compounds.

In polytope geometry, each chord of a polytope is both is a distinct 1-dimensional object, a chord of the sphere of a distinct length &lt;math&gt;l&lt;/math&gt;, and a distinct natural number &lt;math&gt;h&lt;/math&gt;, a unique flavor. If the polytope is regular, it is a noteworthy and successful flavor. The chord's length &lt;math&gt;l&lt;/math&gt; is a square root, related to the natural number &lt;math&gt;h = k/d&lt;/math&gt; and to a set of polytopes they both represent, by a formula discovered by Steinbach.{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden''; Figure 5. Optimal sections and proportions|p=37|ps=; the regular polygons {5}, {7}, {9} and {11} with their diagonals define respectively: {5} the golden bisection proportional to 𝜙; {7} an analogous trisection; {9} an analogous quadrasection; {11} an analogous pentasection.}} The chord length &lt;math&gt;l&lt;/math&gt; is related to the number of sides of the regular polygon formed by the chord (its numerator &lt;math&gt;k&lt;/math&gt;), and to the winding number or density of its regular skew Petrie polygon (its denominator &lt;math&gt;d&lt;/math&gt;).

The regular 4-polytopes do not contain an infinite variety of these chord flavors, but only 30 of them: only 15 unique pairs of 180° degree complementary chords, which combine to make every regular ''n''-dimensional thing found in nature.{{Efn|The 15 ''major chords'' suffice to construct all the regular ''n''-polytopes. Their 180° complements include 15 ''minor chords'' which occur only in the 120-cell, not in any smaller regular polytope.{{Efn|name=Major and minor chords}} Including the complement of the 180° {{radic|4}} digon {2} chord (the {0} chord {{radic|0}}), and the skew variant of the 90° {{radic|2}} square {4} chord (the {30/7} chord), there are 32 distinct natural numbers which are all the chords of all the regular polytopes in all dimensions.|name=32 chords}} All the regular polytopes (including the largest one on the cover of the box) can be built not only from a box containing a number of blocks of 3 shapes glued together by gravity, they can be also be built from a box containing a number of sticks of 15 lengths glued together by rubber joints.

[[File:15 major chords.png|500px|thumb|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the others occur only in the interior of 120-cell, not as edges of any of the regular 4-polytopes. There are 30 distinct 4-space chordal distances between vertices within the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to identify any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=Major and minor chords}} chords are properly labelled #1 — #15 in the order they fan out by size.{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''; Figure 3|ps=; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with a &quot;fan of chords&quot; diagram like the one here.|p=23}} They join vertex pairs which are 1 — 15 edges apart on a &lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon. A #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; chord forms &lt;small&gt;&lt;math&gt;f+1&lt;/math&gt;&lt;/small&gt; regular geodesic polygons which include a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; disjoint regular {{nowrap|&lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt;-gons}} (e.g. &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; a compound of 5 disjoint hexagons) plus a single [[W:Triacontagon#Triacontagram|regular skew {{nowrap|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram}}]] that wraps around the sphere &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times before closing its circle. Notice the maverick #11 chord, with its non-conforming origin point for its {{nowrap|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;}} polygram.]]

These 15 chords (natural numbers) include the integers 2, 3, 4, 5, 6, 10, 11, 15 and 30. The others are the fractions 5/2, 10/3, 13/7, 15/2, 15/4, and 15/7. Represented among the chord lengths (the square roots) measured in ''unit radii'' (not in {{radic|2}} radii, as metrics are often given elsewhere in this article) are the {{background color|paleturquoise|roots of the first 4 positive integers {{radic|1}}, {{radic|2}}, {{radic|3}} and {{radic|4}}}}, the {{background color|#FFCCCC|root of the very distinguished rational fraction {{radic|5/2}}}}, and the roots of various distinguished irrational fractions as Steinbach discovered: for example, 4 of them are {{background color|yellow|roots of golden numbers (a small integer ± 𝜙)}}. The 15 chords each form their regular polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, which is just &lt;small&gt;&lt;math&gt;\{k\}&lt;/math&gt;&lt;/small&gt; in 8 cases. Arranged in order #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; from shortest edge-length square root to longest, the #1 — #15 chords bridge vertex pairs which are &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; vertices apart on a [[W:Triacontagon#Petrie polygons|&lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon]]. Each chord makes a [[W:Triacontagon#Triacontagram|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; polygram]] of its edge length, a compound of the smaller &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. The 15 distinct polygons form a sequence you would not anticipate exactly unless you have Steinbach's formula:

{| style=&quot;background-color:transparent; white-space:nowrap;&quot;
|-
| {30/1} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|.073}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{30} 
| 120-cell edges
|-
| {30/2}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.191}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15}
|- style=&quot;background: yellow;&quot;|
| {30/3}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.382}} 
| &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi}&lt;/math&gt;&lt;/small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{10}''' 
| 600-cell edges
|-
| {30/4} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.573}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/2}
|- style=&quot;background: paleturquoise;&quot;|
| {30/5}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|1}}'''
| &lt;math&gt;1&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{6}''' 
| 24-cell, 8-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/6}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|1.382}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{5}'''
|- style=&quot;background: paleturquoise;&quot;|
| {30/7}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|2}}'''
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{4}''' 
| 16-cell edges
|- style=&quot;background: #FFCCCC;&quot;|
| {30/8}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.5}} 
| &lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;&lt;small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/4} 
| 5-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/9}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.618}} 
| &lt;math&gt;\phi&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{10/3}
|- style=&quot;background: paleturquoise;&quot;|
| {30/10}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|3}}''' 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{3}'''
|-
| {30/11}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.426}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{11/11}
|- style=&quot;background: yellow;&quot;|
| {30/12}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.618}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{5/2}
|-
| {30/13}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.810}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{13/7}
|-
| {30/14}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.928}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/7}
|- style=&quot;background: paleturquoise;&quot;|
| {30/15}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|4}}''' 
| &lt;math&gt;2&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{2}'''
|}

{| class=wikitable style=&quot;white-space:nowrap; float:right;&quot;
!colspan=6|Natural number &lt;small&gt;&lt;math&gt;h=\{k/d\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Fibers&lt;br&gt;&lt;small&gt;&lt;math&gt;f\{h\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Cells&lt;br&gt;&lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|4-tope&lt;br&gt;&lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Verts&lt;br&gt;&lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|[[W:Triacontagon#Triacontagram|30-gram]]&lt;br&gt;&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;
|-
!Chord
!Arc
!colspan=2|&lt;small&gt;&lt;math&gt;l/\sqrt{1}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;l/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;h&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#1
|15.5~°
|{{radic|0.073}}
|0.270
|{{radic|0.146}}
|&lt;small&gt;&lt;math&gt;30&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1\{30\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;5\{3,3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/1\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#2
|25.2~°
|{{radic|0.191}}
|0.437
|{{radic|0.382}}
|&lt;small&gt;&lt;math&gt;15&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;..\{3,3\}..&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/2\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#3
|36°
|{{radic|0.382}}
|0.618
|{{radic|0.764}}
|&lt;small&gt;&lt;math&gt;10&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;30\{3,3\}20&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/3\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#4
|44.5~°
|{{radic|0.573}}
|0.757
|{{radic|1.146}}
|&lt;small&gt;&lt;math&gt;15/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/4\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5
|60°
|{{radic|1}}
|1
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;6&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/5\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#6
|72°
|{{radic|1.382}}
|1.175
|{{radic|2.624}}
|&lt;small&gt;&lt;math&gt;5&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}} 
|&lt;small&gt;&lt;math&gt;10\{5,3\}12&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/6\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#7
|90°
|{{radic|2}}
|1.414
|{{radic|4}}
|&lt;small&gt;&lt;math&gt;4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7\{4\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;4\{4,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;9\{3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/7\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: #FFCCCC;&quot;|
|#8
|104.5~°
|{{radic|2.5}}
|1.581
|{{radic|5}}
|&lt;small&gt;&lt;math&gt;15/4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/4\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}} 
|&lt;small&gt;&lt;math&gt;15\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/8\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#9
|108°
|{{radic|2.618}}
|1.618
|{{radic|5.236}}
|&lt;small&gt;&lt;math&gt;10/3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10/3\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/9\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#10
|120°
|{{radic|3}}
|1.732
|{{radic|6}}
|&lt;small&gt;&lt;math&gt;3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;10\{3\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/10\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#11
|135.5~°
|{{radic|3.426}}
|1.851~
|{{radic|6.852}}
|&lt;small&gt;&lt;math&gt;11/11&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;11\{11/11\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;11\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#12
|144°
|{{radic|3.618}}
|1.902
|{{radic|7.235}}
|&lt;small&gt;&lt;math&gt;5/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2},3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/12\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#13
|154.8~°
|{{radic|3.810}}
|1.952
|{{radic|7.621}}
|&lt;small&gt;&lt;math&gt;13/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;13\{13/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;7\{5,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/13\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#14
|164.5~°
|{{radic|3.928}}
|1.982
|{{radic|7.857}}
|&lt;small&gt;&lt;math&gt;15/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;137\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/14\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{radic|8}}
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15\{2\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐𝜙&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;8\{3,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/15\}&lt;/math&gt;&lt;/small&gt;
|}

The polygons '''{10}''', '''{6}''', '''{5}''', '''{4}''', '''{3}''', and '''{2}''' are flat polygons that each lie on a great circle in a central plane of the 3-sphere. The polygons {30}, {15}, {15/2}, {15/4}, {10/3}, {11/11}, {5/2}, {13/7}, {15/7} and all the {30/''n''} polygons are skew polygrams spiraling through curved 3-space, each successive edge in a different central plane of the 3-sphere. Projected to the 1-sphere (a circle in the plane), they are [[W:Star polygon#Regular star polygon|regular star polygrams]], the simplest being the pentagram {5/2}. In 4-space they wind around the 3-sphere more than once ''without self-intersecting'', before they close their circuit. They lie on helical geodesic circles or ''isoclines'' of circumference greater than 2𝝅''r''.

Even the 6 integer chords which form flat polygons also form skew polygrams. Each of the 15 chords occurs as a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, a fiber bundle of Clifford parallel polygons that totals 30 vertices per bundle. Each chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; also forms a single skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram that winds through another, shorter chord's bundle of parallel polygons, intersecting them in succession multiple times each, and weaving that whole bundle of parallel fibers together as the cross-thread does on a loom. The polygram cross-thread has to wind multiple times through the whole bundle (&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times around the [[W:Triacontagon#Triacontagram|30-gram]] in which each edge connects every &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt;th vertex) to complete an enumeration of all 30 vertices in the bundle. For example, the #5 chord forms hexagon &lt;small&gt;&lt;math&gt;\{6\}&lt;/math&gt;&lt;/small&gt; flat planes in parallel bundles of 5, written &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt;, and also a skew &lt;small&gt;&lt;math&gt;\{15/2\}&lt;/math&gt;&lt;/small&gt;-gram of #5 edges which winds 2 times through the #3 chord's bundle of 3 decagon &lt;small&gt;&lt;math&gt;\{10\}&lt;/math&gt;&lt;/small&gt; flat planes. Every chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; forms both its &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram and its fiber bundle of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, and the edges of one chord's parallel polygons are also the edges of its skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram weaving through some shorter chord's bundle of parallel polygons. Each numbered chord is a natural number, a ratio of two integers and their polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, either flat or skew. Each chord number defines a &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; skew polygon that is a compound bundle of the &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. All these flat or skew polygons are regular polygons made of their numerator instances of a single chord #, whose length is that square root, Steinbach's function of the natural number.

The relationship between each bundle of flat Clifford parallel great circle polygons of a distinct length, and its corresponding bundle of skew polygrams of a longer length, is precisely that of a discrete simple rotation and its corresponding discrete isoclinic rotation. For example, the flat great circle hexagons of edge length {{radic|1}} lie in the invariant rotation planes of the 24-cell's characteristic 60° isoclinic rotation, in which the vertices move on skew hexagram rotation paths of edge length {{radic|3}}. The 24-cell contains 4 bundles of 4 Clifford parallel (disjoint) hexagon central planes each. The 24 vertices of the 24-cell in each bundle are linked by rotation in the 4 hexagon invariant planes in two different ways: simply and isoclinically. In a simple rotation the vertices of a single invariant hexagon exchange places on that great circle hexagon of edge {{radic|1}}. In an isoclinic rotation in which any one of these 4 disjoint hexagon central planes is invariant, all 4 of them are necessarily invariant planes in the rotation, because they rotate in parallel. In that isoclinic rotation the vertices of all 4 hexagon invariant planes (all 24 vertices of the 24-cell) exchange places concurrently, moving along 4 disjoint helical hexagrams of edge {{radic|3}} in parallel. It is a characteristic only of this distinct rotation in hexagon invariant planes that the helical vertex path or ''isocline'' is a hexagram of period 6, edge length {{radic|3}}, and circumference 720°. Rotations in other kinds of polygon invariant planes have different characteristic isoclines. The period of their isoclinic rotation must be ''the same or greater'' than the period of their simple rotation; in most cases (unlike this one) it is greater.

The 15 chords form not only distinct fiber bundles of polygons, but also distinct bundles of polyhedra: cell rings &lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt; and compound vertex figures &lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;. These combine to form their characteristic regular 4-polytope &lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Coxeter|1973|p=269|loc=Compounds|ps=; &quot;It is remarkable that the vertices of {5, 3, 3} include the vertices of all the other fifteen regular polytopes in four dimensions.&quot;}} All these elements of a regular 4-polytope constructed from one chord are themselves regular polytopes, and each of the 16 regular 4-polytopes is a function of one characteristic chord. If the regular 4-polytope is convex, its characteristic chord is its unit-radius edge. For a non-convex regular 4-polytope, the characteristic chord is its unit-radius ''mid-edge diameter'', the distance from the midpoint of each edge to the midpoint of the completely orthogonal edge. Each of the characteristic chords (each distinct natural number) has its expression as one of the 16 regular 4-polytopes and all the products of the distinct regular 1-polytopes, 2-polytopes and 3-polytopes that construct it, and the whole distinct construction from the natural number to the 4-polytope is an [[W:SO(4)|SO(4) symmetry group]] expression of that 4-polytope's characteristic discrete [[W:SO(4)#Isoclinic rotations|isoclinic rotation]], in the invariant central planes of its characteristic chord, by the characteristic chord arc.

...demonstrate this theory by proper enumeration of the chord lengths in the table...

...color-code the △ ☐ 𝜙 ✩ symbols ... chord colors in diagram should match them ... some of the diagram colors are wrong and must be changed, but otherwise diagram is pretty good ... i think in a colored diagram all chords of the same length should have the same size line dash, the polygons are not obvious enough and the 15 fanning-out chords are readily identified by their labels...

... add back the counts of chords in the box (very small font number under each table entry), and present these two images as the front and back a box of building sticks and rubber joints...What's in the other box...

...say the rational-root major chords {{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|5/2}} are the ''n''-edges: their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge...

...abbreviated text of the [[120-cell#Relationships among interior polytopes|''§Relationships among interior polytopes'']] section from the 120-cell article...to explain the color-coding

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius=1}}

See also the elements, properties and metrics of the sequence of regular 4-polytopes summarized in tabular form in the section ''[[#Build with the blocks|§Build with the blocks]]'', below.

== Alicia Boole Stott's original formulation of dimensional analogy ==
&lt;blockquote&gt;&quot;...a method by which bodies having a certain kind of semi-regularity may be derived from regular bodies in an Euclidean space of any number of dimensions...&quot;{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''; presents cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues|pp=12-45}}&lt;/blockquote&gt;

Alica Boole Stott's 1910 paper reads like an exceptionally well-written patent application. It is a comprehensive stepwise description of her original invention, a general method of dimensional analogy between regular and semiregular polytopes of the same edge length.

[[File:The Roots of Three.png|thumb|An illustration of Boole Stott's expansion of unit-edge triangle to hexagon captures the geometric basis in 3 of the &lt;math&gt;A_n&lt;/math&gt;root system.]]

The engineering-like articulation of her new principles is entirely discrete and group-theoretic. Boole Stott repeatedly refers to sets of polytope elements as ''groups''. All her transformations are discrete instances of one or another special case of her &quot;two inverse operations which may be called ''expansion'' and ''contraction''.&quot; Hers is an exploration of ''n''-dimensional space at its most basic: space defined by the discrete objects which fill it, themselves defined directly as groups of elements with a few operations on them, which transform them into each other. This is geometry which seems to precede Euclid's postulates, and does not depend on them. Instead it corresponds directly to the deeper group theory mathematics which underlies geometry, even though Boole Stott, like the ancients, did not have that mathematics.

Boole Stott enumerated her expansion and contraction operations as transformations between regular and semi-regular polytopes of a single edge length. In her 1910 paper she did not consider any polytopes of multiple edge lengths, or expansions of the subject polytope by a distance other than that exactly required to produce new edges of the original edge length, so of course she did not enumerate Moxness's non-uniform rhombicosidodecahedron (Hull #8). This may be why section 8&lt;sub&gt;3&lt;/sub&gt; is missing from her earlier drawings of the sections of {5,3,3}; she may not have had a construction for it.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The Boole Stott expansion (or contraction) operation is as precisely defined a transformation as Coxeter's reflection, rotation and translation operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions). Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;
where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;
where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;
For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} An instance of the expansion (or contraction) operation is an application of the general operation to a particular subject polytope, operating on all of its elements symmetrically at once. The general operation is restricted, such that one may only expand (or contract) the subject polytope by pushing all its edges, faces or cells directly away from (or toward) its center, without changing the size of the moved elements. This splits each vertex into several (or coalesces several vertices into one), and has corresponding effects on the other elements, producing new edges ''of the same size'' between pairs of separated vertices (or reducing some edges to single vertices). The distance all the elements are moved away from (or toward) the center must be such that the vertices separate producing new edges of only the original edge length (or come together reducing edges to single vertices). It is permitted to combine these three operations, e.g. by pushing the edges ''and'' the faces, or by pushing all three element sets, provided the combination results in a semi-regular polytope of only one edge length. The three operations are all commutative with one another, so combined operations can be considered either as a sequence of expansions (or contractions) linking several related polytopes, or as their concurrent movements, a single transformation of the subject polytope. Since there are 8 combinations of one to three operations there are 8 polytope variants (including the original regular polytope), and all are adjacent to one another by a single transformation. The variants of the original regular polytope are named for which element set(s) are moved. By enumerating all 8 variants of expansion (and contraction) for each of the six regular convex 4-polytopes, Boole Stott produced a table of regular and semi-regular 4-polytopes divided into 12 sets of 8: six expansion sets and six contraction sets.{{Sfn|Boole Stott|1910|loc=''Table of Polytopes in S&lt;sub&gt;4&lt;/sub&gt;''; note that Boole Stott's usage of ''S&lt;sub&gt;4&lt;/sub&gt;'' denotes a Euclidean 4-space ''E&lt;sub&gt;4&lt;/sub&gt;'', or a 3-sphere space embedded in it, ''not'' a 4-sphere space embedded in a Euclidean 5-space|p=30}} Each set includes just one of the six regular convex 4-polytopes, with its distinct unit-radius edge length, and the 7 possible semi-regular expansions (or contractions) of it, which all preserve that characteristic edge length. The polytopes in each set of 8 all have the same total number of elements as their original regular polytope, but each polytope in the set has a distinct combination of cell, face, and vertex figures. The 12 sets are not completely disjoint, and even within one set the 8 operations do not necessarily produce 8 unique polytopes. For example, the regular 5-cell (the 4-simplex polytope) has 5 cells, 10 faces, 10 edges, and 5 vertices (30 elements). Expanding the 5-cell produces 7 related semi-regular 4-polytopes of 30 elements each (two of which happen to be the same polytope). In each expansion, the 5 cells have expanded into semi-regular or regular polyhedra, the 10 faces and 10 edges ''may'' have expanded into prisms, and the 5 original vertices have expanded into regular or semi-regular cells.

== The regular 11-cell 4-polytope ==

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosadodecahedron). Each non-convex Jessen's icosahedron is a set of 6 reflex edges that are {{radic|5}} 5-cell edges, each set symmetrically arranged in 3 parallel pairs, not as far apart ({{radic|3}}) as they are long, in 3 orthogonal central planes. The 12-point (Jessen's sets of 6 reflex 5-cell edges) are inscribed in the 60-point with their endpoints as a vertex of a pentagon face. The 12 pentagon faces are joined to 5 other pentagon faces by the 5-cell edges; not to their nearest neighbor pentagons but to their next-to-nearest neighbor pentagons; and since there are 5 vertices to each pentagon face, there are 10 12-point (Jessen's sets of 6 reflex 5-cell edges). Each pentagon face is joined to another pentagon face by 5 reflex edges of five distinct Jessen's. The 5 reflex edges are Clifford parallel, the 5 edges of a pentagonal prism column that has a twist to it. The contraction of those 5 edges into 1, and those 10 vertices (two pentagons) into 2, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing all their edges as a single Jessen's. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (octahedron) by conflating its parallel pairs of edges, leaving the 11-cell's abstract 6-point (hexad cell). The hexad's situation is that it occurs in bundles of 5, disjoint but close together in the sense of adjacent, like the great circle rings in a Hopf fiber bundle. 

To summarize, the 12-point (2-hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell). We shall now be able to see how Fuller's perfect hexads combine with Todd's perfect pentads to form the perfect heptads of the 11-cell and 137-cell.  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=alias the hemi-cuboctahedron}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces. (The ''heptad'' arrives at the party, anonymously; they are our distinguished anti-celebrity, who most of the other guests have never even heard of.) The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. This 12-point (truncated tetrahedron) is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 100 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp: Truncated tetrahedron|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp: Truncated tetrahedron|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}} Of course Moxness has been there, seen that, already with his graphics software;{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} see his parameterized illustration of it exactly as it occurs in the 11-cell, below in his bestiary of A3, B3, and H3 Archimedian and Catalan polyhedra in the 120-cell.]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:Archimedean_and_Catalan_solid_hulls_with_their_Weyl_orbit_definitions.svg|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of the 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. In this catalog of Archimedean and Catalan polyhedra sighted in the 120-cell by Moxness's software,{{Sfn|Moxness: Archimedean and Catalan hulls|2023|loc=Hull #1 Archimedean Name A3 110 Truncated Tetrahedron A (upper left)}} it is the first entry (upper left), Moxness's Hull # = 1 with 12 vertices of 3D norm {{radic|11}}/2. On a {{radic|2}}-radius 3-sphere, 5 of these with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell. 20 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges are cells in the [[#The rings of the 11-cells|cell rings of the quasi-regular 11-cell]], which also has 12 regular 5-cells as cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

...

Both kinds of cells in the 11-cell have expressions in two different dimensions (on the 2-sphere and 3-sphere), because they are all centered on the center of the 3-sphere, unlike the cells of most 4-polytopes, which are centered on various off-center points within the 4-polytope's interior. 

One kind of 11-cell cell appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (semi-regular truncated cuboctahedron), a.k.a. the truncated tetrahedron. The other kind of cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now, except in the 120-cell. But this is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron, and a regular tetrahedron behind each face. The 11-cell is thus a honeycomb of mixed regular tetrahedra and truncated cuboctahedra (a.k.a. truncated tetrahedra) that meet at their shared 5-cell faces. 

Both the 5-cells (decahedra) and the truncated cuboctahedra (truncated tetrahedra) are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks in their 4-polytopes (off-center).{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} Therefore all the 11-cell's cells are concentric, which indeed makes a different kind of honeycomb, an inside-out cellular honeycomb with all the cells piled on top of each other overlapping in 4-space. They do not overlap in 3-sphere space however; their 3-volumes (but not their 4-contents) are disjoint. They make a proper 4D honeycomb like the cells of any 4-polytope, with volumetrically disjoint 3D cells that meet each other in the proper way at their shared 2D faces.

An 11-cell contains 6 completely disjoint 5-point (1-pentad 5-cell decahedron cells), and 5 volumetrically disjoint 12-point (2-hexad truncated cuboctahedron cells), 11 volumetrically disjoint cells altogether. Of course it is supposed to have 11 identical 11-point (elevenad hemi-icosahedral cells), and that is also true in its abstract sense, which is not the same sense as disjoint cellular containment: each pentad or hexad cell is inscribed in one of 11 larger 60-point (Moxness's rhombicosadodecahedron cells). The abstract pentads and hexads are only hemi-polyhedra, so the 11-cell's pentad cells are 12 (out of 120) disjoint instances of a pentad 5-point, and its hexad cells are 10 (out of 100) disjoint instances of a hexad 6-point, 1/10th of the 120-cell's cellular volume in both cases. The 11-cell's hemi-icosahedral cells are 11 (out of 120) non-disjoint instances of an elevenad 60-point. Each 11-cell shares each of its hexad cells with 9 other 11-cells, and each of its 11 vertices (out of 600) occurs in 9 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.|name=Tetrahedral vertex figures}}

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

...insert image thumb of 11-cell...

...Construct the 11-cell from the heptad....

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...Four concentric 4-polytopes may be mutually orthogonal. In any set of 4 mutually orthogonal regular 5-cells, one pair of faces in each orthogonal pair are completely orthogonal to each other. Thus three edges in one 5-cell in an orthogonal pair are opposite (completely orthogonal) to a face of the orthogonal 5-cell. .... A 5-cell has 5 vertices in its abstract hemi-icosahedral cells, but only 4 vertices in one concrete icosahedral cell; the 5th lies in an icosahedral cell of another 11-cell (not in another icosahedral cell of the same 11-cell).

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

....

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 11-cell ===

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon.... (wrong, 11-cell's Hopf map is the truncated icosahedron)

....

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk: Truncated icosahedron|2018}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

...

Since the rhombicosadodecahedron has 120 5-cell edges, the 120-cell has 120 completely disjoint 5-cells, and the rhombicosadodecahedron shares each vertex with 11 other rhombicosadodecahedra, how a rhombicosadodecahedron is bound to its neighbor 5-cells is constrained. A further constraint is that the abstract hemi-icosahedron has only 15 5-cell edges, so the 120 actual 5-cell edges of the rhombicosadodecahedron are conflated 8-to-1 in the hemi-icosahedron. To be conflated by contraction into a single edge, those 8 edges must lie parallel to each other in the 120-cell; either actually parallel in the same plane, or Clifford parallel in separate planes, as two edges in completely orthogonal planes may be parallel and orthogonal at the same time. Each 5-cell inscribed in the 120-cell contains 5 such tetrahedra occur in parallel pairs in 4 orthogonal planes (none of them completely orthogonal planes). 

.... 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
Each conflated vertex is the 10 vertices of two orthogonal regular 5-cells...
The 60-point contains 6 pairs of these orthogonal regular 5-cell cells...
...its 20 hexad cells are ..., a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere.

== The regular 137-cell 4-polytope ==

(description originally written for the 11-cell, which applies only to a fibration of 11 11-cells)

It is quasi-regular with two kinds of cells, the 12-point (regular icosahedron) and the 5-point (regular 5-cell), which have just one kind of regular face between them. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of regular facets (it is semi-regular with regular facets), it also means that its facets have expressions of two different dimensionalities because they are concentric in the 3-sphere. The 11-cell has two kinds of regular cells. One appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (regular icosahedron), but since it is embedded at the center of the 3-sphere (unlike the cells of most 4-polytopes) it also acts as an inscribed 4-polytope.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} The other cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now except in the 120-cell. This is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron. The 11-cell is a honeycomb of mixed icosahedron and half-icosahedron cells. Like the icosahedra, the decahedra are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks, off-center in their 4-polytopes. All the 11-cell's cells are concentric, which indeed makes an unusual honeycomb of cells, an inside-out honeycomb. It is a proper 4D honeycomb however, with volumetrically disjoint 3D cells that meet each other in the proper way at shared 2D faces. The embedded 12-point (regular icosahedron) cell is an icosahedral pyramid (as a 4-polytope), which is filled by 20 4-point (irregular tetrahedra) that meet at its center; the center point of the 11-cell acts as a 12th vertex, just as the center point of the 24-cell acts as a 25th vertex in some honeycombs. The 5-point (regular 5-cell) cell contains five 4-point (regular tetrahedra) which meet the 12-point (regular icosahedra)'s irregular tetrahedral cells face-to-face, so the 11-cell is quasi-regular in the usual sense of having two varieties of regular 3-polytope cells, but additionally both varieties of of its cells are bi-dimensional: another clue that the 11-cell has something to do with dimensional analogy

....

The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

....

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur to make an 11-cell. 11 of them occur in a 137-cell, all 120 of them occur in the 120-cell, and they occur nowhere else.

How can this regular 4-polytope possibly exist? By Schläfli's criterion {3, 3, ..} exists, but not as a finite polytope.{{Sfn|Coxeter|1973|loc=§7.7 Schläfli's criterion}} Coxeter found {3, 5, 3} as a finite hyperbolic honeycomb, but not in Euclidean 4-space.{{Sfn|Coxeter|1968|title=The Beauty of Geometry: Twelve Geometric Essays|loc=''Regular Honeycombs in Hyperbolic Space''|ps=; &quot;The honeycombs considered by the above authors [Schlegel, Schläfli, Klein, Sommerville] have finite cells and finite vertex figures. ... A further extension allows a cell or vertex figure to be a ''star''-polytope, so that the honeycomb covers the space several times. .... We shall find that there are four regular star-honeycombs in hyperbolic 4-space, as well as two infinite families of them in the hyperbolic plane.&quot;}} We find that {3, 3, ..} is a central expression of the &lt;math&gt;H_4&lt;/math&gt; root system, being the regular convex hull in 4 dimensions of 137 points, 96 pentads and 600 hexads. The minimal expression of &lt;math&gt;H_4&lt;/math&gt; is the regular convex hull {3, 3, 5} of 120 points, 600 tetrads and 75 hexads, and its maximal expression is the regular convex hull {5, 3, 3} of 600 points, 120 pentads and 675 hexads. {3, 3, ..} is a third regular &lt;math&gt;H_4&lt;/math&gt; polytope which fits between them. The properties and metrics of this regular sequence are presented in tabular form below in ''[[#Build with the blocks|§Build with the blocks]]''.

{|class=&quot;wikitable&quot;
!colspan=3|Regular &lt;math&gt;H_4&lt;/math&gt; polytopes
|-
|[[File:600-cell.gif|200px]]
|[[File:5-cell.gif|200px]]
|[[File:120-cell.gif|200px]]
|-
!120-point 600-cell
!137-point 137-cell
!600-point 120-cell
|}

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 137-cell ===
....

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=Apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|2022|loc=''[[Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]''}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie: On Fuller's use of language|2024|loc=''[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry'']]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center, 1948-1949|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with their mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the gender neutral shad can open their six beaks all the way, until they become flat squares and they becomes a cuboctahedron, or they can shut them all tight like a turtle retracting into their octahedron shell. The six mouths always move in unison. This is [[Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do their ''really'' odd trick -- where they flip their 6 jaws 90 degrees in their 6 faces and shut their 6 beaks on the opposite axis of their squares than the one they opened them on -- or whether they will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than their normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make their edge length exactly the same size as their radius, when they open their mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing their mouths in spherical synchrony, their 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron (the hexad), then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=''[[W:SO(4)|SO(4)]]''}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=Isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its orthogonal hexad's planes (wz, wx, wy).{{Efn|name=Six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of (expanded octahederon 16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh &amp; Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== Legendre's 12-point hexad binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a 12-point double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's perfectly realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, this 2-hexad expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre hexad) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction.{{Efn|name=Concentric polyhedra}} In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== ''n''-simplexes ===
....

The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== ''n''-orthoplexes ===
....
[[File:Great 5-cell √5 digons rectangle.png|thumb|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} ''regular'' great hexagons. These are the same 200 dodecagon central planes illustrated above, which also contain 6 120-cell edges and two opposing ''irregular'' great hexagons (truncated {{radic|6}} triangles). The 1200 {{radic|5}} 5-cell face triangles occupy 1200 other, non-central planes.]]

...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's Hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[16--cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[16--cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=Concentric polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[16--cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== ''n''-taliesins ===
The 4-simplex pentad building block combines with the 4-orthoplex hexad building block to make the 4-taliesin heptad building block. The heptad is the essential building block of the ''taliesin'' family of polytopes. The 11-cell and 57-cell hemi-polytopes and the 137-cell and 120-cell regular 4-polytopes are nuclear members of this trans-dimensional family. The 24-cell and 600-cell are extended members of it, because they also contain instances of the heptad building block.

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its spherical symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* [[W:Taliesin Myrddin Namkai-Meche|Taliesin Myrddin Namkai-Meche]], a young Reed College scholar who was stabbed to death by a crazy man in 2017 on a Portland, Oregon light rail train, when he rose to the nonviolent defense of two black teenagers who were being violently assaulted in racist and anti-Muslim language by a right-wing extremist.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in orthogonal [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].
* '''Taliesan polytope''', an (''n+1)''-polytope whose ''n''-polytope dimensional analogue inscribed concentrically in the ''n''-sphere forms a convex honeycomb of itself which is the (''n+1)''-polytope.{{Efn|name=Embedding point}}
* The taliesan polytope's honeycomb, a Hopf fibration of its great circles.
* The ''n''-dimensional family of dimensionally analogous polytopes related to a taliesan polytope by expansion and contraction operations.
* The characteristic [[w:Orthoscheme|orthoscheme]] of a teliesan polytope, an irregular simplex that is the complete gene for the entire expression of trans-dimensional symmetry relationships among the polytopes of the ''n''-dimensional family, as well as the complete gene for the teliesan polytope's rotational symmetry.{{Efn|name=Characteristic orthoscheme}}

=== ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== ''n''-equilaterals ===
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed at four different [[#n-leviathon|chord]] lengths from each other. In the {{radic|2}}-radius polytopes we have using as examples in this paper they are {{radic|2}}, {{radic|4}}, {{radic|6}} and {{radic|8}}, but in a ''unit-radius'' polytope they are: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}.

Each vertex is joined to 8 others by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}. The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== ''n''-pentagonals ===

[[File:600-cell.gif|thumb|The 120-point (600-cell), &lt;math&gt;H^4&lt;/math&gt;.]]
...

[[File:600-cell vertex geometry.png|thumb|Vertex geometry of the 600-cell, showing the 5 regular great circle polygons and the 8 vertex-to-vertex chord lengths with angles of arc. The golden ratio{{Efn|name=golden chords|group=}} governs the fractional roots of every other chord,{{Efn|name=fractional root chords}} and the radial golden triangles which meet at the center.|alt=|400x400px]]
The 120 vertices of the 600-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏&lt;sup&gt;−1&lt;/sup&gt;) beginning with a vertex; see column ''a''}} at eight different [[#n-leviathon|chord]] lengths from each other. These edges and chords of the 600-cell are simply the edges and chords of its five great circle polygons. In ascending order of length in a ''unit-radius'' polytope (not in a {{radic|2}} radius polytopes like our usual examples), they are {{radic|0.𝚫}}, {{radic|1}}, {{radic|1.𝚫}}, {{radic|2}}, {{radic|2.𝚽}}, {{radic|3}}, {{radic|3.𝚽}}, and {{radic|4}}.{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=Decimal fractions of square roots|group=}}

Notice that the four [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#''n-''equilaterals|hypercubic chords]] of the 24-cell ({{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|4}}) alternate with the four new chords of the 600-cell's additional great circles, the decagons and pentagons. The new ''golden chord'' lengths are necessarily square roots of fractions, but very special fractions related to the golden ratio{{Efn|1=The fractional-root ''golden chords'' are irrational fractions that are functions of {{radic|5}}. They exemplify that the [[W:golden ratio|golden ratio]] &lt;big&gt;𝜙&lt;/big&gt; {{=}} {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618 is a circle ratio related to &lt;big&gt;𝜋&lt;/big&gt;:&lt;br&gt;
: {{sfrac|𝜋|5}} = arccos (𝜙/2)
is one decagon edge, the 𝚽 = {{radic|0.𝚫}} = {{radic|0.382~}} ≈ 0.618 chord.
Reciprocally, in this function discovered by Robert Everest expressing &lt;big&gt;𝜙&lt;/big&gt; as a function of &lt;big&gt;𝜋&lt;/big&gt; and the numbers 1, 2, 3 and 5 of the Fibonacci series:&lt;br&gt;
: &lt;big&gt;𝜙&lt;/big&gt; = 1 – 2 cos ({{sfrac|3𝜋|5}})
{{sfrac|3𝜋|5}} is the arc length of the &lt;big&gt;𝜙&lt;/big&gt; = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618 chord.|name=golden chords|group=}} including the two [[W:golden section|golden section]]s of {{radic|5}}, as shown in the diagram.{{Efn|The 600-cell edges are decagon edges of length {{radic|0.𝚫}}, which is 𝚽, the ''smaller'' golden section of {{radic|5}}; the edges are in the inverse [[W:golden ratio|golden ratio]] {{sfrac|1|φ}} to the {{radic|1}} hexagon chords (the 24-cell edges).
The other fractional-root chords exhibit golden relationships as well. The chord of length {{radic|1.𝚫}} is a pentagon edge.
The next fractional-root chord is a decagon diagonal of length {{radic|2.𝚽}} which is &lt;big&gt;𝜙&lt;/big&gt;, the ''larger'' golden section of {{radic|5}}; it is in the golden ratio{{Efn|name=golden chords|group=}} to the {{radic|1}} chord (and the radius).{{Efn|Notice in the diagram how the &lt;big&gt;𝜙&lt;/big&gt; chord (the ''larger'' golden section) sums with the adjacent 𝚽 edge (the ''smaller'' golden section) to {{radic|5}}, as if together they were a {{radic|5}} chord bent to fit inside the {{radic|4}} diameter.}}
The last fractional-root chord is the pentagon diagonal of length {{radic|3.𝚽}}.
The [[W:Pentagon#Side length is given|diagonal of a regular pentagon]] is always in the golden ratio to its edge, and indeed &lt;big&gt;𝜙&lt;/big&gt;{{radic|1.𝚫}} is {{radic|3.𝚽}}.|name=fractional root chords|group=}}

....


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== ''n''-leviathon ===
...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of its &lt;math&gt;10^2&lt;/math&gt; disjoint instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction, before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]].{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}} .... It is only because she did not have the 11-cell to start with that she did not discover the regular 137-cell {3, 3, ..} by her method.

Between Alicia Boole Stott and her identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and her identification of symmetry groups with conservation laws (in her 1st theorem), we owe the deepest mathematics of geometry and physics originally to two pioneering women, colleagues of their contemporaries the greatest men of the academy, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just an [[W:abstract polytope|abstract 4-polytope]], not just a singleton convex 4-polytope, and not just a honeycomb.{{Sfn|Coxeter|1970|loc=''Twisted Honeycombs''}} The 11-point (11-cell) has a concrete ...regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}&lt;nowiki/&gt;} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, 3, 3} as the 137-point (137-cell).

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 11-cell, 137-cell and 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=''Clifford Spinors and Root System Induction: H4 and the Grand Antiprism''}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=Apology}}

== Build with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Duveneck|1978|loc=Carl Jung, quoted in ''Life on Two Levels''|p=ii|ps=.{{Sfn|Jung|1961|ps=: &quot;The best of truths is of no use unless it has become one's most personal inner experience. It is the duty of everyone who takes a solitary path to share with society what he finds on his journey of discovery.&quot;}}}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1954|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius={{radic|2}}}}

== Acknowledgements ==
...

== Notes ==
{{Regular convex 4-polytopes Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Citation|last=Boole Stott|first=Alicia|author-link=W:Alicia Boole Stott|title=Geometrical deduction of semiregular from regular polytopes and space fillings|date=1910|pages=12-45|publisher=Johannes Muller|place=Amsterdam|url=https://dwc.knaw.nl/DL/publications/PU00011492.pdf}}
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      <text bytes="208360" sha1="tt39v866v1lqopciq5lbqw9wcwccd0o" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|June 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a regular 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the &lt;s&gt;120-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 5, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 11 [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] cells. Eleven 11-cells (plural) are the 137-cell, the &lt;s&gt;600-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 3, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 137 [[W:Rhombic triacontahedron|triacontahedron]] cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=''Regularity of Graphs, Complexes and Designs''}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=''A Symmetrical Arrangement of Eleven Hemi-Icosahedra''}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]]
The 4-dimensional regular polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:Simplex|simplex]] (called the [[5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:Orthoplex|orthoplex]] (called the [[16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular convex 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the [[w:Tesseract|16-point (8-cell) tesseract]], the 4-cubic, which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found within any of the other regular convex 4-polytopes, except in the largest and most complex one, the [[120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). However, we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the 16-cell is their very starting point, and the most frequently used tool in the box.

== 5-cells and hemi-icosahedra in the 11-cell ==
[[File:Symmetrical_5-set_Venn_diagram.svg|thumb|The 5-point (10-face) regular 5-cell (the regular 4-simplex). Grünbaum's rotationally symmetrical 5-set Venn diagram{{Sfn|Grünbaum|1975|loc=''Rotationally symmetrical 5-set Venn diagram'', Fig 1 (e)|ps=; partitions the individual elements of the 5-cell.}} is an illustration of the 5-cell labeling each of its &lt;math&gt;2^5&lt;/math&gt; elements.{{Sfn|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled; image includes the Python code to render it, optimising for maximum area of the smallest regions.}}]]
[[File:Hemi-icosahedron.png|thumb|The 6-point (10 face) [[W:hemi-icosahedron|hemi-icosahedron]], an abstraction of the regular icosahedron, has half as many faces, edges and vertices. Each element of the abstract polyhedron represents two real elements found in different places in the 11-cell's realization.{{Sfn|Ruen: hemi-icosahedron|2007}}]]
The most apparent relationship between the pentad 5-cell and the hexad hemi-icosahedron is that they both have 10 triangular faces. When we see a facet congruence between a 4-polytope and a 3-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 6-point 5-simplex. It is also the same size as the 6-point 3-orthoplex; thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no chord congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals, pentad and hexad, who are famously stand-offish with each other. Interesting!

The cell of the 11-cell is an abstract hexad hemi-icosahedron containing pentads in its interior, most handsomely illustrated by Séquin.{{Sfn|Séquin &amp; Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing rings of polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The 11-cell and 5-cell of radius {{radic|2}} both have edge length {{radic|5}}, so their triangular faces are congruent. The 11-cell's 11 hemi-icosahedral cells have 10 triangle faces each, and a 5-cell is behind each face. Only 3 of the 5-cell's 5 vertices are visible on the surface of the hemi-icosahedron as the exposed face, but its other two vertices are visible on the opposite side of the hemi-icosahedron as an exposed edge.{{Sfn|Coxeter|1984|p=110|loc=§6. The Petrie polygon [of the 11-cell]|ps=; &quot;We may reasonably call this edge and face ''opposites''. It is easy to find the face opposite to a given edge by looking at the faces to which a given edge belongs. ... Conversely, given a face, we can find the opposite edge by seeing which vertices belong to neither of the hemi-icosahedra which share that face. The ten edges opposite to the ten faces of one hemi-icosahedron are the edges of the complementary &lt;math&gt;a_4&lt;/math&gt; [4-simplex], that is, the joins of all pairs of the five vertices [of the 11-cell] not belonging to the given hemi-icosahedron.&quot;}} The other parts of the 5-cell (its other 6 edges, its other 9 faces and its 5 tetrahedral cells) lie outside the hyperplane occupied by the hemi-icosahedron.

There are 11 disjoint 5-cells in each 11-cell, which is comprised of 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. The real 11-cell is harder to see than the abstract 11-cell that represents it, because the real hemi-icosahedron that the abstract hemi-icosahedron represents is hard to see, but seeing these objects will be easier once we have identified the real hemi-icosahedron and exactly where the 11-cell's real elements reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each hemi-icosahedron's 10 faces come from 10 disjoint 5-cells, and there is just one 5-cell with which it does not share a face.{{Efn|As Coxeter observes (in the previous citation), that unrepresented 5-point (5-cell) is the other 5 vertices of the 11-point (11-cell) that are not vertices of this 6-point (hemi-icosahedron), the hemi-icosahedron's disjoint complement.}} Each 5-cell has 10 faces from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share a face.

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in two different planes which are completely orthogonal to each other. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must be sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness: 8 concentric hulls|2022|loc=Hull #8 (lower right)|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell of the 11-cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Compounds in the 120-cell|§Compounds in the 120-cell]]'' that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software which uses Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s to render the polyhedra which are found in the interior of ''n''-dimensional polytopes.{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the tetrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is a real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (Hull #8) is a concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this {12} irregular great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing {6} irregular great hexagons (truncated triangles) of alternating 120-cell edges and 5-cell edges. The great dodecagon is a great circle of the hemi-icosahedral cell, and also a great circle of the 120-cell, because unlike the cells of most 4-polytopes, the 11-cell's cells are embedded in 4-space at the exact center of the 4-polytope, making them inscribed 4-polytopes themselves.{{Efn|name=Embedding point}} There are 25 great dodecagons and 120 5-cell edges in Moxness's 60-point (Hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (Hull #8) polyhedron altogether.

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point (Hull #8) is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] similar to the one from the catalog shown here,{{Sfn|Piesk: Rhombicosidodecahedron|2018}} but a slightly shallower truncation with smaller red pentagons (120-cell faces) and narrower rhombs. Rhombicosidodecahedra are made by truncating the [[W:Rhombic triacontahedron|rhombic triacontahedron]], which is the 30-sided polyhedron with only one kind of face, the dual of the 30-point icosidodecahedron. The 120-cell contains 120 of these 11-cell cells.]]
Moxness's 60-point (Hull #8) is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (Hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|The 5-point (regular 5-cell) can be another abstraction of Moxness's 60-point (Hull #8), 12-vertices-into-1, but it is not strictly the result of a contraction operation as described by Boole Stott, since she did not apply her expansion and contraction operations to uniform polytopes with more than one edge length.}} In fact Moxness was not the first to find a rhombicosidodecahedron in the 120-cell; Alicia Boole Stott was. It is the 6th section of {5,3,3} beginning with a cell, which Boole Stott identified as the ''e&lt;sub&gt;2&lt;/sub&gt; expansion'' of the icosahedron (or equivalently of its dual polyhedron the dodecahedron).{{Sfn|Boole Stott|1910|loc=§Examples of the e&lt;sub&gt;2&lt;/sub&gt; expansion|p=7}} But that rhombicosidodecahedron identified by Boole Stott is not Moxness's Hull #8: rather, it is the semi-regular Archimedean solid, with a single edge length and square faces. Moxness's Hull #8 with its two distinct edge lengths and rectangular faces is Coxeter's 8th section of {5,3,3} beginning with a cell, which is missing from the sections illustrated by Boole Stott.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than tetrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. The 120-cell contains 120 of Moxness's 60-point (11-cell cells). They are non-disjoint, with 12 60-points sharing each vertex.

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (Hull #8) is the real hemi-icosahedron cell, with 120 5-cell edges in it as chords lurking under its surface. The rhombicosadodecahedron has enough of those 5-cell edges in its interior to make 20 disjoint tetrahedral cells which belong to 20 distinct 5-cells.

The 600-point (120-cell) may be constructed as a semi-regular honeycomb of two kinds of cells, the 120 non-disjoint 60-point (Moxness's rhombicosadodecahedra) and their neighbors the 120 disjoint 5-point (5-cells), cell-bonded together by shared tetrahedral cells of edge-length {{radic|5}}. Each rhombicosadodecahedron is bonded to 20 islanded 5-cells, sharing with each a 5-cell tetrahedron (4 of its interior 5-cell faces, and 6 of its interior 5-cell edges). The rhombicosadodecahedron is also bonded directly to 10 other rhombicosadodecahedra (since each 11-cell has 11 rhombicosadodecahedron cells), again not by sharing one of its exterior faces, but this time by sharing an interior face, a {{radic|6}} triangle which lies in a central plane.{{Efn|The {{radic|6}} chord is another hidden chord lurking beneath the 60-point rhombicosadodecahedron's surface, like the {{radic|5}} 5-cell edge. Both these hidden chords lie on the same great circle with the 120-cell edges (the pentagon face edges). Three {{radic|6}} chords make a triangle which lies in a central plane. This {{radic|6}} great triangle must not be confused with the {{radic|5}} face triangle of the 5-cell, which does not lie in a central plane.}} The 120 rhombicosadodecahedra are also bonded directly together at ''all'' their exterior faces: at their pentagon faces (which are 120-cell faces of dodecahedral cells); at their small triangular faces (which are 600-cell faces of small tetrahedral cells, 10 of which are inscribed in each dodecahedral cell); and at their rectangular faces (which are faces of 10-point [[w:Pentagonal_prism|pentagonal prism]] cells, where the pentagonal prism is a column inscribed in a dodecahedral cell, joining its opposite pentagon faces). The 120 rhombicosadodecahedra are bonded directly to each other at their own exterior faces; no dodecahedra, small tetrahedra, or pentagonal prisms lie between them, and those objects are not cells of this honeycomb; we mention them here only to help locate the rhombicosadodecahedra within the 120-cell. The only objects which fill holes between the rhombicosadodecahedra are the 120 islanded 5-cells.{{Efn|This honeycomb of the 120-cell resembles the [[600-cell#Icosahedra|honeycomb of the 600-cell consisting of 24 face-bonded regular icosahedra]] with 24 clusters of 5 tetrahedra filling the holes between them. In the 120-cell there are 5 times as many of everything as in the 600-cell, so there are 120 (rhombicosadodecahedron) icosahedral cells with 600 tetrahedra filling the holes between them.}}

It is possible for rhombicosadodecahedra to meet each other at more than one kind of exterior face because they are not meeting there as cells of the same 11-cell; they belong to different 11-cells. The 11 rhombicosadodecahedra in the same 11-cell do not meet at their own exterior faces, they meet at their interior {{radic|6}} triangle faces. As a cellular 4-polytope, the 11-cell is inside-out. Its cells meet each other at interior planes, sharing interior {{radic|6}} triangle faces, instead of in the usual way that 4-polytopes' cells share an exterior face. It is a deeper kind of intertwining to be sure, but the 11 rhombicosadodecahedra in the same 11-cell are still volumetrically disjoint cells, and the 11-cell is a perfectly real 4-polytope with real cells. It is a semi-regular 4-polytope, because it has two different kinds of cells: 11 rhombicosadodecahedra and 11 5-cells. The 11 5-cells are islanded (not bound to each other), and the 11 rhombicosadodecahedra are bound both to 5-cells (at {{radic|5}} 5-cell faces) and to each other (at {{radic|6}} central triangle faces). In the section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#What's in the box|§What's in the box]]'', below, we will look closely at the relation between the {{radic|5}} chord and the {{radic|6}} chord, which is the relation between the pentad and the hexad, and at the general phenomenon of hemi-polytopes which meet each other at a central plane. Then in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Cell rings of the 11-cells|§Cell rings of the 11-cells]]'' we will look at the two-cell cellular structure of the 11-cell in more detail.

Considered as a single complex of overlapping cells rather than as distinct honeycombs of volumetrically disjoint cells, the 120-cell contains 60-point (rhombicosadodecahedral cells), islanded 5-point (5-cells), 20-point (dodecahedral cells), and 10-point ([[w:Pentagonal_prism|pentagonal prism]] cells). The 60-points meet each other in pairs at their pentagon faces, meet regular dodecahedra at those pentagon faces also, meet each other in quads at their 600-cell faces where they share an interior 4-point (600-cell tetrahedral cell) and its four faces (just as they share a larger interior 5-cell tetrahedron and its faces), and meet each other and the 10-point (pentagonal prisms) at their rectangle faces. We might try to imagine such a complex with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with parts truncated from 30 neighboring truncated dodecahedra pentagonal prisms). There are 120 of each of these 3 kinds of dodecahedron. 

Moxness's 60-point (rhombicosadodecahedron cell) is a central object of the 120-cell, and contains many wonders. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra by expansion and contraction operations, as Alicia Boole Stott discovered before 1910,{{Sfn|Polo-Blanco: ''Theory and history of geometric models of Alicia Boole Stott''|2007|loc=§5.3.2 1910 paper on semi-regular polytopes|pp=152-158|ps=; summarizes Boole Stott's method and results from {{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}}, including her tables of expansion-contraction dimensional analogies and a few of her illustrations.}} and those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, rediscovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter &amp; Petrie|1938|p=4|loc=''[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]''}}|name=Concentric polyhedra}} Before we move on from the 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice one more thing about the hidden {{radic|5}} chords lurking below its surface. The 12 little pentagon faces are spanned by sets of 6 {{radic|5}} chords (5-cell edges). The 6 chords of each set are disjoint (they don't touch or form 5-cell faces), but they are symmetrically arranged in 3 parallel pairs, {{radic|3}} apart, which lie in 3 orthogonal central planes. It turns out they are the 6 reflex edges of a 12-point non-convex polyhedron called the [[w:Jessen's_icosahedron|Jessen's icosahedron]], inscribed in the 60-point cell. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again and work our way outwards, before eventually in ''[[#The perfection of Fuller's cyclic design|§The perfection of Fuller's cyclic design]]'' we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cell)s ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks, showing its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Pentahemidemicube.png|120px]]
|[[File:Tetrahemihexahedron_rotation.gif|120px]]
|[[File:Pentahemicosahedron.png|120px]]
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts needed to build the 120-cell (the picture on the cover of the box), and everything else that's invisible in that picture which fits inside the 120-cell: all the astonishing regular 4-polytopes. The pentad building block is half a pair of orthogonal 5-point (regular 5-cells), the hexad building block is half an 8-point (16-cell), and the heptad building block is a quarter of an 11-point (11-cell). Everything can be built from these blocks by embedding them at the center of the 3-sphere, inscribing them as 4-polytopes to make compounds as with the 16-cell building block, rather than by assembling them face-to-face as 3-polytopes to make honeycombs in which they are cells or vertex figures.{{Efn|name=Embedding point}}

The pentad building block is the 5-point '''pentahemidemicube''', an abstract polyhedron with 10 equilateral triangle faces, 10 {{radic|5}} edges, and 5 vertices.{{Sfn|Christie: Pentahemidemicube|2024|loc=File:Pentahemidemicube.png|ps=; &quot;The 5-point ''pentahemidemicube'' is an abstract polyhedron with 10 equilateral triangle faces, 10 edges, and 5 vertices. It has a real presentation as the regular 5-cell.&quot;}} It has a real presentation as the 5-point (regular 5-cell 4-polytope), a regular tetrahedron expanded by addition of a fifth vertex equidistant from all the others in a fourth dimension. The pentad block is a tetrahedral pyramid (a 5-cell) inscribed in a cube: a tetrahedron inscribed in a cube (a demicube) expanded by addition of a fifth vertex, one of the cube's other 4 vertices. Each pentad (5-cell) building block lies in the 120-cell orthogonal to three other pentad (5-cell) building blocks. They are four completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges. Each regular 5-cell in the 120-cell is a pentad (pentahemidemicube) building block, in one of 30 orthogonal sets of 4. There are 120 pentad building blocks in the box.

Every vertex of the 16-cell, and therefore every vertex of the 8-cell, 24-cell, 600-cell and 120-cell, is surrounded by a 6-point (octahedral vertex figure). The hexad building block is that 6-point octahedron embedded at the center of the 4-polytope instead of at a vertex, that is, inscribed in the 3-sphere as a 4-polytope, the 6-point '''[[w:Tetrahemihexahedron|tetrahemihexahedron]]''' or '''hemi-cuboctahedron'''.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; &quot;The tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U&lt;sub&gt;4&lt;/sub&gt;. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices.&quot;}} The hexads have 4 red triangle 5-cell faces, bounded by 12 {{radic|5}} edges. Their 3 yellow square faces lie in central planes, because the building block is an abstract 4-polytope. The hexad is an irregular abstract polyhedron that also has 6 {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

Two orthogonal 5-point (pentad) building blocks intersect in a 4-point (inscribed tetrahedron) to form a 6-point (hexad) building block. Two orthogonal 6-point (hexad) building blocks intersect in a 4-point (central square) to form an 8-point (16-cell) building block, the most usable building block, from which everything larger can be built by compounding.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 120 pentads to pair with a completely orthogonal hexad, and although there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed) they only result in 120 pairs two different ways, so there are 120 11-cells. The 11-cell is 6 pentads' and 5 hexads' smallest joint compound, and the 120-cell is their largest, 120 pentads and 100 hexads.

The heptad{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''|pages=22–31}} building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|4}} edges and 9 {{radic|5}} edges), and 7 vertices.{{Sfn|Christie: Pentahemicosahedron|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;}} It is a compound of a 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]) and a 6-point (hexad [[W:hemi-icosahedron|hemi-icosahedron]]) intersecting in a 4-point (tetrahedron), that is, of three orthogonal 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]&lt;nowiki/&gt;s) intersecting in their common 4-point (tetrahedron). It is also the abstract cube contracted by removal of one vertex, which reduces one of the cube's two inscribed tetrahedra (demicubes) to a single remaining triangular face. As with all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Four &lt;small&gt;&lt;math&gt;(11-7)&lt;/math&gt;&lt;/small&gt; orthogonal 7-point (pentahemicosahedron) blocks can be joined at a common Petrie polygon (a skew hexagon which contains their orthogonal 6-edge paths) to construct a real 11-point ([[W:11-cell|11-cell]] semi-regular 4-polytope), which magically contains six &lt;small&gt;&lt;math&gt;(11-5)&lt;/math&gt;&lt;/small&gt; 5-point (1-pentad regular decahedra) as cells and five &lt;small&gt;&lt;math&gt;(11-6)&lt;/math&gt;&lt;/small&gt; 12-point (2-hexad semi-regular cuboctahedra) as cells. There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like [[W:Euclid's postulates|Euclid's postulates]], and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as honeycombs of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles. It possesses the complete symmetry of the polytope without any redundancy, because it contains one of each of the polytope's characteristic root elements.{{Efn|A [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be partitioned radially by its planes of symmetry (Coxeter's &quot;reflecting circles&quot;) into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated rotationally to generate its regular 4-polytope because it is the complete [[W:gene|gene]] for it, containing all of its elements and capturing all of its symmetry without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring that triggers the reflections.|name=Characteristic orthoscheme}} It is the gene for the polytope, which can be replicated to construct the polytope.

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its &lt;math&gt;A_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the regular compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;small&gt;&lt;math&gt;120\times 120 = 14400&lt;/math&gt;&lt;/small&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[16--cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its &lt;math&gt;B_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 384. The 120-cell is the convex hull of the regular compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 24-point (24-cell) is subdivided into 1152 instances of its [[24-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;F_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 1152. The 120-cell is the convex hull of the regular compound of 25 disjoint 24-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;25\times 1152 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has three equivalent origins in distinct root systems, though most are obviously constructed from one root, and not so obviously from the others. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything larger than the pentad can be constructed by compounding hexads, and we shall see that everything as large as a 24-cell has a construction from heptads which are a product of a pentad and a hexad. Surely every uniform polytope can be constructed by some function of expansions and contractions from any of three root systems, however indirect the recipe. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes, related by an equals sign between them, then give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics by [[W:Noether's theorem|Noether's theorem]].

=== Pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 octahedral cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell contains the 600 octahedral cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It also contains the 600 cubical cells of its 75 8-cells. It is a 600-cell in three ways, although the octahedra and cubes are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell and the 8-cell, respectively.|name=600 cells three ways}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (1-pentad 5-cell), out of the 8-point (2-hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (1-pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope that contains 5-point (1-pentad 5-cell) building blocks beside the largest one, it just isn't a ''regular'' 4-polytope. It is semi-regular with two kinds of cells, the 4-point (regular tetrahedron) and the 12-point (truncated cuboctahedron), which have just one kind of regular face between them, a 5-cell face. That semi-regular 4-polytope is the 11-cell.

We said earlier that the 11-cell had 11 disjoint 5-cells and 11 hemi-icosahedral cells in it, and that its construction from them might be hard to see, but now we are going to account numerically, at least, for how it can have 6 pentad cells plus 5 hexad cells magically adding to 11 of the same kind of elevenad cell. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, 11. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (5-cell regular decahedron) cells, and the 12 6-point (hexad) building blocks into 100 12-point (regular icosahedron) cells, and then magically adding one whole 5-point (5-cell regular decahedron) cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 12-point (regular icosahedron) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such (cell bundles of 6 + 5 = 11 cells), with each 11-cell bundle containing 11 of both kinds of cell. Very, very strange!{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden'': §II To Add is to Multiply|p=38|ps=; the sequence of regular polygons with harmonious proportions beginning {5}, {7}, {9}, {11} as a family of natural numbers; &quot;The numbers in this optimal family have remarkable properties [that allow] similar figures to be arranged easily or allow a figure to be dissected into a set of similar figures. This is the origin of the Greek idea of geometric progression — that ''multiplication can be accomplished instead by addition''.... For instance, it is well known that the square of the golden ratio is one more than itself (written above as 𝜙&lt;sup&gt;2&lt;/sup&gt; = 𝜙 + 1) and one less than 𝜙 is its reciprocal (1/𝜙 = 𝜙 - 1). The first relation equates multiplication with addition, while the second accomplishes division through subtraction. [These are pentagon {5} bisectional golden ratios, but] the heptagonal [{7} tri-sectional] ratios also behave in surprising ways — adding to multiply, subtracting to divide.... There are analogous constructions for [nonagon {9}] quadrisectional and [hendecagon {11}] pentasectional systems.&quot;}}

The 120-cell has three interchangeable cyclic expansion-contraction cycles, the pentad, hexad and heptad cyclic designs, rooted in the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems, respectively. Each of them constructs the whole sequence of regular 4-polytopes from a single building block. The key to all three cycles is that 5 + 6 = 11, and there is an 11-point quasi-regular 4-polytope in the sequence of convex 4-polytopes. It lies between the 8-point (16-cell) and the 16-point (8-cell), and is itself a &lt;math&gt;B_4&lt;/math&gt; polytope. Its role is to be a unified single building block common to all three cyclic expansion-contraction cycles. It is involved with the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems and expresses in one building block their relations to each other.

We have already seen how the hexad building cycle works. It is plain compounding of 4-orthoplexes at the beginning, increasing to densely intricate non-disjoint symmetry toward the end, making it difficult to see why it works once past the 24-cell. This is how the build works, in terms of sixes. The 16-cell is a 2-hexad (3 pairs of completely orthogonal great square planes). The 24-cell is 3 16-cells and a 4-hexad, 4 different ways: 16 non-disjoint hexads (great hexagon planes). The 120-point (600-cell) is 25 24-cells and a 20-hexad, 10 different ways: 200 non-disjoint hexads (great hexagon planes). The 600-point (120-cell) is 225 (9 times 25) 24-cells. ....

The pentad building cycle doesn't do plain compounding to regular 4-polytopes until the last one, the 120-cell. Fives don't even appear as pentagons until after the 24-cell. We have seen how there are 11 5-cells in the quasi-regular 11-cell, which occurs before the 8-cell, and also how everything can be compounded from irregular 5-cells (4-orthoschemes specialized for each 4-polytope). It is still not evident how a pentad cycle can work to build all the regular 4-polytopes from the regular 4-simplex. This is how is how the build works, in terms of fives. ....

This is how the build works, in terms of elevens. The 11-cell is 4 orthogonal heptads (intersecting in their skew hexagon Petrie polygon). The 24-cell is also 4 heptads (intersecting in 4 orthogonal digon axes, so each heptad contributes 6 unique points to a single fibration of 24 points). Heptads are quarter elevenads, so the 11-cell and 24-cell are both 1 elevenad. ....

See the tabular sequence of convex 4-polytopes in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Build with the blocks|§Build with the blocks]]'', below: table rows ''Inscribed, Pentad, Hexad, Heptad''.

=== 11 of them ===
There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of eleven 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum eleven of them. There is only a single Hopf fibration of the 11 great circles of an 11-cell. 

The 5-cell and the 11-cell are limit cases. The 5-cells are completely disjoint, but the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 10 other instances of itself.

== Cell rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer &amp; Segerman|2013|loc=''Puzzling the 120-cell''}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his section of the Wikipedia article on the 120-cell entitled ''[[120-cell#Visualization|§Visualization]]''.{{Sfn|Ruen &amp; Goucher et al. eds. 120-cell|2024|loc=Goucher's ''[[120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious place to look for the 11-cell's Hopf map is in Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other place it could be hiding. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The quasi-regular convex 11-cell|§The quasi-regular convex 11-cell]]'', and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen: triacontagon|2011|loc=''Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them except his illustrations. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Four orthogonal projections of the 120-cell to a central plane, selected from the table of 3 regular and 11 compound 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|100 icosahedra&lt;br&gt;120 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|5 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 icosahedra&lt;br&gt;6 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|120
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|4
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|50
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|7.2°
|align=center|36°
|align=center|6°
|align=center|7.2°
|}

....

Here we summarize our findings on the 11-cells so far, including in advance some results to be obtained in the next section two sections.

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section....

== The 15 major chords ring everything ==
The natural numbers are each a distinct flavor.

Each integer has its own distinct taste. The first six integers {0..5} are the six strongest, most familiar tastes in nature, instantly distinguished and utterly unique.

The seventh flavor {6} is the first ratio of two of the others. Thereafter follows in strange sequence all the ratios of all the flavors, each marriage of every two flavors, most of which are fractions &lt;math&gt;\{k/d\}&lt;/math&gt;, but some of which are integers &lt;math&gt;\{k\}&lt;/math&gt;.

Some of these ratios are successful flavors in their own right, as distinct and inimitable as the marriage of chocolate and oranges, but the marriage of many flavor pairs is mud. It is a rule of thumb among flavor specialists that nearly any taste can be successfully alloyed with either onions or chocolate, but the direct marriage of these two flavors is notably unsuccessful. Some flavors make famous pairings or infamous ones, and subtle new pairings are always to be discovered that work surprisingly well for some purpose, but the majority of flavor marriages are undistinguished compounds.

In polytope geometry, each chord of a polytope is both is a distinct 1-dimensional object, a chord of the sphere of a distinct length &lt;math&gt;l&lt;/math&gt;, and a distinct natural number &lt;math&gt;h&lt;/math&gt;, a unique flavor. If the polytope is regular, it is a noteworthy and successful flavor. The chord's length &lt;math&gt;l&lt;/math&gt; is a square root, related to the natural number &lt;math&gt;h = k/d&lt;/math&gt; and to a set of polytopes they both represent, by a formula discovered by Steinbach.{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden''; Figure 5. Optimal sections and proportions|p=37|ps=; the regular polygons {5}, {7}, {9} and {11} with their diagonals define respectively: {5} the golden bisection proportional to 𝜙; {7} an analogous trisection; {9} an analogous quadrasection; {11} an analogous pentasection.}} The chord length &lt;math&gt;l&lt;/math&gt; is related to the number of sides of the regular polygon formed by the chord (its numerator &lt;math&gt;k&lt;/math&gt;), and to the winding number or density of its regular skew Petrie polygon (its denominator &lt;math&gt;d&lt;/math&gt;).

The regular 4-polytopes do not contain an infinite variety of these chord flavors, but only 30 of them: only 15 unique pairs of 180° degree complementary chords, which combine to make every regular ''n''-dimensional thing found in nature.{{Efn|The 15 ''major chords'' suffice to construct all the regular ''n''-polytopes. Their 180° complements include 15 ''minor chords'' which occur only in the 120-cell, not in any smaller regular polytope.{{Efn|name=Major and minor chords}} Including the complement of the 180° {{radic|4}} digon {2} chord (the {0} chord {{radic|0}}), and the skew variant of the 90° {{radic|2}} square {4} chord (the {30/7} chord), there are 32 distinct natural numbers which are all the chords of all the regular polytopes in all dimensions.|name=32 chords}} All the regular polytopes (including the largest one on the cover of the box) can be built not only from a box containing a number of blocks of 3 shapes glued together by gravity, they can be also be built from a box containing a number of sticks of 15 lengths glued together by rubber joints.

[[File:15 major chords.png|500px|thumb|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the others occur only in the interior of 120-cell, not as edges of any of the regular 4-polytopes. There are 30 distinct 4-space chordal distances between vertices within the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to identify any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=Major and minor chords}} chords are properly labelled #1 — #15 in the order they fan out by size.{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''; Figure 3|ps=; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with a &quot;fan of chords&quot; diagram like the one here.|p=23}} They join vertex pairs which are 1 — 15 edges apart on a &lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon. A #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; chord forms &lt;small&gt;&lt;math&gt;f+1&lt;/math&gt;&lt;/small&gt; regular geodesic polygons which include a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; disjoint regular {{nowrap|&lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt;-gons}} (e.g. &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; a compound of 5 disjoint hexagons) plus a single [[W:Triacontagon#Triacontagram|regular skew {{nowrap|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram}}]] that wraps around the sphere &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times before closing its circle. Notice the maverick #11 chord, with its non-conforming origin point for its {{nowrap|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;}} polygram.]]

These 15 chords (natural numbers) include the integers 2, 3, 4, 5, 6, 10, 11, 15 and 30. The others are the fractions 5/2, 10/3, 13/7, 15/2, 15/4, and 15/7. Represented among the chord lengths (the square roots) measured in ''unit radii'' (not in {{radic|2}} radii, as metrics are often given elsewhere in this article) are the {{background color|paleturquoise|roots of the first 4 positive integers {{radic|1}}, {{radic|2}}, {{radic|3}} and {{radic|4}}}}, the {{background color|#FFCCCC|root of the very distinguished rational fraction {{radic|5/2}}}}, and the roots of various distinguished irrational fractions as Steinbach discovered: for example, 4 of them are {{background color|yellow|roots of golden numbers (a small integer ± 𝜙)}}. The 15 chords each form their regular polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, which is just &lt;small&gt;&lt;math&gt;\{k\}&lt;/math&gt;&lt;/small&gt; in 8 cases. Arranged in order #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; from shortest edge-length square root to longest, the #1 — #15 chords bridge vertex pairs which are &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; vertices apart on a [[W:Triacontagon#Petrie polygons|&lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon]]. Each chord makes a [[W:Triacontagon#Triacontagram|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; polygram]] of its edge length, a compound of the smaller &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. The 15 distinct polygons form a sequence you would not anticipate exactly unless you have Steinbach's formula:

{| style=&quot;background-color:transparent; white-space:nowrap;&quot;
|-
| {30/1} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|.073}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{30} 
| 120-cell edges
|-
| {30/2}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.191}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15}
|- style=&quot;background: yellow;&quot;|
| {30/3}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.382}} 
| &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi}&lt;/math&gt;&lt;/small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{10}''' 
| 600-cell edges
|-
| {30/4} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.573}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/2}
|- style=&quot;background: paleturquoise;&quot;|
| {30/5}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|1}}'''
| &lt;math&gt;1&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{6}''' 
| 24-cell, 8-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/6}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|1.382}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{5}'''
|- style=&quot;background: paleturquoise;&quot;|
| {30/7}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|2}}'''
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{4}''' 
| 16-cell edges
|- style=&quot;background: #FFCCCC;&quot;|
| {30/8}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.5}} 
| &lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;&lt;small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/4} 
| 5-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/9}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.618}} 
| &lt;math&gt;\phi&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{10/3}
|- style=&quot;background: paleturquoise;&quot;|
| {30/10}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|3}}''' 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{3}'''
|-
| {30/11}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.426}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{11/11}
|- style=&quot;background: yellow;&quot;|
| {30/12}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.618}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{5/2}
|-
| {30/13}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.810}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{13/7}
|-
| {30/14}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.928}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/7}
|- style=&quot;background: paleturquoise;&quot;|
| {30/15}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|4}}''' 
| &lt;math&gt;2&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{2}'''
|}

{| class=wikitable style=&quot;white-space:nowrap; float:right;&quot;
!colspan=6|Natural number &lt;small&gt;&lt;math&gt;h=\{k/d\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Fibers&lt;br&gt;&lt;small&gt;&lt;math&gt;f\{h\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Cells&lt;br&gt;&lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|4-tope&lt;br&gt;&lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Verts&lt;br&gt;&lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|[[W:Triacontagon#Triacontagram|30-gram]]&lt;br&gt;&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;
|-
!Chord
!Arc
!colspan=2|&lt;small&gt;&lt;math&gt;l/\sqrt{1}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;l/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;h&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#1
|15.5~°
|{{radic|0.073}}
|0.270
|{{radic|0.146}}
|&lt;small&gt;&lt;math&gt;30&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1\{30\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;5\{3,3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/1\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#2
|25.2~°
|{{radic|0.191}}
|0.437
|{{radic|0.382}}
|&lt;small&gt;&lt;math&gt;15&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;..\{3,3\}..&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/2\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#3
|36°
|{{radic|0.382}}
|0.618
|{{radic|0.764}}
|&lt;small&gt;&lt;math&gt;10&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;30\{3,3\}20&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/3\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#4
|44.5~°
|{{radic|0.573}}
|0.757
|{{radic|1.146}}
|&lt;small&gt;&lt;math&gt;15/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/4\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5
|60°
|{{radic|1}}
|1
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;6&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/5\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#6
|72°
|{{radic|1.382}}
|1.175
|{{radic|2.624}}
|&lt;small&gt;&lt;math&gt;5&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}} 
|&lt;small&gt;&lt;math&gt;10\{5,3\}12&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/6\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#7
|90°
|{{radic|2}}
|1.414
|{{radic|4}}
|&lt;small&gt;&lt;math&gt;4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7\{4\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;4\{4,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;9\{3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/7\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: #FFCCCC;&quot;|
|#8
|104.5~°
|{{radic|2.5}}
|1.581
|{{radic|5}}
|&lt;small&gt;&lt;math&gt;15/4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/4\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}} 
|&lt;small&gt;&lt;math&gt;15\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/8\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#9
|108°
|{{radic|2.618}}
|1.618
|{{radic|5.236}}
|&lt;small&gt;&lt;math&gt;10/3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10/3\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/9\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#10
|120°
|{{radic|3}}
|1.732
|{{radic|6}}
|&lt;small&gt;&lt;math&gt;3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;10\{3\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/10\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#11
|135.5~°
|{{radic|3.426}}
|1.851~
|{{radic|6.852}}
|&lt;small&gt;&lt;math&gt;11/11&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;11\{11/11\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;11\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#12
|144°
|{{radic|3.618}}
|1.902
|{{radic|7.235}}
|&lt;small&gt;&lt;math&gt;5/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2},3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/12\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#13
|154.8~°
|{{radic|3.810}}
|1.952
|{{radic|7.621}}
|&lt;small&gt;&lt;math&gt;13/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;13\{13/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;7\{5,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/13\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#14
|164.5~°
|{{radic|3.928}}
|1.982
|{{radic|7.857}}
|&lt;small&gt;&lt;math&gt;15/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;137\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/14\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{radic|8}}
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15\{2\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐𝜙&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;8\{3,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/15\}&lt;/math&gt;&lt;/small&gt;
|}

The polygons '''{10}''', '''{6}''', '''{5}''', '''{4}''', '''{3}''', and '''{2}''' are flat polygons that each lie on a great circle in a central plane of the 3-sphere. The polygons {30}, {15}, {15/2}, {15/4}, {10/3}, {11/11}, {5/2}, {13/7}, {15/7} and all the {30/''n''} polygons are skew polygrams spiraling through curved 3-space, each successive edge in a different central plane of the 3-sphere. Projected to the 1-sphere (a circle in the plane), they are [[W:Star polygon#Regular star polygon|regular star polygrams]], the simplest being the pentagram {5/2}. In 4-space they wind around the 3-sphere more than once ''without self-intersecting'', before they close their circuit. They lie on helical geodesic circles or ''isoclines'' of circumference greater than 2𝝅''r''.

Even the 6 integer chords which form flat polygons also form skew polygrams. Each of the 15 chords occurs as a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, a fiber bundle of Clifford parallel polygons that totals 30 vertices per bundle. Each chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; also forms a single skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram that winds through another, shorter chord's bundle of parallel polygons, intersecting them in succession multiple times each, and weaving that whole bundle of parallel fibers together as the cross-thread does on a loom. The polygram cross-thread has to wind multiple times through the whole bundle (&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times around the [[W:Triacontagon#Triacontagram|30-gram]] in which each edge connects every &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt;th vertex) to complete an enumeration of all 30 vertices in the bundle. For example, the #5 chord forms hexagon &lt;small&gt;&lt;math&gt;\{6\}&lt;/math&gt;&lt;/small&gt; flat planes in parallel bundles of 5, written &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt;, and also a skew &lt;small&gt;&lt;math&gt;\{15/2\}&lt;/math&gt;&lt;/small&gt;-gram of #5 edges which winds 2 times through the #3 chord's bundle of 3 decagon &lt;small&gt;&lt;math&gt;\{10\}&lt;/math&gt;&lt;/small&gt; flat planes. Every chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; forms both its &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram and its fiber bundle of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, and the edges of one chord's parallel polygons are also the edges of its skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram weaving through some shorter chord's bundle of parallel polygons. Each numbered chord is a natural number, a ratio of two integers and their polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, either flat or skew. Each chord number defines a &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; skew polygon that is a compound bundle of the &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. All these flat or skew polygons are regular polygons made of their numerator instances of a single chord #, whose length is that square root, Steinbach's function of the natural number.

The relationship between each bundle of flat Clifford parallel great circle polygons of a distinct length, and its corresponding bundle of skew polygrams of a longer length, is precisely that of a discrete simple rotation and its corresponding discrete isoclinic rotation. For example, the flat great circle hexagons of edge length {{radic|1}} lie in the invariant rotation planes of the 24-cell's characteristic 60° isoclinic rotation, in which the vertices move on skew hexagram rotation paths of edge length {{radic|3}}. The 24-cell contains 4 bundles of 4 Clifford parallel (disjoint) hexagon central planes each. The 24 vertices of the 24-cell in each bundle are linked by rotation in the 4 hexagon invariant planes in two different ways: simply and isoclinically. In a simple rotation the vertices of a single invariant hexagon exchange places on that great circle hexagon of edge {{radic|1}}. In an isoclinic rotation in which any one of these 4 disjoint hexagon central planes is invariant, all 4 of them are necessarily invariant planes in the rotation, because they rotate in parallel. In that isoclinic rotation the vertices of all 4 hexagon invariant planes (all 24 vertices of the 24-cell) exchange places concurrently, moving along 4 disjoint helical hexagrams of edge {{radic|3}} in parallel. It is a characteristic only of this distinct rotation in hexagon invariant planes that the helical vertex path or ''isocline'' is a hexagram of period 6, edge length {{radic|3}}, and circumference 720°. Rotations in other kinds of polygon invariant planes have different characteristic isoclines. The period of their isoclinic rotation must be ''the same or greater'' than the period of their simple rotation; in most cases (unlike this one) it is greater.

The 15 chords form not only distinct fiber bundles of polygons, but also distinct bundles of polyhedra: cell rings &lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt; and compound vertex figures &lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;. These combine to form their characteristic regular 4-polytope &lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Coxeter|1973|p=269|loc=Compounds|ps=; &quot;It is remarkable that the vertices of {5, 3, 3} include the vertices of all the other fifteen regular polytopes in four dimensions.&quot;}} All these elements of a regular 4-polytope constructed from one chord are themselves regular polytopes, and each of the 16 regular 4-polytopes is a function of one characteristic chord. If the regular 4-polytope is convex, its characteristic chord is its unit-radius edge. For a non-convex regular 4-polytope, the characteristic chord is its unit-radius ''mid-edge diameter'', the distance from the midpoint of each edge to the midpoint of the completely orthogonal edge. Each of the characteristic chords (each distinct natural number) has its expression as one of the 16 regular 4-polytopes and all the products of the distinct regular 1-polytopes, 2-polytopes and 3-polytopes that construct it, and the whole distinct construction from the natural number to the 4-polytope is an [[W:SO(4)|SO(4) symmetry group]] expression of that 4-polytope's characteristic discrete [[W:SO(4)#Isoclinic rotations|isoclinic rotation]], in the invariant central planes of its characteristic chord, by the characteristic chord arc.

...demonstrate this theory by proper enumeration of the chord lengths in the table...

...color-code the △ ☐ 𝜙 ✩ symbols ... chord colors in diagram should match them ... some of the diagram colors are wrong and must be changed, but otherwise diagram is pretty good ... i think in a colored diagram all chords of the same length should have the same size line dash, the polygons are not obvious enough and the 15 fanning-out chords are readily identified by their labels...

... add back the counts of chords in the box (very small font number under each table entry), and present these two images as the front and back a box of building sticks and rubber joints...What's in the other box...

...say the rational-root major chords {{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|5/2}} are the ''n''-edges: their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge...

...abbreviated text of the [[120-cell#Relationships among interior polytopes|''§Relationships among interior polytopes'']] section from the 120-cell article...to explain the color-coding

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius=1}}

See also the elements, properties and metrics of the sequence of regular 4-polytopes summarized in tabular form in the section ''[[#Build with the blocks|§Build with the blocks]]'', below.

== Alicia Boole Stott's original formulation of dimensional analogy ==
&lt;blockquote&gt;&quot;...a method by which bodies having a certain kind of semi-regularity may be derived from regular bodies in an Euclidean space of any number of dimensions...&quot;{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''; presents cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues|pp=12-45}}&lt;/blockquote&gt;

Alica Boole Stott's 1910 paper reads like an exceptionally well-written patent application. It is a comprehensive stepwise description of her original invention, a general method of dimensional analogy between regular and semiregular polytopes of the same edge length.

[[File:The Roots of Three.png|thumb|An illustration of Boole Stott's expansion of unit-edge triangle to hexagon captures the geometric basis in 3 of the &lt;math&gt;A_n&lt;/math&gt;root system.]]

The engineering-like articulation of her new principles is entirely discrete and group-theoretic. Boole Stott repeatedly refers to sets of polytope elements as ''groups''. All her transformations are discrete instances of one or another special case of her &quot;two inverse operations which may be called ''expansion'' and ''contraction''.&quot; Hers is an exploration of ''n''-dimensional space at its most basic: space defined by the discrete objects which fill it, themselves defined directly as groups of elements with a few operations on them, which transform them into each other. This is geometry which seems to precede Euclid's postulates, and does not depend on them. Instead it corresponds directly to the deeper group theory mathematics which underlies geometry, even though Boole Stott, like the ancients, did not have that mathematics.

Boole Stott enumerated her expansion and contraction operations as transformations between regular and semi-regular polytopes of a single edge length. In her 1910 paper she did not consider any polytopes of multiple edge lengths, or expansions of the subject polytope by a distance other than that exactly required to produce new edges of the original edge length, so of course she did not enumerate Moxness's non-uniform rhombicosidodecahedron (Hull #8). This may be why section 8&lt;sub&gt;3&lt;/sub&gt; is missing from her earlier drawings of the sections of {5,3,3}; she may not have had a construction for it.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The Boole Stott expansion (or contraction) operation is as precisely defined a transformation as Coxeter's reflection, rotation and translation operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions). Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;
where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;
where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;
For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} An instance of the expansion (or contraction) operation is an application of the general operation to a particular subject polytope, operating on all of its elements symmetrically at once. The general operation is restricted, such that one may only expand (or contract) the subject polytope by pushing all its edges, faces or cells directly away from (or toward) its center, without changing the size of the moved elements. This splits each vertex into several (or coalesces several vertices into one), and has corresponding effects on the other elements, producing new edges ''of the same size'' between pairs of separated vertices (or reducing some edges to single vertices). The distance all the elements are moved away from (or toward) the center must be such that the vertices separate producing new edges of only the original edge length (or come together reducing edges to single vertices). It is permitted to combine these three operations, e.g. by pushing the edges ''and'' the faces, or by pushing all three element sets, provided the combination results in a semi-regular polytope of only one edge length. The three operations are all commutative with one another, so combined operations can be considered either as a sequence of expansions (or contractions) linking several related polytopes, or as their concurrent movements, a single transformation of the subject polytope. Since there are 8 combinations of one to three operations there are 8 polytope variants (including the original regular polytope), and all are adjacent to one another by a single transformation. The variants of the original regular polytope are named for which element set(s) are moved. By enumerating all 8 variants of expansion (and contraction) for each of the six regular convex 4-polytopes, Boole Stott produced a table of regular and semi-regular 4-polytopes divided into 12 sets of 8: six expansion sets and six contraction sets.{{Sfn|Boole Stott|1910|loc=''Table of Polytopes in S&lt;sub&gt;4&lt;/sub&gt;''; note that Boole Stott's usage of ''S&lt;sub&gt;4&lt;/sub&gt;'' denotes a Euclidean 4-space ''E&lt;sub&gt;4&lt;/sub&gt;'', or a 3-sphere space embedded in it, ''not'' a 4-sphere space embedded in a Euclidean 5-space|p=30}} Each set includes just one of the six regular convex 4-polytopes, with its distinct unit-radius edge length, and the 7 possible semi-regular expansions (or contractions) of it, which all preserve that characteristic edge length. The polytopes in each set of 8 all have the same total number of elements as their original regular polytope, but each polytope in the set has a distinct combination of cell, face, and vertex figures. The 12 sets are not completely disjoint, and even within one set the 8 operations do not necessarily produce 8 unique polytopes. For example, the regular 5-cell (the 4-simplex polytope) has 5 cells, 10 faces, 10 edges, and 5 vertices (30 elements). Expanding the 5-cell produces 7 related semi-regular 4-polytopes of 30 elements each (two of which happen to be the same polytope). In each expansion, the 5 cells have expanded into semi-regular or regular polyhedra, the 10 faces and 10 edges ''may'' have expanded into prisms, and the 5 original vertices have expanded into regular or semi-regular cells.

== The regular 11-cell 4-polytope ==

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosadodecahedron). Each non-convex Jessen's icosahedron is a set of 6 reflex edges that are {{radic|5}} 5-cell edges, each set symmetrically arranged in 3 parallel pairs, not as far apart ({{radic|3}}) as they are long, in 3 orthogonal central planes. The 12-point (Jessen's sets of 6 reflex 5-cell edges) are inscribed in the 60-point with their endpoints as a vertex of a pentagon face. The 12 pentagon faces are joined to 5 other pentagon faces by the 5-cell edges; not to their nearest neighbor pentagons but to their next-to-nearest neighbor pentagons; and since there are 5 vertices to each pentagon face, there are 10 12-point (Jessen's sets of 6 reflex 5-cell edges). Each pentagon face is joined to another pentagon face by 5 reflex edges of five distinct Jessen's. The 5 reflex edges are Clifford parallel, the 5 edges of a pentagonal prism column that has a twist to it. The contraction of those 5 edges into 1, and those 10 vertices (two pentagons) into 2, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing all their edges as a single Jessen's. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (octahedron) by conflating its parallel pairs of edges, leaving the 11-cell's abstract 6-point (hexad cell). The hexad's situation is that it occurs in bundles of 5, disjoint but close together in the sense of adjacent, like the great circle rings in a Hopf fiber bundle. 

To summarize, the 12-point (2-hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell). We shall now be able to see how Fuller's perfect hexads combine with Todd's perfect pentads to form the perfect heptads of the 11-cell and 137-cell.  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=alias the hemi-cuboctahedron}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces. (The ''heptad'' arrives at the party, anonymously; they are our distinguished anti-celebrity, who most of the other guests have never even heard of.) The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. This 12-point (truncated tetrahedron) is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 100 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp: Truncated tetrahedron|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp: Truncated tetrahedron|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}} Of course Moxness has been there, seen that, already with his graphics software;{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} see his parameterized illustration of it exactly as it occurs in the 11-cell, below in his bestiary of A3, B3, and H3 Archimedian and Catalan polyhedra in the 120-cell.]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:Archimedean_and_Catalan_solid_hulls_with_their_Weyl_orbit_definitions.svg|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of the 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. In this catalog of Archimedean and Catalan polyhedra sighted in the 120-cell by Moxness's software,{{Sfn|Moxness: Archimedean and Catalan hulls|2023|loc=Hull #1 Archimedean Name A3 110 Truncated Tetrahedron A (upper left)}} it is the first entry (upper left), Moxness's Hull # = 1 with 12 vertices of 3D norm {{radic|11}}/2. On a {{radic|2}}-radius 3-sphere, 5 of these with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell. 20 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges are cells in the [[#The rings of the 11-cells|cell rings of the quasi-regular 11-cell]], which also has 12 regular 5-cells as cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

...

Both kinds of cells in the 11-cell have expressions in two different dimensions (on the 2-sphere and 3-sphere), because they are all centered on the center of the 3-sphere, unlike the cells of most 4-polytopes, which are centered on various off-center points within the 4-polytope's interior. 

One kind of 11-cell cell appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (semi-regular truncated cuboctahedron), a.k.a. the truncated tetrahedron. The other kind of cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now, except in the 120-cell. But this is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron, and a regular tetrahedron behind each face. The 11-cell is thus a honeycomb of mixed regular tetrahedra and truncated cuboctahedra (a.k.a. truncated tetrahedra) that meet at their shared 5-cell faces. 

Both the 5-cells (decahedra) and the truncated cuboctahedra (truncated tetrahedra) are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks in their 4-polytopes (off-center).{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} Therefore all the 11-cell's cells are concentric, which indeed makes a different kind of honeycomb, an inside-out cellular honeycomb with all the cells piled on top of each other overlapping in 4-space. They do not overlap in 3-sphere space however; their 3-volumes (but not their 4-contents) are disjoint. They make a proper 4D honeycomb like the cells of any 4-polytope, with volumetrically disjoint 3D cells that meet each other in the proper way at their shared 2D faces.

An 11-cell contains 6 completely disjoint 5-point (1-pentad 5-cell decahedron cells), and 5 volumetrically disjoint 12-point (2-hexad truncated cuboctahedron cells), 11 volumetrically disjoint cells altogether. Of course it is supposed to have 11 identical 11-point (elevenad hemi-icosahedral cells), and that is also true in its abstract sense, which is not the same sense as disjoint cellular containment: each pentad or hexad cell is inscribed in one of 11 larger 60-point (Moxness's rhombicosadodecahedron cells). The abstract pentads and hexads are only hemi-polyhedra, so the 11-cell's pentad cells are 12 (out of 120) disjoint instances of a pentad 5-point, and its hexad cells are 10 (out of 100) disjoint instances of a hexad 6-point, 1/10th of the 120-cell's cellular volume in both cases. The 11-cell's hemi-icosahedral cells are 11 (out of 120) non-disjoint instances of an elevenad 60-point. Each 11-cell shares each of its hexad cells with 9 other 11-cells, and each of its 11 vertices (out of 600) occurs in 9 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.|name=Tetrahedral vertex figures}}

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

...insert image thumb of 11-cell...

...Construct the 11-cell from the heptad....

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...Four concentric 4-polytopes may be mutually orthogonal. In any set of 4 mutually orthogonal regular 5-cells, one pair of faces in each orthogonal pair are completely orthogonal to each other. Thus three edges in one 5-cell in an orthogonal pair are opposite (completely orthogonal) to a face of the orthogonal 5-cell. .... A 5-cell has 5 vertices in its abstract hemi-icosahedral cells, but only 4 vertices in one concrete icosahedral cell; the 5th lies in an icosahedral cell of another 11-cell (not in another icosahedral cell of the same 11-cell).

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

....

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 11-cell ===

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon.... (wrong, 11-cell's Hopf map is the truncated icosahedron)

....

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk: Truncated icosahedron|2018}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

...

Since the rhombicosadodecahedron has 120 5-cell edges, the 120-cell has 120 completely disjoint 5-cells, and the rhombicosadodecahedron shares each vertex with 11 other rhombicosadodecahedra, how a rhombicosadodecahedron is bound to its neighbor 5-cells is constrained. A further constraint is that the abstract hemi-icosahedron has only 15 5-cell edges, so the 120 actual 5-cell edges of the rhombicosadodecahedron are conflated 8-to-1 in the hemi-icosahedron. To be conflated by contraction into a single edge, those 8 edges must lie parallel to each other in the 120-cell; either actually parallel in the same plane, or Clifford parallel in separate planes, as two edges in completely orthogonal planes may be parallel and orthogonal at the same time. Each 5-cell inscribed in the 120-cell contains 5 such tetrahedra occur in parallel pairs in 4 orthogonal planes (none of them completely orthogonal planes). 

.... 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
Each conflated vertex is the 10 vertices of two orthogonal regular 5-cells...
The 60-point contains 6 pairs of these orthogonal regular 5-cell cells...
...its 20 hexad cells are ..., a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere.

== The regular 137-cell 4-polytope ==

(description originally written for the 11-cell, which applies only to a fibration of 11 11-cells)

It is quasi-regular with two kinds of cells, the 12-point (regular icosahedron) and the 5-point (regular 5-cell), which have just one kind of regular face between them. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of regular facets (it is semi-regular with regular facets), it also means that its facets have expressions of two different dimensionalities because they are concentric in the 3-sphere. The 11-cell has two kinds of regular cells. One appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (regular icosahedron), but since it is embedded at the center of the 3-sphere (unlike the cells of most 4-polytopes) it also acts as an inscribed 4-polytope.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} The other cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now except in the 120-cell. This is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron. The 11-cell is a honeycomb of mixed icosahedron and half-icosahedron cells. Like the icosahedra, the decahedra are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks, off-center in their 4-polytopes. All the 11-cell's cells are concentric, which indeed makes an unusual honeycomb of cells, an inside-out honeycomb. It is a proper 4D honeycomb however, with volumetrically disjoint 3D cells that meet each other in the proper way at shared 2D faces. The embedded 12-point (regular icosahedron) cell is an icosahedral pyramid (as a 4-polytope), which is filled by 20 4-point (irregular tetrahedra) that meet at its center; the center point of the 11-cell acts as a 12th vertex, just as the center point of the 24-cell acts as a 25th vertex in some honeycombs. The 5-point (regular 5-cell) cell contains five 4-point (regular tetrahedra) which meet the 12-point (regular icosahedra)'s irregular tetrahedral cells face-to-face, so the 11-cell is quasi-regular in the usual sense of having two varieties of regular 3-polytope cells, but additionally both varieties of of its cells are bi-dimensional: another clue that the 11-cell has something to do with dimensional analogy

....

The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

....

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur to make an 11-cell. 11 of them occur in a 137-cell, all 120 of them occur in the 120-cell, and they occur nowhere else.

How can this regular 4-polytope possibly exist? By Schläfli's criterion {3, 3, ..} exists, but not as a finite polytope.{{Sfn|Coxeter|1973|loc=§7.7 Schläfli's criterion}} Coxeter found {3, 5, 3} as a finite hyperbolic honeycomb, but not in Euclidean 4-space.{{Sfn|Coxeter|1968|title=The Beauty of Geometry: Twelve Geometric Essays|loc=''Regular Honeycombs in Hyperbolic Space''|ps=; &quot;The honeycombs considered by the above authors [Schlegel, Schläfli, Klein, Sommerville] have finite cells and finite vertex figures. ... A further extension allows a cell or vertex figure to be a ''star''-polytope, so that the honeycomb covers the space several times. .... We shall find that there are four regular star-honeycombs in hyperbolic 4-space, as well as two infinite families of them in the hyperbolic plane.&quot;}} We find that {3, 3, ..} is a central expression of the &lt;math&gt;H_4&lt;/math&gt; root system, being the regular convex hull in 4 dimensions of 137 points, 96 pentads and 600 hexads. The minimal expression of &lt;math&gt;H_4&lt;/math&gt; is the regular convex hull {3, 3, 5} of 120 points, 600 tetrads and 75 hexads, and its maximal expression is the regular convex hull {5, 3, 3} of 600 points, 120 pentads and 675 hexads. {3, 3, ..} is a third regular &lt;math&gt;H_4&lt;/math&gt; polytope which fits between them. The properties and metrics of this regular sequence are presented in tabular form below in ''[[#Build with the blocks|§Build with the blocks]]''.

{|class=&quot;wikitable&quot;
!colspan=3|Regular &lt;math&gt;H_4&lt;/math&gt; polytopes
|-
|[[File:600-cell.gif|200px]]
|[[File:5-cell.gif|200px]]
|[[File:120-cell.gif|200px]]
|-
!120-point 600-cell
!137-point 137-cell
!600-point 120-cell
|}

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 137-cell ===
....

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=Apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|2022|loc=''[[Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]''}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie: On Fuller's use of language|2024|loc=''[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry'']]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center, 1948-1949|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with their mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the gender neutral shad can open their six beaks all the way, until they become flat squares and they becomes a cuboctahedron, or they can shut them all tight like a turtle retracting into their octahedron shell. The six mouths always move in unison. This is [[Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do their ''really'' odd trick -- where they flip their 6 jaws 90 degrees in their 6 faces and shut their 6 beaks on the opposite axis of their squares than the one they opened them on -- or whether they will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than their normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make their edge length exactly the same size as their radius, when they open their mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing their mouths in spherical synchrony, their 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron (the hexad), then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=''[[W:SO(4)|SO(4)]]''}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=Isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its orthogonal hexad's planes (wz, wx, wy).{{Efn|name=Six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of (expanded octahederon 16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh &amp; Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== Legendre's 12-point hexad binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a 12-point double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's perfectly realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, this 2-hexad expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre hexad) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction.{{Efn|name=Concentric polyhedra}} In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== ''n''-simplexes ===
....

The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== ''n''-orthoplexes ===
....
[[File:Great 5-cell √5 digons rectangle.png|thumb|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} ''regular'' great hexagons. These are the same 200 dodecagon central planes illustrated above, which also contain 6 120-cell edges and two opposing ''irregular'' great hexagons (truncated {{radic|6}} triangles). The 1200 {{radic|5}} 5-cell face triangles occupy 1200 other, non-central planes.]]

...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's Hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[16--cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[16--cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=Concentric polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[16--cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== ''n''-taliesins ===
The 4-simplex pentad building block combines with the 4-orthoplex hexad building block to make the 4-taliesin heptad building block. The heptad is the essential building block of the ''taliesin'' family of polytopes. The 11-cell and 57-cell hemi-polytopes and the 137-cell and 120-cell regular 4-polytopes are nuclear members of this trans-dimensional family. The 24-cell and 600-cell are extended members of it, because they also contain instances of the heptad building block.

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its spherical symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* [[W:Taliesin Myrddin Namkai-Meche|Taliesin Myrddin Namkai-Meche]], a young Reed College scholar who was stabbed to death by a crazy man in 2017 on a Portland, Oregon light rail train, when he rose to the nonviolent defense of two black teenagers who were being violently assaulted in racist and anti-Muslim language by a right-wing extremist.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in orthogonal [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].
* '''Taliesan polytope''', an (''n+1)''-polytope whose ''n''-polytope dimensional analogue inscribed concentrically in the ''n''-sphere forms a convex honeycomb of itself which is the (''n+1)''-polytope.{{Efn|name=Embedding point}}
* The taliesan polytope's honeycomb, a Hopf fibration of its great circles.
* The ''n''-dimensional family of dimensionally analogous polytopes related to a taliesan polytope by expansion and contraction operations.
* The characteristic [[w:Orthoscheme|orthoscheme]] of a teliesan polytope, an irregular simplex that is the complete gene for the entire expression of trans-dimensional symmetry relationships among the polytopes of the ''n''-dimensional family, as well as the complete gene for the teliesan polytope's rotational symmetry.{{Efn|name=Characteristic orthoscheme}}

=== ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== ''n''-equilaterals ===
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed at four different [[#n-leviathon|chord]] lengths from each other. In the {{radic|2}}-radius polytopes we have using as examples in this paper they are {{radic|2}}, {{radic|4}}, {{radic|6}} and {{radic|8}}, but in a ''unit-radius'' polytope they are: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}.

Each vertex is joined to 8 others by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}. The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== ''n''-pentagonals ===

[[File:600-cell.gif|thumb|The 120-point (600-cell), &lt;math&gt;H^4&lt;/math&gt;.]]
...

[[File:600-cell vertex geometry.png|thumb|Vertex geometry of the 600-cell, showing the 5 regular great circle polygons and the 8 vertex-to-vertex chord lengths with angles of arc. The golden ratio{{Efn|name=golden chords|group=}} governs the fractional roots of every other chord,{{Efn|name=fractional root chords}} and the radial golden triangles which meet at the center.|alt=|400x400px]]
The 120 vertices of the 600-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏&lt;sup&gt;−1&lt;/sup&gt;) beginning with a vertex; see column ''a''}} at eight different [[#n-leviathon|chord]] lengths from each other. These edges and chords of the 600-cell are simply the edges and chords of its five great circle polygons. In ascending order of length in a ''unit-radius'' polytope (not in a {{radic|2}} radius polytopes like our usual examples), they are {{radic|0.𝚫}}, {{radic|1}}, {{radic|1.𝚫}}, {{radic|2}}, {{radic|2.𝚽}}, {{radic|3}}, {{radic|3.𝚽}}, and {{radic|4}}.{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=Decimal fractions of square roots|group=}}

Notice that the four [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#''n-''equilaterals|hypercubic chords]] of the 24-cell ({{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|4}}) alternate with the four new chords of the 600-cell's additional great circles, the decagons and pentagons. The new ''golden chord'' lengths are necessarily square roots of fractions, but very special fractions related to the golden ratio{{Efn|1=The fractional-root ''golden chords'' are irrational fractions that are functions of {{radic|5}}. They exemplify that the [[W:golden ratio|golden ratio]] &lt;big&gt;𝜙&lt;/big&gt; {{=}} {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618 is a circle ratio related to &lt;big&gt;𝜋&lt;/big&gt;:&lt;br&gt;
: {{sfrac|𝜋|5}} = arccos (𝜙/2)
is one decagon edge, the 𝚽 = {{radic|0.𝚫}} = {{radic|0.382~}} ≈ 0.618 chord.
Reciprocally, in this function discovered by Robert Everest expressing &lt;big&gt;𝜙&lt;/big&gt; as a function of &lt;big&gt;𝜋&lt;/big&gt; and the numbers 1, 2, 3 and 5 of the Fibonacci series:&lt;br&gt;
: &lt;big&gt;𝜙&lt;/big&gt; = 1 – 2 cos ({{sfrac|3𝜋|5}})
{{sfrac|3𝜋|5}} is the arc length of the &lt;big&gt;𝜙&lt;/big&gt; = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618 chord.|name=golden chords|group=}} including the two [[W:golden section|golden section]]s of {{radic|5}}, as shown in the diagram.{{Efn|The 600-cell edges are decagon edges of length {{radic|0.𝚫}}, which is 𝚽, the ''smaller'' golden section of {{radic|5}}; the edges are in the inverse [[W:golden ratio|golden ratio]] {{sfrac|1|φ}} to the {{radic|1}} hexagon chords (the 24-cell edges).
The other fractional-root chords exhibit golden relationships as well. The chord of length {{radic|1.𝚫}} is a pentagon edge.
The next fractional-root chord is a decagon diagonal of length {{radic|2.𝚽}} which is &lt;big&gt;𝜙&lt;/big&gt;, the ''larger'' golden section of {{radic|5}}; it is in the golden ratio{{Efn|name=golden chords|group=}} to the {{radic|1}} chord (and the radius).{{Efn|Notice in the diagram how the &lt;big&gt;𝜙&lt;/big&gt; chord (the ''larger'' golden section) sums with the adjacent 𝚽 edge (the ''smaller'' golden section) to {{radic|5}}, as if together they were a {{radic|5}} chord bent to fit inside the {{radic|4}} diameter.}}
The last fractional-root chord is the pentagon diagonal of length {{radic|3.𝚽}}.
The [[W:Pentagon#Side length is given|diagonal of a regular pentagon]] is always in the golden ratio to its edge, and indeed &lt;big&gt;𝜙&lt;/big&gt;{{radic|1.𝚫}} is {{radic|3.𝚽}}.|name=fractional root chords|group=}}

....


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== ''n''-leviathon ===
...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of its &lt;math&gt;10^2&lt;/math&gt; disjoint instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction, before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]].{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}} .... It is only because she did not have the 11-cell to start with that she did not discover the regular 137-cell {3, 3, ..} by her method.

Between Alicia Boole Stott and her identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and her identification of symmetry groups with conservation laws (in her 1st theorem), we owe the deepest mathematics of geometry and physics originally to two pioneering women, colleagues of their contemporaries the greatest men of the academy, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just an [[W:abstract polytope|abstract 4-polytope]], not just a singleton convex 4-polytope, and not just a honeycomb.{{Sfn|Coxeter|1970|loc=''Twisted Honeycombs''}} The 11-point (11-cell) has a concrete ...regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}&lt;nowiki/&gt;} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, 3, 3} as the 137-point (137-cell).

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 11-cell, 137-cell and 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=''Clifford Spinors and Root System Induction: H4 and the Grand Antiprism''}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=Apology}}

== Build with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Duveneck|1978|loc=Carl Jung, quoted in ''Life on Two Levels''|p=ii|ps=.{{Sfn|Jung|1961|ps=: &quot;The best of truths is of no use unless it has become one's most personal inner experience. It is the duty of everyone who takes a solitary path to share with society what he finds on his journey of discovery.&quot;}}}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1954|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius={{radic|2}}}}

== Acknowledgements ==
...

== Notes ==
{{Regular convex 4-polytopes Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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{{Refend}}</text>
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      <text bytes="208291" sha1="gdp20bzb90wbah5xlyo9ojk1wdubom2" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|June 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a regular 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the &lt;s&gt;120-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 5, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 11 [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] cells. Eleven 11-cells (plural) are the 137-cell, the &lt;s&gt;600-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 3, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 137 [[W:Rhombic triacontahedron|triacontahedron]] cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=''Regularity of Graphs, Complexes and Designs''}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=''A Symmetrical Arrangement of Eleven Hemi-Icosahedra''}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]]
The 4-dimensional regular polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:Simplex|simplex]] (called the [[5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:Orthoplex|orthoplex]] (called the [[16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular convex 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the [[w:Tesseract|16-point (8-cell) tesseract]], the 4-cubic, which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found within any of the other regular convex 4-polytopes, except in the largest and most complex one, the [[120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). However, we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the 16-cell is their very starting point, and the most frequently used tool in the box.

== 5-cells and hemi-icosahedra in the 11-cell ==
[[File:Symmetrical_5-set_Venn_diagram.svg|thumb|The 5-point (10-face) regular 5-cell (the regular 4-simplex). Grünbaum's rotationally symmetrical 5-set Venn diagram{{Sfn|Grünbaum|1975|loc=''Rotationally symmetrical 5-set Venn diagram'', Fig 1 (e)|ps=; partitions the individual elements of the 5-cell.}} is an illustration of the 5-cell labeling each of its &lt;math&gt;2^5&lt;/math&gt; elements.{{Sfn|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled; image includes the Python code to render it, optimising for maximum area of the smallest regions.}}]]
[[File:Hemi-icosahedron.png|thumb|The 6-point (10 face) [[W:hemi-icosahedron|hemi-icosahedron]], an abstraction of the regular icosahedron, has half as many faces, edges and vertices. Each element of the abstract polyhedron represents two real elements found in different places in the 11-cell's realization.{{Sfn|Ruen: hemi-icosahedron|2007}}]]
The most apparent relationship between the pentad 5-cell and the hexad hemi-icosahedron is that they both have 10 triangular faces. When we see a facet congruence between a 4-polytope and a 3-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 6-point 5-simplex. It is also the same size as the 6-point 3-orthoplex; thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no chord congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals, pentad and hexad, who are famously stand-offish with each other. Interesting!

The cell of the 11-cell is an abstract hexad hemi-icosahedron containing pentads in its interior, most handsomely illustrated by Séquin.{{Sfn|Séquin &amp; Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing rings of polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The 11-cell and 5-cell of radius {{radic|2}} both have edge length {{radic|5}}, so their triangular faces are congruent. The 11-cell's 11 hemi-icosahedral cells have 10 triangle faces each, and a 5-cell is behind each face. Only 3 of the 5-cell's 5 vertices are visible on the surface of the hemi-icosahedron as the exposed face, but its other two vertices are visible on the opposite side of the hemi-icosahedron as an exposed edge.{{Sfn|Coxeter|1984|p=110|loc=§6. The Petrie polygon [of the 11-cell]|ps=; &quot;We may reasonably call this edge and face ''opposites''. It is easy to find the face opposite to a given edge by looking at the faces to which a given edge belongs. ... Conversely, given a face, we can find the opposite edge by seeing which vertices belong to neither of the hemi-icosahedra which share that face. The ten edges opposite to the ten faces of one hemi-icosahedron are the edges of the complementary &lt;math&gt;a_4&lt;/math&gt; [4-simplex], that is, the joins of all pairs of the five vertices [of the 11-cell] not belonging to the given hemi-icosahedron.&quot;}} The other parts of the 5-cell (its other 6 edges, its other 9 faces and its 5 tetrahedral cells) lie outside the hyperplane occupied by the hemi-icosahedron.

There are 11 disjoint 5-cells in each 11-cell, which is comprised of 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. The real 11-cell is harder to see than the abstract 11-cell that represents it, because the real hemi-icosahedron that the abstract hemi-icosahedron represents is hard to see, but seeing these objects will be easier once we have identified the real hemi-icosahedron and exactly where the 11-cell's real elements reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each hemi-icosahedron's 10 faces come from 10 disjoint 5-cells, and there is just one 5-cell with which it does not share a face.{{Efn|As Coxeter observes (in the previous citation), that unrepresented 5-point (5-cell) is the other 5 vertices of the 11-point (11-cell) that are not vertices of this 6-point (hemi-icosahedron), the hemi-icosahedron's disjoint complement.}} Each 5-cell has 10 faces from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share a face.

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in two different planes which are completely orthogonal to each other. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must be sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness: 8 concentric hulls|2022|loc=Hull #8 (lower right)|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell of the 11-cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Compounds in the 120-cell|§Compounds in the 120-cell]]'' that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software which uses Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s to render the polyhedra which are found in the interior of ''n''-dimensional polytopes.{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the tetrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is a real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (Hull #8) is a concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this {12} irregular great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing {6} irregular great hexagons (truncated triangles) of alternating 120-cell edges and 5-cell edges. The great dodecagon is a great circle of the hemi-icosahedral cell, and also a great circle of the 120-cell, because unlike the cells of most 4-polytopes, the 11-cell's cells are embedded in 4-space at the exact center of the 4-polytope, making them inscribed 4-polytopes themselves.{{Efn|name=Embedding point}} There are 25 great dodecagons and 120 5-cell edges in Moxness's 60-point (Hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (Hull #8) polyhedron altogether.

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point (Hull #8) is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] similar to the one from the catalog shown here,{{Sfn|Piesk: Rhombicosidodecahedron|2018}} but a slightly shallower truncation with smaller red pentagons (120-cell faces) and narrower rhombs. Rhombicosidodecahedra are made by truncating the [[W:Rhombic triacontahedron|rhombic triacontahedron]], which is the 30-sided polyhedron with only one kind of face, the dual of the 30-point icosidodecahedron. The 120-cell contains 120 of these 11-cell cells.]]
Moxness's 60-point (Hull #8) is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (Hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|The 5-point (regular 5-cell) can be another abstraction of Moxness's 60-point (Hull #8), 12-vertices-into-1, but it is not strictly the result of a contraction operation as described by Boole Stott, since she did not apply her expansion and contraction operations to uniform polytopes with more than one edge length.}} In fact Moxness was not the first to find a rhombicosidodecahedron in the 120-cell; Alicia Boole Stott was. It is the 6th section of {5,3,3} beginning with a cell, which Boole Stott identified as the ''e&lt;sub&gt;2&lt;/sub&gt; expansion'' of the icosahedron (or equivalently of its dual polyhedron the dodecahedron).{{Sfn|Boole Stott|1910|loc=§Examples of the e&lt;sub&gt;2&lt;/sub&gt; expansion|p=7}} But that rhombicosidodecahedron identified by Boole Stott is not Moxness's Hull #8: rather, it is the semi-regular Archimedean solid, with a single edge length and square faces. Moxness's Hull #8 with its two distinct edge lengths and rectangular faces is Coxeter's 8th section of {5,3,3} beginning with a cell, which is missing from the sections illustrated by Boole Stott.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than tetrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. The 120-cell contains 120 of Moxness's 60-point (11-cell cells). They are non-disjoint, with 12 60-points sharing each vertex.

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (Hull #8) is the real hemi-icosahedron cell, with 120 5-cell edges in it as chords lurking under its surface. The rhombicosadodecahedron has enough of those 5-cell edges in its interior to make 20 disjoint tetrahedral cells which belong to 20 distinct 5-cells.

The 600-point (120-cell) may be constructed as a semi-regular honeycomb of two kinds of cells, the 120 non-disjoint 60-point (Moxness's rhombicosadodecahedra) and their neighbors the 120 disjoint 5-point (5-cells), cell-bonded together by shared tetrahedral cells of edge-length {{radic|5}}. Each rhombicosadodecahedron is bonded to 20 islanded 5-cells, sharing with each a 5-cell tetrahedron (4 of its interior 5-cell faces, and 6 of its interior 5-cell edges). The rhombicosadodecahedron is also bonded directly to 10 other rhombicosadodecahedra (since each 11-cell has 11 rhombicosadodecahedron cells), again not by sharing one of its exterior faces, but this time by sharing an interior face, a {{radic|6}} triangle which lies in a central plane.{{Efn|The {{radic|6}} chord is another hidden chord lurking beneath the 60-point rhombicosadodecahedron's surface, like the {{radic|5}} 5-cell edge. Both these hidden chords lie on the same great circle with the 120-cell edges (the pentagon face edges). Three {{radic|6}} chords make a triangle which lies in a central plane. This {{radic|6}} great triangle must not be confused with the {{radic|5}} face triangle of the 5-cell, which does not lie in a central plane.}} The 120 rhombicosadodecahedra are also bonded directly together at ''all'' their exterior faces: at their pentagon faces (which are 120-cell faces of dodecahedral cells); at their small triangular faces (which are 600-cell faces of small tetrahedral cells, 10 of which are inscribed in each dodecahedral cell); and at their rectangular faces (which are faces of 10-point [[w:Pentagonal_prism|pentagonal prism]] cells, where the pentagonal prism is a column inscribed in a dodecahedral cell, joining its opposite pentagon faces). The 120 rhombicosadodecahedra are bonded directly to each other at their own exterior faces; no dodecahedra, small tetrahedra, or pentagonal prisms lie between them, and those objects are not cells of this honeycomb; we mention them here only to help locate the rhombicosadodecahedra within the 120-cell. The only objects which fill holes between the rhombicosadodecahedra are the 120 islanded 5-cells.{{Efn|This honeycomb of the 120-cell resembles the [[600-cell#Icosahedra|honeycomb of the 600-cell consisting of 24 face-bonded regular icosahedra]] with 24 clusters of 5 tetrahedra filling the holes between them. In the 120-cell there are 5 times as many of everything as in the 600-cell, so there are 120 (rhombicosadodecahedron) icosahedral cells with 600 tetrahedra filling the holes between them.}}

It is possible for rhombicosadodecahedra to meet each other at more than one kind of exterior face because they are not meeting there as cells of the same 11-cell; they belong to different 11-cells. The 11 rhombicosadodecahedra in the same 11-cell do not meet at their own exterior faces, they meet at their interior {{radic|6}} triangle faces. As a cellular 4-polytope, the 11-cell is inside-out. Its cells meet each other at interior planes, sharing interior {{radic|6}} triangle faces, instead of in the usual way that 4-polytopes' cells share an exterior face. It is a deeper kind of intertwining to be sure, but the 11 rhombicosadodecahedra in the same 11-cell are still volumetrically disjoint cells, and the 11-cell is a perfectly real 4-polytope with real cells. It is a semi-regular 4-polytope, because it has two different kinds of cells: 11 rhombicosadodecahedra and 11 5-cells. The 11 5-cells are islanded (not bound to each other), and the 11 rhombicosadodecahedra are bound both to 5-cells (at {{radic|5}} 5-cell faces) and to each other (at {{radic|6}} central triangle faces). In the section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#What's in the box|§What's in the box]]'', below, we will look closely at the relation between the {{radic|5}} chord and the {{radic|6}} chord, which is the relation between the pentad and the hexad, and at the general phenomenon of hemi-polytopes which meet each other at a central plane. Then in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Cell rings of the 11-cells|§Cell rings of the 11-cells]]'' we will look at the two-cell cellular structure of the 11-cell in more detail.

Considered as a single complex of overlapping cells rather than as distinct honeycombs of volumetrically disjoint cells, the 120-cell contains 60-point (rhombicosadodecahedral cells), islanded 5-point (5-cells), 20-point (dodecahedral cells), and 10-point ([[w:Pentagonal_prism|pentagonal prism]] cells). The 60-points meet each other in pairs at their pentagon faces, meet regular dodecahedra at those pentagon faces also, meet each other in quads at their 600-cell faces where they share an interior 4-point (600-cell tetrahedral cell) and its four faces (just as they share a larger interior 5-cell tetrahedron and its faces), and meet each other and the 10-point (pentagonal prisms) at their rectangle faces. We might try to imagine such a complex with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with parts truncated from 30 neighboring truncated dodecahedra pentagonal prisms). There are 120 of each of these 3 kinds of dodecahedron. 

Moxness's 60-point (rhombicosadodecahedron cell) is a central object of the 120-cell, and contains many wonders. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra by expansion and contraction operations, as Alicia Boole Stott discovered before 1910,{{Sfn|Polo-Blanco: ''Theory and history of geometric models of Alicia Boole Stott''|2007|loc=§5.3.2 1910 paper on semi-regular polytopes|pp=152-158|ps=; summarizes Boole Stott's method and results from {{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}}, including her tables of expansion-contraction dimensional analogies and a few of her illustrations.}} and those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, rediscovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter &amp; Petrie|1938|p=4|loc=''[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]''}}|name=Concentric polyhedra}} Before we move on from the 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice one more thing about the hidden {{radic|5}} chords lurking below its surface. The 12 little pentagon faces are spanned by sets of 6 {{radic|5}} chords (5-cell edges). The 6 chords of each set are disjoint (they don't touch or form 5-cell faces), but they are symmetrically arranged in 3 parallel pairs, {{radic|3}} apart, which lie in 3 orthogonal central planes. It turns out they are the 6 reflex edges of a 12-point non-convex polyhedron called the [[w:Jessen's_icosahedron|Jessen's icosahedron]], inscribed in the 60-point cell. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again and work our way outwards, before eventually in ''[[#The perfection of Fuller's cyclic design|§The perfection of Fuller's cyclic design]]'' we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cell)s ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks, showing its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Pentahemidemicube.png|120px]]
|[[File:Tetrahemihexahedron_rotation.gif|120px]]
|[[File:Pentahemicosahedron.png|120px]]
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts needed to build the 120-cell (the picture on the cover of the box), and everything else that's invisible in that picture which fits inside the 120-cell: all the astonishing regular 4-polytopes. The pentad building block is half a pair of orthogonal 5-point (regular 5-cells), the hexad building block is half an 8-point (16-cell), and the heptad building block is a quarter of an 11-point (11-cell). Everything can be built from these blocks by embedding them at the center of the 3-sphere, inscribing them as 4-polytopes to make compounds as with the 16-cell building block, rather than by assembling them face-to-face as 3-polytopes to make honeycombs in which they are cells or vertex figures.{{Efn|name=Embedding point}}

The pentad building block is the 5-point '''pentahemidemicube''', an abstract polyhedron with 10 equilateral triangle faces, 10 {{radic|5}} edges, and 5 vertices.{{Sfn|Christie: Pentahemidemicube|2024|loc=File:Pentahemidemicube.png|ps=; &quot;The 5-point ''pentahemidemicube'' is an abstract polyhedron with 10 equilateral triangle faces, 10 edges, and 5 vertices. It has a real presentation as the regular 5-cell.&quot;}} It has a real presentation as the 5-point (regular 5-cell 4-polytope), a regular tetrahedron expanded by addition of a fifth vertex equidistant from all the others in a fourth dimension. The pentad block is a tetrahedral pyramid (a 5-cell) inscribed in a cube: a tetrahedron inscribed in a cube (a demicube) expanded by addition of a fifth vertex, one of the cube's other 4 vertices. Each pentad (5-cell) building block lies in the 120-cell orthogonal to three other pentad (5-cell) building blocks. They are four completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges. Each regular 5-cell in the 120-cell is a pentad (pentahemidemicube) building block, in one of 30 orthogonal sets of 4. There are 120 pentad building blocks in the box.

Every vertex of the 16-cell, and therefore every vertex of the 8-cell, 24-cell, 600-cell and 120-cell, is surrounded by a 6-point (octahedral vertex figure). The hexad building block is that 6-point octahedron embedded at the center of the 4-polytope instead of at a vertex, that is, inscribed in the 3-sphere as a 4-polytope, the 6-point '''[[w:Tetrahemihexahedron|tetrahemihexahedron]]''' or '''hemi-cuboctahedron'''.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; &quot;The tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U&lt;sub&gt;4&lt;/sub&gt;. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices.&quot;}} The hexads have 4 red triangle 5-cell faces, bounded by 12 {{radic|5}} edges. Their 3 yellow square faces lie in central planes, because the building block is an abstract 4-polytope. The hexad is an irregular abstract polyhedron that also has 6 {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

Two orthogonal 5-point (pentad) building blocks intersect in a 4-point (inscribed tetrahedron) to form a 6-point (hexad) building block. Two orthogonal 6-point (hexad) building blocks intersect in a 4-point (central square) to form an 8-point (16-cell) building block, the most usable building block, from which everything larger can be built by compounding.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 120 pentads to pair with a completely orthogonal hexad, and although there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed) they only result in 120 pairs two different ways, so there are 120 11-cells. The 11-cell is 6 pentads' and 5 hexads' smallest joint compound, and the 120-cell is their largest, 120 pentads and 100 hexads.

The heptad{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''|pages=22–31}} building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|4}} edges and 9 {{radic|5}} edges), and 7 vertices.{{Sfn|Christie: Pentahemicosahedron|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;}} It is a compound of a 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]) and a 6-point (hexad [[W:hemi-icosahedron|hemi-icosahedron]]) intersecting in a 4-point (tetrahedron), that is, of three orthogonal 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]&lt;nowiki/&gt;s) intersecting in their common 4-point (tetrahedron). It is also the abstract cube contracted by removal of one vertex, which reduces one of the cube's two inscribed tetrahedra (demicubes) to a single remaining triangular face. As with all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Four &lt;small&gt;&lt;math&gt;(11-7)&lt;/math&gt;&lt;/small&gt; orthogonal 7-point (pentahemicosahedron) blocks can be joined at a common Petrie polygon (a skew hexagon which contains their orthogonal 6-edge paths) to construct a real 11-point ([[W:11-cell|11-cell]] semi-regular 4-polytope), which magically contains six &lt;small&gt;&lt;math&gt;(11-5)&lt;/math&gt;&lt;/small&gt; 5-point (1-pentad regular decahedra) as cells and five &lt;small&gt;&lt;math&gt;(11-6)&lt;/math&gt;&lt;/small&gt; 12-point (2-hexad semi-regular cuboctahedra) as cells. There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like [[W:Euclid's postulates|Euclid's postulates]], and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as honeycombs of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles. It possesses the complete symmetry of the polytope without any redundancy, because it contains one of each of the polytope's characteristic root elements.{{Efn|A [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be partitioned radially by its planes of symmetry (Coxeter's &quot;reflecting circles&quot;) into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated rotationally to generate its regular 4-polytope because it is the complete [[W:gene|gene]] for it, containing all of its elements and capturing all of its symmetry without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring that triggers the reflections.|name=Characteristic orthoscheme}} It is the gene for the polytope, which can be replicated to construct the polytope.

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its &lt;math&gt;A_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the regular compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;small&gt;&lt;math&gt;120\times 120 = 14400&lt;/math&gt;&lt;/small&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[16--cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its &lt;math&gt;B_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 384. The 120-cell is the convex hull of the regular compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 24-point (24-cell) is subdivided into 1152 instances of its [[24-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;F_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 1152. The 120-cell is the convex hull of the regular compound of 25 disjoint 24-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;25\times 1152 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has three equivalent origins in distinct root systems, though most are obviously constructed from one root, and not so obviously from the others. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything larger than the pentad can be constructed by compounding hexads, and we shall see that everything as large as a 24-cell has a construction from heptads which are a product of a pentad and a hexad. Surely every uniform polytope can be constructed by some function of expansions and contractions from any of three root systems, however indirect the recipe. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes, related by an equals sign between them, then give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics by [[W:Noether's theorem|Noether's theorem]].

=== Pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 octahedral cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell contains the 600 octahedral cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It also contains the 600 cubical cells of its 75 8-cells. It is a 600-cell in three ways, although the octahedra and cubes are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell and the 8-cell, respectively.|name=600 cells three ways}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (1-pentad 5-cell), out of the 8-point (2-hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (1-pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope that contains 5-point (1-pentad 5-cell) building blocks beside the largest one, it just isn't a ''regular'' 4-polytope. It is semi-regular with two kinds of cells, the 4-point (regular tetrahedron) and the 12-point (truncated cuboctahedron), which have just one kind of regular face between them, a 5-cell face. That semi-regular 4-polytope is the 11-cell.

We said earlier that the 11-cell had 11 disjoint 5-cells and 11 hemi-icosahedral cells in it, and that its construction from them might be hard to see, but now we are going to account numerically, at least, for how it can have 6 pentad cells plus 5 hexad cells magically adding to 11 of the same kind of elevenad cell. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, 11. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (5-cell regular decahedron) cells, and the 12 6-point (hexad) building blocks into 100 12-point (regular icosahedron) cells, and then magically adding one whole 5-point (5-cell regular decahedron) cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 12-point (regular icosahedron) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such (cell bundles of 6 + 5 = 11 cells), with each 11-cell bundle containing 11 of both kinds of cell. Very, very strange!{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden'': §II To Add is to Multiply|p=38|ps=; the sequence of regular polygons with harmonious proportions beginning {5}, {7}, {9}, {11} as a family of natural numbers; &quot;The numbers in this optimal family have remarkable properties [that allow] similar figures to be arranged easily or allow a figure to be dissected into a set of similar figures. This is the origin of the Greek idea of geometric progression — that ''multiplication can be accomplished instead by addition''.... For instance, it is well known that the square of the golden ratio is one more than itself (written above as 𝜙&lt;sup&gt;2&lt;/sup&gt; = 𝜙 + 1) and one less than 𝜙 is its reciprocal (1/𝜙 = 𝜙 - 1). The first relation equates multiplication with addition, while the second accomplishes division through subtraction. [These are pentagon {5} bisectional golden ratios, but] the heptagonal [{7} tri-sectional] ratios also behave in surprising ways — adding to multiply, subtracting to divide.... There are analogous constructions for [nonagon {9}] quadrisectional and [hendecagon {11}] pentasectional systems.&quot;}}

The 120-cell has three interchangeable cyclic expansion-contraction cycles, the pentad, hexad and heptad cyclic designs, rooted in the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems, respectively. Each of them constructs the whole sequence of regular 4-polytopes from a single building block. The key to all three cycles is that 5 + 6 = 11, and there is an 11-point quasi-regular 4-polytope in the sequence of convex 4-polytopes. It lies between the 8-point (16-cell) and the 16-point (8-cell), and is itself a &lt;math&gt;B_4&lt;/math&gt; polytope. Its role is to be a unified single building block common to all three cyclic expansion-contraction cycles. It is involved with the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems and expresses in one building block their relations to each other.

We have already seen how the hexad building cycle works. It is plain compounding of 4-orthoplexes at the beginning, increasing to densely intricate non-disjoint symmetry toward the end, making it difficult to see why it works once past the 24-cell. This is how the build works, in terms of sixes. The 16-cell is a 2-hexad (3 pairs of completely orthogonal great square planes). The 24-cell is 3 16-cells and a 4-hexad, 4 different ways: 16 non-disjoint hexads (great hexagon planes). The 120-point (600-cell) is 25 24-cells and a 20-hexad, 10 different ways: 200 non-disjoint hexads (great hexagon planes). The 600-point (120-cell) is 225 (9 times 25) 24-cells. ....

The pentad building cycle doesn't do plain compounding to regular 4-polytopes until the last one, the 120-cell. Fives don't even appear as pentagons until after the 24-cell. We have seen how there are 11 5-cells in the quasi-regular 11-cell, which occurs before the 8-cell, and also how everything can be compounded from irregular 5-cells (4-orthoschemes specialized for each 4-polytope). It is still not evident how a pentad cycle can work to build all the regular 4-polytopes from the regular 4-simplex. This is how is how the build works, in terms of fives. ....

This is how the build works, in terms of elevens. The 11-cell is 4 orthogonal heptads (intersecting in their skew hexagon Petrie polygon). The 24-cell is also 4 heptads (intersecting in 4 orthogonal digon axes, so each heptad contributes 6 unique points to a single fibration of 24 points). Heptads are quarter elevenads, so the 11-cell and 24-cell are both 1 elevenad. ....

See the tabular sequence of convex 4-polytopes in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Build with the blocks|§Build with the blocks]]'', below: table rows ''Inscribed, Pentad, Hexad, Heptad''.

=== 11 of them ===
There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of eleven 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum eleven of them. There is only a single Hopf fibration of the 11 great circles of an 11-cell. 

The 5-cell and the 11-cell are limit cases. The 5-cells are completely disjoint, but the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 10 other instances of itself.

== Cell rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer &amp; Segerman|2013|loc=''Puzzling the 120-cell''}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his section of the Wikipedia article on the 120-cell entitled ''[[120-cell#Visualization|§Visualization]]''.{{Sfn|Ruen &amp; Goucher et al. eds. 120-cell|2024|loc=Goucher's ''[[120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious place to look for the 11-cell's Hopf map is in Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other place it could be hiding. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The quasi-regular convex 11-cell|§The quasi-regular convex 11-cell]]'', and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen: triacontagon|2011|loc=''Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them except his illustrations. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Four orthogonal projections of the 120-cell to a central plane, selected from the table of 3 regular and 11 compound 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|100 icosahedra&lt;br&gt;120 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|5 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 icosahedra&lt;br&gt;6 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|120
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|4
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|50
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|7.2°
|align=center|36°
|align=center|6°
|align=center|7.2°
|}

....

Here we summarize our findings on the 11-cells so far, including in advance some results to be obtained in the next section two sections.

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section....

== The 15 major chords ring everything ==
The natural numbers are each a distinct flavor.

Each integer has its own distinct taste. The first six integers {0..5} are the six strongest, most familiar tastes in nature, instantly distinguished and utterly unique.

The seventh flavor {6} is the first ratio of two of the others. Thereafter follows in strange sequence all the ratios of all the flavors, each marriage of every two flavors, most of which are fractions &lt;math&gt;\{k/d\}&lt;/math&gt;, but some of which are integers &lt;math&gt;\{k\}&lt;/math&gt;.

Some of these ratios are successful flavors in their own right, as distinct and inimitable as the marriage of chocolate and oranges, but the marriage of many flavor pairs is mud. It is a rule of thumb among flavor specialists that nearly any taste can be successfully alloyed with either onions or chocolate, but the direct marriage of these two flavors is notably unsuccessful. Some flavors make famous pairings or infamous ones, and subtle new pairings are always to be discovered that work surprisingly well for some purpose, but the majority of flavor marriages are undistinguished compounds.

In polytope geometry, each chord of a polytope is both is a distinct 1-dimensional object, a chord of the sphere of a distinct length &lt;math&gt;l&lt;/math&gt;, and a distinct natural number &lt;math&gt;h&lt;/math&gt;, a unique flavor. If the polytope is regular, it is a noteworthy and successful flavor. The chord's length &lt;math&gt;l&lt;/math&gt; is a square root, related to the natural number &lt;math&gt;h = k/d&lt;/math&gt; and to a set of polytopes they both represent, by a formula discovered by Steinbach.{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden''; Figure 5. Optimal sections and proportions|p=37|ps=; the regular polygons {5}, {7}, {9} and {11} with their diagonals define respectively: {5} the golden bisection proportional to 𝜙; {7} an analogous trisection; {9} an analogous quadrasection; {11} an analogous pentasection.}} The chord length &lt;math&gt;l&lt;/math&gt; is related to the number of sides of the regular polygon formed by the chord (its numerator &lt;math&gt;k&lt;/math&gt;), and to the winding number or density of its regular skew Petrie polygon (its denominator &lt;math&gt;d&lt;/math&gt;).

The regular 4-polytopes do not contain an infinite variety of these chord flavors, but only 30 of them: only 15 unique pairs of 180° degree complementary chords, which combine to make every regular ''n''-dimensional thing found in nature.{{Efn|The 15 ''major chords'' suffice to construct all the regular ''n''-polytopes. Their 180° complements include 15 ''minor chords'' which occur only in the 120-cell, not in any smaller regular polytope.{{Efn|name=Major and minor chords}} Including the complement of the 180° {{radic|4}} digon {2} chord (the {0} chord {{radic|0}}), and the skew variant of the 90° {{radic|2}} square {4} chord (the {30/7} chord), there are 32 distinct natural numbers which are all the chords of all the regular polytopes in all dimensions.|name=32 chords}} All the regular polytopes (including the largest one on the cover of the box) can be built not only from a box containing a number of blocks of 3 shapes glued together by gravity, they can be also be built from a box containing a number of sticks of 15 lengths glued together by rubber joints.

[[File:15 major chords.png|500px|thumb|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the others occur only in the interior of 120-cell, not as edges of any of the regular 4-polytopes. There are 30 distinct 4-space chordal distances between vertices within the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to identify any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=Major and minor chords}} chords are properly labelled #1 — #15 in the order they fan out by size.{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''; Figure 3|ps=; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with a &quot;fan of chords&quot; diagram like the one here.|p=23}} They join vertex pairs which are 1 — 15 edges apart on a &lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon. A #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; chord forms &lt;small&gt;&lt;math&gt;f+1&lt;/math&gt;&lt;/small&gt; regular geodesic polygons which include a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; disjoint regular {{nowrap|&lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt;-gons}} (e.g. &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; a compound of 5 disjoint hexagons) plus a single [[W:Triacontagon#Triacontagram|regular skew {{nowrap|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram}}]] that wraps around the sphere &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times before closing its circle. Notice the maverick #11 chord, with its non-conforming origin point for its {{nowrap|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;}} polygram.]]

These 15 chords (natural numbers) include the integers 2, 3, 4, 5, 6, 10, 11, 15 and 30. The others are the fractions 5/2, 10/3, 13/7, 15/2, 15/4, and 15/7. Represented among the chord lengths (the square roots) measured in ''unit radii'' (not in {{radic|2}} radii, as metrics are often given elsewhere in this article) are the {{background color|paleturquoise|roots of the first 4 positive integers {{radic|1}}, {{radic|2}}, {{radic|3}} and {{radic|4}}}}, the {{background color|#FFCCCC|root of the very distinguished rational fraction {{radic|5/2}}}}, and the roots of various distinguished irrational fractions as Steinbach discovered: for example, 4 of them are {{background color|yellow|roots of golden numbers (a small integer ± 𝜙)}}. The 15 chords each form their regular polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, which is just &lt;small&gt;&lt;math&gt;\{k\}&lt;/math&gt;&lt;/small&gt; in 8 cases. Arranged in order #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; from shortest edge-length square root to longest, the #1 — #15 chords bridge vertex pairs which are &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; vertices apart on a [[W:Triacontagon#Petrie polygons|&lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon]]. Each chord makes a [[W:Triacontagon#Triacontagram|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; polygram]] of its edge length, a compound of the smaller &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. The 15 distinct polygons form a sequence you would not anticipate exactly unless you have Steinbach's formula:

{| style=&quot;background-color:transparent; white-space:nowrap;&quot;
|-
| {30/1} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|.073}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{30} 
| 120-cell edges
|-
| {30/2}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.191}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15}
|- style=&quot;background: yellow;&quot;|
| {30/3}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.382}} 
| &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi}&lt;/math&gt;&lt;/small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{10}''' 
| 600-cell edges
|-
| {30/4} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.573}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/2}
|- style=&quot;background: paleturquoise;&quot;|
| {30/5}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|1}}'''
| &lt;math&gt;1&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{6}''' 
| 24-cell, 8-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/6}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|1.382}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{5}'''
|- style=&quot;background: paleturquoise;&quot;|
| {30/7}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|2}}'''
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{4}''' 
| 16-cell edges
|- style=&quot;background: #FFCCCC;&quot;|
| {30/8}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.5}} 
| &lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;&lt;small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/4} 
| 5-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/9}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.618}} 
| &lt;math&gt;\phi&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{10/3}
|- style=&quot;background: paleturquoise;&quot;|
| {30/10}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|3}}''' 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{3}'''
|-
| {30/11}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.426}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{11/11}
|- style=&quot;background: yellow;&quot;|
| {30/12}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.618}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{5/2}
|-
| {30/13}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.810}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{13/7}
|-
| {30/14}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.928}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/7}
|- style=&quot;background: paleturquoise;&quot;|
| {30/15}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|4}}''' 
| &lt;math&gt;2&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{2}'''
|}

{| class=wikitable style=&quot;white-space:nowrap; float:right;&quot;
!colspan=6|Natural number &lt;small&gt;&lt;math&gt;h=\{k/d\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Fibers&lt;br&gt;&lt;small&gt;&lt;math&gt;f\{h\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Cells&lt;br&gt;&lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|4-tope&lt;br&gt;&lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Verts&lt;br&gt;&lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|[[W:Triacontagon#Triacontagram|30-gram]]&lt;br&gt;&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;
|-
!Chord
!Arc
!colspan=2|&lt;small&gt;&lt;math&gt;l/\sqrt{1}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;l/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;h&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#1
|15.5~°
|{{radic|0.073}}
|0.270
|{{radic|0.146}}
|&lt;small&gt;&lt;math&gt;30&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1\{30\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;5\{3,3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/1\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#2
|25.2~°
|{{radic|0.191}}
|0.437
|{{radic|0.382}}
|&lt;small&gt;&lt;math&gt;15&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;..\{3,3\}..&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/2\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#3
|36°
|{{radic|0.382}}
|0.618
|{{radic|0.764}}
|&lt;small&gt;&lt;math&gt;10&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;30\{3,3\}20&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/3\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#4
|44.5~°
|{{radic|0.573}}
|0.757
|{{radic|1.146}}
|&lt;small&gt;&lt;math&gt;15/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/4\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5
|60°
|{{radic|1}}
|1
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;6&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/5\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#6
|72°
|{{radic|1.382}}
|1.175
|{{radic|2.624}}
|&lt;small&gt;&lt;math&gt;5&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}} 
|&lt;small&gt;&lt;math&gt;10\{5,3\}12&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/6\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#7
|90°
|{{radic|2}}
|1.414
|{{radic|4}}
|&lt;small&gt;&lt;math&gt;4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7\{4\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;4\{4,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;9\{3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/7\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: #FFCCCC;&quot;|
|#8
|104.5~°
|{{radic|2.5}}
|1.581
|{{radic|5}}
|&lt;small&gt;&lt;math&gt;15/4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/4\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}} 
|&lt;small&gt;&lt;math&gt;15\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/8\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#9
|108°
|{{radic|2.618}}
|1.618
|{{radic|5.236}}
|&lt;small&gt;&lt;math&gt;10/3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10/3\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/9\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#10
|120°
|{{radic|3}}
|1.732
|{{radic|6}}
|&lt;small&gt;&lt;math&gt;3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;10\{3\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/10\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#11
|135.5~°
|{{radic|3.426}}
|1.851~
|{{radic|6.852}}
|&lt;small&gt;&lt;math&gt;11/11&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;11\{11/11\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;11\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#12
|144°
|{{radic|3.618}}
|1.902
|{{radic|7.235}}
|&lt;small&gt;&lt;math&gt;5/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2},3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/12\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#13
|154.8~°
|{{radic|3.810}}
|1.952
|{{radic|7.621}}
|&lt;small&gt;&lt;math&gt;13/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;13\{13/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;7\{5,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/13\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#14
|164.5~°
|{{radic|3.928}}
|1.982
|{{radic|7.857}}
|&lt;small&gt;&lt;math&gt;15/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;137\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/14\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{radic|8}}
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15\{2\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐𝜙&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;8\{3,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/15\}&lt;/math&gt;&lt;/small&gt;
|}

The polygons '''{10}''', '''{6}''', '''{5}''', '''{4}''', '''{3}''', and '''{2}''' are flat polygons that each lie on a great circle in a central plane of the 3-sphere. The polygons {30}, {15}, {15/2}, {15/4}, {10/3}, {11/11}, {5/2}, {13/7}, {15/7} and all the {30/''n''} polygons are skew polygrams spiraling through curved 3-space, each successive edge in a different central plane of the 3-sphere. Projected to the 1-sphere (a circle in the plane), they are [[W:Star polygon#Regular star polygon|regular star polygrams]], the simplest being the pentagram {5/2}. In 4-space they wind around the 3-sphere more than once ''without self-intersecting'', before they close their circuit. They lie on helical geodesic circles or ''isoclines'' of circumference greater than 2𝝅''r''.

Even the 6 integer chords which form flat polygons also form skew polygrams. Each of the 15 chords occurs as a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, a fiber bundle of Clifford parallel polygons that totals 30 vertices per bundle. Each chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; also forms a single skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram that winds through another, shorter chord's bundle of parallel polygons, intersecting them in succession multiple times each, and weaving that whole bundle of parallel fibers together as the cross-thread does on a loom. The polygram cross-thread has to wind multiple times through the whole bundle (&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times around the [[W:Triacontagon#Triacontagram|30-gram]] in which each edge connects every &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt;th vertex) to complete an enumeration of all 30 vertices in the bundle. For example, the #5 chord forms hexagon &lt;small&gt;&lt;math&gt;\{6\}&lt;/math&gt;&lt;/small&gt; flat planes in parallel bundles of 5, written &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt;, and also a skew &lt;small&gt;&lt;math&gt;\{15/2\}&lt;/math&gt;&lt;/small&gt;-gram of #5 edges which winds 2 times through the #3 chord's bundle of 3 decagon &lt;small&gt;&lt;math&gt;\{10\}&lt;/math&gt;&lt;/small&gt; flat planes. Every chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; forms both its &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram and its fiber bundle of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, and the edges of one chord's parallel polygons are also the edges of its skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram weaving through some shorter chord's bundle of parallel polygons. Each numbered chord is a natural number, a ratio of two integers and their polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, either flat or skew. Each chord number defines a &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; skew polygon that is a compound bundle of the &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. All these flat or skew polygons are regular polygons made of their numerator instances of a single chord #, whose length is that square root, Steinbach's function of the natural number.

The relationship between each bundle of flat Clifford parallel great circle polygons of a distinct length, and its corresponding bundle of skew polygrams of a longer length, is precisely that of a discrete simple rotation and its corresponding discrete isoclinic rotation. For example, the flat great circle hexagons of edge length {{radic|1}} lie in the invariant rotation planes of the 24-cell's characteristic 60° isoclinic rotation, in which the vertices move on skew hexagram rotation paths of edge length {{radic|3}}. The 24-cell contains 4 bundles of 4 Clifford parallel (disjoint) hexagon central planes each. The 24 vertices of the 24-cell in each bundle are linked by rotation in the 4 hexagon invariant planes in two different ways: simply and isoclinically. In a simple rotation the vertices of a single invariant hexagon exchange places on that great circle hexagon of edge {{radic|1}}. In an isoclinic rotation in which any one of these 4 disjoint hexagon central planes is invariant, all 4 of them are necessarily invariant planes in the rotation, because they rotate in parallel. In that isoclinic rotation the vertices of all 4 hexagon invariant planes (all 24 vertices of the 24-cell) exchange places concurrently, moving along 4 disjoint helical hexagrams of edge {{radic|3}} in parallel. It is a characteristic only of this distinct rotation in hexagon invariant planes that the helical vertex path or ''isocline'' is a hexagram of period 6, edge length {{radic|3}}, and circumference 720°. Rotations in other kinds of polygon invariant planes have different characteristic isoclines. The period of their isoclinic rotation must be ''the same or greater'' than the period of their simple rotation; in most cases (unlike this one) it is greater.

The 15 chords form not only distinct fiber bundles of polygons, but also distinct bundles of polyhedra: cell rings &lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt; and compound vertex figures &lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;. These combine to form their characteristic regular 4-polytope &lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Coxeter|1973|p=269|loc=Compounds|ps=; &quot;It is remarkable that the vertices of {5, 3, 3} include the vertices of all the other fifteen regular polytopes in four dimensions.&quot;}} All these elements of a regular 4-polytope constructed from one chord are themselves regular polytopes, and each of the 16 regular 4-polytopes is a function of one characteristic chord. If the regular 4-polytope is convex, its characteristic chord is its unit-radius edge. For a non-convex regular 4-polytope, the characteristic chord is its unit-radius ''mid-edge diameter'', the distance from the midpoint of each edge to the midpoint of the completely orthogonal edge. Each of the characteristic chords (each distinct natural number) has its expression as one of the 16 regular 4-polytopes and all the products of the distinct regular 1-polytopes, 2-polytopes and 3-polytopes that construct it, and the whole distinct construction from the natural number to the 4-polytope is an [[W:SO(4)|SO(4) symmetry group]] expression of that 4-polytope's characteristic discrete [[W:SO(4)#Isoclinic rotations|isoclinic rotation]], in the invariant central planes of its characteristic chord, by the characteristic chord arc.

...demonstrate this theory by proper enumeration of the chord lengths in the table...

...color-code the △ ☐ 𝜙 ✩ symbols ... chord colors in diagram should match them ... some of the diagram colors are wrong and must be changed, but otherwise diagram is pretty good ... i think in a colored diagram all chords of the same length should have the same size line dash, the polygons are not obvious enough and the 15 fanning-out chords are readily identified by their labels...

... add back the counts of chords in the box (very small font number under each table entry), and present these two images as the front and back a box of building sticks and rubber joints...What's in the other box...

...say the rational-root major chords {{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|5/2}} are the ''n''-edges: their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge...

...abbreviated text of the [[120-cell#Relationships among interior polytopes|''§Relationships among interior polytopes'']] section from the 120-cell article...to explain the color-coding

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius=1}}

See also the elements, properties and metrics of the sequence of regular 4-polytopes summarized in tabular form in the section ''[[#Build with the blocks|§Build with the blocks]]'', below.

== Alicia Boole Stott's original formulation of dimensional analogy ==
&lt;blockquote&gt;&quot;...a method by which bodies having a certain kind of semi-regularity may be derived from regular bodies in an Euclidean space of any number of dimensions...&quot;{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''; presents cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues|pp=12-45}}&lt;/blockquote&gt;

Alica Boole Stott's 1910 paper reads like an exceptionally well-written patent application. It is a comprehensive stepwise description of her original invention, a general method of dimensional analogy between regular and semiregular polytopes of the same edge length.

[[File:The Roots of Three.png|thumb|An illustration of Boole Stott's expansion of unit-edge triangle to hexagon captures the geometric basis in 3 of the &lt;math&gt;A_n&lt;/math&gt;root system.]]

The engineering-like articulation of her new principles is entirely discrete and group-theoretic. Boole Stott repeatedly refers to sets of polytope elements as ''groups''. All her transformations are discrete instances of one or another special case of her &quot;two inverse operations which may be called ''expansion'' and ''contraction''.&quot; Hers is an exploration of ''n''-dimensional space at its most basic: space defined by the discrete objects which fill it, themselves defined directly as groups of elements with a few operations on them, which transform them into each other. This is geometry which seems to precede Euclid's postulates, and does not depend on them. Instead it corresponds directly to the deeper group theory mathematics which underlies geometry, even though Boole Stott, like the ancients, did not have that mathematics.

Boole Stott enumerated her expansion and contraction operations as transformations between regular and semi-regular polytopes of a single edge length. In her 1910 paper she did not consider any polytopes of multiple edge lengths, or expansions of the subject polytope by a distance other than that exactly required to produce new edges of the original edge length, so of course she did not enumerate Moxness's non-uniform rhombicosidodecahedron (Hull #8). This may be why section 8&lt;sub&gt;3&lt;/sub&gt; is missing from her earlier drawings of the sections of {5,3,3}; she may not have had a construction for it.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The Boole Stott expansion (or contraction) operation is as precisely defined a transformation as Coxeter's reflection, rotation and translation operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions). Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;
where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;
where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;
For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} An instance of the expansion (or contraction) operation is an application of the general operation to a particular subject polytope, operating on all of its elements symmetrically at once. The general operation is restricted, such that one may only expand (or contract) the subject polytope by pushing all its edges, faces or cells directly away from (or toward) its center, without changing the size of the moved elements. This splits each vertex into several (or coalesces several vertices into one), and has corresponding effects on the other elements, producing new edges ''of the same size'' between pairs of separated vertices (or reducing some edges to single vertices). The distance all the elements are moved away from (or toward) the center must be such that the vertices separate producing new edges of only the original edge length (or come together reducing edges to single vertices). It is permitted to combine these three operations, e.g. by pushing the edges ''and'' the faces, or by pushing all three element sets, provided the combination results in a semi-regular polytope of only one edge length. The three operations are all commutative with one another, so combined operations can be considered either as a sequence of expansions (or contractions) linking several related polytopes, or as their concurrent movements, a single transformation of the subject polytope. Since there are 8 combinations of one to three operations there are 8 polytope variants (including the original regular polytope), and all are adjacent to one another by a single transformation. The variants of the original regular polytope are named for which element set(s) are moved. By enumerating all 8 variants of expansion (and contraction) for each of the six regular convex 4-polytopes, Boole Stott produced a table of regular and semi-regular 4-polytopes divided into 12 sets of 8: six expansion sets and six contraction sets.{{Sfn|Boole Stott|1910|loc=''Table of Polytopes in S&lt;sub&gt;4&lt;/sub&gt;''; note that Boole Stott's usage of ''S&lt;sub&gt;4&lt;/sub&gt;'' denotes a Euclidean 4-space ''E&lt;sub&gt;4&lt;/sub&gt;'', or a 3-sphere space embedded in it, ''not'' a 4-sphere space embedded in a Euclidean 5-space|p=30}} Each set includes just one of the six regular convex 4-polytopes, with its distinct unit-radius edge length, and the 7 possible semi-regular expansions (or contractions) of it, which all preserve that characteristic edge length. The polytopes in each set of 8 all have the same total number of elements as their original regular polytope, but each polytope in the set has a distinct combination of cell, face, and vertex figures. The 12 sets are not completely disjoint, and even within one set the 8 operations do not necessarily produce 8 unique polytopes. For example, the regular 5-cell (the 4-simplex polytope) has 5 cells, 10 faces, 10 edges, and 5 vertices (30 elements). Expanding the 5-cell produces 7 related semi-regular 4-polytopes of 30 elements each (two of which happen to be the same polytope). In each expansion, the 5 cells have expanded into semi-regular or regular polyhedra, the 10 faces and 10 edges ''may'' have expanded into prisms, and the 5 original vertices have expanded into regular or semi-regular cells.

== The regular 11-cell 4-polytope ==

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosadodecahedron). Each non-convex Jessen's icosahedron is a set of 6 reflex edges that are {{radic|5}} 5-cell edges, each set symmetrically arranged in 3 parallel pairs, not as far apart ({{radic|3}}) as they are long, in 3 orthogonal central planes. The 12-point (Jessen's sets of 6 reflex 5-cell edges) are inscribed in the 60-point with their endpoints as a vertex of a pentagon face. The 12 pentagon faces are joined to 5 other pentagon faces by the 5-cell edges; not to their nearest neighbor pentagons but to their next-to-nearest neighbor pentagons; and since there are 5 vertices to each pentagon face, there are 10 12-point (Jessen's sets of 6 reflex 5-cell edges). Each pentagon face is joined to another pentagon face by 5 reflex edges of five distinct Jessen's. The 5 reflex edges are Clifford parallel, the 5 edges of a pentagonal prism column that has a twist to it. The contraction of those 5 edges into 1, and those 10 vertices (two pentagons) into 2, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing all their edges as a single Jessen's. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (octahedron) by conflating its parallel pairs of edges, leaving the 11-cell's abstract 6-point (hexad cell). The hexad's situation is that it occurs in bundles of 5, disjoint but close together in the sense of adjacent, like the great circle rings in a Hopf fiber bundle. 

To summarize, the 12-point (2-hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell). We shall now be able to see how Fuller's perfect hexads combine with Todd's perfect pentads to form the perfect heptads of the 11-cell and 137-cell.  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=alias the hemi-cuboctahedron}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces. (The ''heptad'' arrives at the party, anonymously; they are our distinguished anti-celebrity, who most of the other guests have never even heard of.) The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. This 12-point (truncated tetrahedron) is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 100 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp: Truncated tetrahedron|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp: Truncated tetrahedron|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}} Of course Moxness has been there, seen that, already with his graphics software;{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} see his parameterized illustration of it exactly as it occurs in the 11-cell, below in his bestiary of A3, B3, and H3 Archimedian and Catalan polyhedra in the 120-cell.]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:Archimedean_and_Catalan_solid_hulls_with_their_Weyl_orbit_definitions.svg|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of the 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. In this catalog of Archimedean and Catalan polyhedra sighted in the 120-cell by Moxness's software,{{Sfn|Moxness: Archimedean and Catalan hulls|2023|loc=Hull #1 Archimedean Name A3 110 Truncated Tetrahedron A (upper left)}} it is the first entry (upper left), Moxness's Hull # = 1 with 12 vertices of 3D norm {{radic|11}}/2. On a {{radic|2}}-radius 3-sphere, 5 of these with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell. 20 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges are cells in the [[#The rings of the 11-cells|cell rings of the quasi-regular 11-cell]], which also has 12 regular 5-cells as cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

...

Both kinds of cells in the 11-cell have expressions in two different dimensions (on the 2-sphere and 3-sphere), because they are all centered on the center of the 3-sphere, unlike the cells of most 4-polytopes, which are centered on various off-center points within the 4-polytope's interior. 

One kind of 11-cell cell appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (semi-regular truncated cuboctahedron), a.k.a. the truncated tetrahedron. The other kind of cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now, except in the 120-cell. But this is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron, and a regular tetrahedron behind each face. The 11-cell is thus a honeycomb of mixed regular tetrahedra and truncated cuboctahedra (a.k.a. truncated tetrahedra) that meet at their shared 5-cell faces. 

Both the 5-cells (decahedra) and the truncated cuboctahedra (truncated tetrahedra) are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks in their 4-polytopes (off-center).{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} Therefore all the 11-cell's cells are concentric, which indeed makes a different kind of honeycomb, an inside-out cellular honeycomb with all the cells piled on top of each other overlapping in 4-space. They do not overlap in 3-sphere space however; their 3-volumes (but not their 4-contents) are disjoint. They make a proper 4D honeycomb like the cells of any 4-polytope, with volumetrically disjoint 3D cells that meet each other in the proper way at their shared 2D faces.

An 11-cell contains 6 completely disjoint 5-point (1-pentad 5-cell decahedron cells), and 5 volumetrically disjoint 12-point (2-hexad truncated cuboctahedron cells), 11 volumetrically disjoint cells altogether. Of course it is supposed to have 11 identical 11-point (elevenad hemi-icosahedral cells), and that is also true in its abstract sense, which is not the same sense as disjoint cellular containment: each pentad or hexad cell is inscribed in one of 11 larger 60-point (Moxness's rhombicosadodecahedron cells). The abstract pentads and hexads are only hemi-polyhedra, so the 11-cell's pentad cells are 12 (out of 120) disjoint instances of a pentad 5-point, and its hexad cells are 10 (out of 100) disjoint instances of a hexad 6-point, 1/10th of the 120-cell's cellular volume in both cases. The 11-cell's hemi-icosahedral cells are 11 (out of 120) non-disjoint instances of an elevenad 60-point. Each 11-cell shares each of its hexad cells with 9 other 11-cells, and each of its 11 vertices (out of 600) occurs in 9 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.|name=Tetrahedral vertex figures}}

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

...insert image thumb of 11-cell...

...Construct the 11-cell from the heptad....

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...Four concentric 4-polytopes may be mutually orthogonal. In any set of 4 mutually orthogonal regular 5-cells, one pair of faces in each orthogonal pair are completely orthogonal to each other. Thus three edges in one 5-cell in an orthogonal pair are opposite (completely orthogonal) to a face of the orthogonal 5-cell. .... A 5-cell has 5 vertices in its abstract hemi-icosahedral cells, but only 4 vertices in one concrete icosahedral cell; the 5th lies in an icosahedral cell of another 11-cell (not in another icosahedral cell of the same 11-cell).

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

....

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 11-cell ===

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon.... (wrong, 11-cell's Hopf map is the truncated icosahedron)

....

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk: Truncated icosahedron|2018}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

...

Since the rhombicosadodecahedron has 120 5-cell edges, the 120-cell has 120 completely disjoint 5-cells, and the rhombicosadodecahedron shares each vertex with 11 other rhombicosadodecahedra, how a rhombicosadodecahedron is bound to its neighbor 5-cells is constrained. A further constraint is that the abstract hemi-icosahedron has only 15 5-cell edges, so the 120 actual 5-cell edges of the rhombicosadodecahedron are conflated 8-to-1 in the hemi-icosahedron. To be conflated by contraction into a single edge, those 8 edges must lie parallel to each other in the 120-cell; either actually parallel in the same plane, or Clifford parallel in separate planes, as two edges in completely orthogonal planes may be parallel and orthogonal at the same time. Each 5-cell inscribed in the 120-cell contains 5 such tetrahedra occur in parallel pairs in 4 orthogonal planes (none of them completely orthogonal planes). 

.... 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
Each conflated vertex is the 10 vertices of two orthogonal regular 5-cells...
The 60-point contains 6 pairs of these orthogonal regular 5-cell cells...
...its 20 hexad cells are ..., a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere.

== The regular 137-cell 4-polytope ==

(description originally written for the 11-cell, which applies only to a fibration of 11 11-cells)

It is quasi-regular with two kinds of cells, the 12-point (regular icosahedron) and the 5-point (regular 5-cell), which have just one kind of regular face between them. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of regular facets (it is semi-regular with regular facets), it also means that its facets have expressions of two different dimensionalities because they are concentric in the 3-sphere. The 11-cell has two kinds of regular cells. One appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (regular icosahedron), but since it is embedded at the center of the 3-sphere (unlike the cells of most 4-polytopes) it also acts as an inscribed 4-polytope.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} The other cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now except in the 120-cell. This is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron. The 11-cell is a honeycomb of mixed icosahedron and half-icosahedron cells. Like the icosahedra, the decahedra are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks, off-center in their 4-polytopes. All the 11-cell's cells are concentric, which indeed makes an unusual honeycomb of cells, an inside-out honeycomb. It is a proper 4D honeycomb however, with volumetrically disjoint 3D cells that meet each other in the proper way at shared 2D faces. The embedded 12-point (regular icosahedron) cell is an icosahedral pyramid (as a 4-polytope), which is filled by 20 4-point (irregular tetrahedra) that meet at its center; the center point of the 11-cell acts as a 12th vertex, just as the center point of the 24-cell acts as a 25th vertex in some honeycombs. The 5-point (regular 5-cell) cell contains five 4-point (regular tetrahedra) which meet the 12-point (regular icosahedra)'s irregular tetrahedral cells face-to-face, so the 11-cell is quasi-regular in the usual sense of having two varieties of regular 3-polytope cells, but additionally both varieties of of its cells are bi-dimensional: another clue that the 11-cell has something to do with dimensional analogy

....

The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

....

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur to make an 11-cell. 11 of them occur in a 137-cell, all 120 of them occur in the 120-cell, and they occur nowhere else.

How can this regular 4-polytope possibly exist? By Schläfli's criterion {3, 3, ..} exists, but not as a finite polytope.{{Sfn|Coxeter|1973|loc=§7.7 Schläfli's criterion}} Coxeter found {3, 5, 3} as a finite hyperbolic honeycomb, but not in Euclidean 4-space.{{Sfn|Coxeter|1968|title=The Beauty of Geometry: Twelve Geometric Essays|loc=''Regular Honeycombs in Hyperbolic Space''|ps=; &quot;The honeycombs considered by the above authors [Schlegel, Schläfli, Klein, Sommerville] have finite cells and finite vertex figures. ... A further extension allows a cell or vertex figure to be a ''star''-polytope, so that the honeycomb covers the space several times. .... We shall find that there are four regular star-honeycombs in hyperbolic 4-space, as well as two infinite families of them in the hyperbolic plane.&quot;}} We find that {3, 3, ..} is a central expression of the &lt;math&gt;H_4&lt;/math&gt; root system, being the regular convex hull in 4 dimensions of 137 points, 96 pentads and 600 hexads. The minimal expression of &lt;math&gt;H_4&lt;/math&gt; is the regular convex hull {3, 3, 5} of 120 points, 600 tetrads and 75 hexads, and its maximal expression is the regular convex hull {5, 3, 3} of 600 points, 120 pentads and 675 hexads. {3, 3, ..} is a third regular &lt;math&gt;H_4&lt;/math&gt; polytope which fits between them. The properties and metrics of this regular sequence are presented in tabular form below in ''[[#Build with the blocks|§Build with the blocks]]''.

{|class=&quot;wikitable&quot;
!colspan=3|Regular &lt;math&gt;H_4&lt;/math&gt; polytopes
|-
|[[File:600-cell.gif|200px]]
|[[File:5-cell.gif|200px]]
|[[File:120-cell.gif|200px]]
|-
!120-point 600-cell
!137-point 137-cell
!600-point 120-cell
|}

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 137-cell ===
....

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=Apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|2022|loc=''[[Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]''}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie: On Fuller's use of language|2024|loc=''[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry'']]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center, 1948-1949|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with their mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the gender neutral shad can open their six beaks all the way, until they become flat squares and they becomes a cuboctahedron, or they can shut them all tight like a turtle retracting into their octahedron shell. The six mouths always move in unison. This is [[Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do their ''really'' odd trick -- where they flip their 6 jaws 90 degrees in their 6 faces and shut their 6 beaks on the opposite axis of their squares than the one they opened them on -- or whether they will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than their normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make their edge length exactly the same size as their radius, when they open their mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing their mouths in spherical synchrony, their 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron (the hexad), then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=''[[W:SO(4)|SO(4)]]''}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=Isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its orthogonal hexad's planes (wz, wx, wy).{{Efn|name=Six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of (expanded octahederon 16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh &amp; Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== Legendre's 12-point hexad binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a 12-point double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's perfectly realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, this 2-hexad expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre hexad) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction.{{Efn|name=Concentric polyhedra}} In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== ''n''-simplexes ===
....

The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== ''n''-orthoplexes ===
....
[[File:Great 5-cell √5 digons rectangle.png|thumb|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} ''regular'' great hexagons. These are the same 200 dodecagon central planes illustrated above, which also contain 6 120-cell edges and two opposing ''irregular'' great hexagons (truncated {{radic|6}} triangles). The 1200 {{radic|5}} 5-cell face triangles occupy 1200 other, non-central planes.]]

...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's Hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[16--cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[16--cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=Concentric polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[16--cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== ''n''-taliesins ===
The 4-simplex pentad building block combines with the 4-orthoplex hexad building block to make the 4-taliesin heptad building block. The heptad is the essential building block of the ''taliesin'' family of polytopes. The 11-cell and 57-cell hemi-polytopes and the 137-cell and 120-cell regular 4-polytopes are nuclear members of this trans-dimensional family. The 24-cell and 600-cell are extended members of it, because they also contain instances of the heptad building block.

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its spherical symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* [[W:Taliesin Myrddin Namkai-Meche|Taliesin Myrddin Namkai-Meche]], a young Reed College scholar who was stabbed to death by a crazy man in 2017 on a Portland, Oregon light rail train, when he rose to the nonviolent defense of two black teenagers who were being violently assaulted in racist and anti-Muslim language by a right-wing extremist.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in orthogonal [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].
* '''Taliesan polytope''', an (''n+1)''-polytope whose ''n''-polytope dimensional analogue inscribed concentrically in the ''n''-sphere forms a convex honeycomb of itself which is the (''n+1)''-polytope.{{Efn|name=Embedding point}}
* The taliesan polytope's honeycomb, a Hopf fibration of its great circles.
* The ''n''-dimensional family of dimensionally analogous polytopes related to a taliesan polytope by expansion and contraction operations.
* The characteristic [[w:Orthoscheme|orthoscheme]] of a teliesan polytope, an irregular simplex that is the complete gene for the entire expression of trans-dimensional symmetry relationships among the polytopes of the ''n''-dimensional family, as well as the complete gene for the teliesan polytope's rotational symmetry.{{Efn|name=Characteristic orthoscheme}}

=== ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== ''n''-equilaterals ===
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed at four different [[#n-leviathon|chord]] lengths from each other. In the {{radic|2}}-radius polytopes we have using as examples in this paper they are {{radic|2}}, {{radic|4}}, {{radic|6}} and {{radic|8}}, but in a ''unit-radius'' polytope they are: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}.

Each vertex is joined to 8 others by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}. The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== ''n''-pentagonals ===

[[File:600-cell.gif|thumb|The 120-point (600-cell), &lt;math&gt;H^4&lt;/math&gt;.]]
...

[[File:600-cell vertex geometry.png|thumb|Vertex geometry of the 600-cell, showing the 5 regular great circle polygons and the 8 vertex-to-vertex chord lengths with angles of arc. The golden ratio{{Efn|name=golden chords|group=}} governs the fractional roots of every other chord,{{Efn|name=fractional root chords}} and the radial golden triangles which meet at the center.|alt=|400x400px]]
The 120 vertices of the 600-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏&lt;sup&gt;−1&lt;/sup&gt;) beginning with a vertex; see column ''a''}} at eight different [[#n-leviathon|chord]] lengths from each other. These edges and chords of the 600-cell are simply the edges and chords of its five great circle polygons. In ascending order of length in a ''unit-radius'' polytope (not in a {{radic|2}} radius polytopes like our usual examples), they are {{radic|0.𝚫}}, {{radic|1}}, {{radic|1.𝚫}}, {{radic|2}}, {{radic|2.𝚽}}, {{radic|3}}, {{radic|3.𝚽}}, and {{radic|4}}.{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=Decimal fractions of square roots|group=}}

Notice that the four [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#''n-''equilaterals|hypercubic chords]] of the 24-cell ({{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|4}}) alternate with the four new chords of the 600-cell's additional great circles, the decagons and pentagons. The new ''golden chord'' lengths are necessarily square roots of fractions, but very special fractions related to the golden ratio{{Efn|1=The fractional-root ''golden chords'' are irrational fractions that are functions of {{radic|5}}. They exemplify that the [[W:golden ratio|golden ratio]] &lt;big&gt;𝜙&lt;/big&gt; {{=}} {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618 is a circle ratio related to &lt;big&gt;𝜋&lt;/big&gt;:&lt;br&gt;
: {{sfrac|𝜋|5}} = arccos (𝜙/2)
is one decagon edge, the 𝚽 = {{radic|0.𝚫}} = {{radic|0.382~}} ≈ 0.618 chord.
Reciprocally, in this function discovered by Robert Everest expressing &lt;big&gt;𝜙&lt;/big&gt; as a function of &lt;big&gt;𝜋&lt;/big&gt; and the numbers 1, 2, 3 and 5 of the Fibonacci series:&lt;br&gt;
: &lt;big&gt;𝜙&lt;/big&gt; = 1 – 2 cos ({{sfrac|3𝜋|5}})
{{sfrac|3𝜋|5}} is the arc length of the &lt;big&gt;𝜙&lt;/big&gt; = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618 chord.|name=golden chords|group=}} including the two [[W:golden section|golden section]]s of {{radic|5}}, as shown in the diagram.{{Efn|The 600-cell edges are decagon edges of length {{radic|0.𝚫}}, which is 𝚽, the ''smaller'' golden section of {{radic|5}}; the edges are in the inverse [[W:golden ratio|golden ratio]] {{sfrac|1|φ}} to the {{radic|1}} hexagon chords (the 24-cell edges).
The other fractional-root chords exhibit golden relationships as well. The chord of length {{radic|1.𝚫}} is a pentagon edge.
The next fractional-root chord is a decagon diagonal of length {{radic|2.𝚽}} which is &lt;big&gt;𝜙&lt;/big&gt;, the ''larger'' golden section of {{radic|5}}; it is in the golden ratio{{Efn|name=golden chords|group=}} to the {{radic|1}} chord (and the radius).{{Efn|Notice in the diagram how the &lt;big&gt;𝜙&lt;/big&gt; chord (the ''larger'' golden section) sums with the adjacent 𝚽 edge (the ''smaller'' golden section) to {{radic|5}}, as if together they were a {{radic|5}} chord bent to fit inside the {{radic|4}} diameter.}}
The last fractional-root chord is the pentagon diagonal of length {{radic|3.𝚽}}.
The [[W:Pentagon#Side length is given|diagonal of a regular pentagon]] is always in the golden ratio to its edge, and indeed &lt;big&gt;𝜙&lt;/big&gt;{{radic|1.𝚫}} is {{radic|3.𝚽}}.|name=fractional root chords|group=}}

....


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== ''n''-leviathon ===
...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of its &lt;math&gt;10^2&lt;/math&gt; disjoint instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction, before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]].{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}} .... It is only because she did not have the 11-cell to start with that she did not discover the regular 137-cell {3, 3, ..} by her method.

Between Alicia Boole Stott and her identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and her identification of symmetry groups with conservation laws (in her 1st theorem), we owe the deepest mathematics of geometry and physics originally to two pioneering women, colleagues of their contemporaries the greatest men of the academy, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just an [[W:abstract polytope|abstract 4-polytope]], not just a singleton convex 4-polytope, and not just a honeycomb.{{Sfn|Coxeter|1970|loc=''Twisted Honeycombs''}} The 11-point (11-cell) has a concrete ...regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}&lt;nowiki/&gt;} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, 3, 3} as the 137-point (137-cell).

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 11-cell, 137-cell and 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=''Clifford Spinors and Root System Induction: H4 and the Grand Antiprism''}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=Apology}}

== Build with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Duveneck|1978|loc=Carl Jung, quoted in ''Life on Two Levels''|p=ii|ps=.{{Sfn|Jung|1961|ps=: &quot;The best of truths is of no use unless it has become one's most personal inner experience. It is the duty of everyone who takes a solitary path to share with society what he finds on his journey of discovery.&quot;}}}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1954|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius={{radic|2}}}}

== Acknowledgements ==
...

== Notes ==
{{Regular convex 4-polytopes Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{Refend}}</text>
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      <text bytes="208259" sha1="97wudzf4j247ovozabma5egojfnmir4" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|June 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a regular 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the &lt;s&gt;120-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 5, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 11 [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] cells. Eleven 11-cells (plural) are the 137-cell, the &lt;s&gt;600-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 3, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 137 [[W:Rhombic triacontahedron|triacontahedron]] cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=''Regularity of Graphs, Complexes and Designs''}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=''A Symmetrical Arrangement of Eleven Hemi-Icosahedra''}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]]
The 4-dimensional regular polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:Simplex|simplex]] (called the [[5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:Orthoplex|orthoplex]] (called the [[16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular convex 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the [[w:Tesseract|16-point (8-cell) tesseract]], the 4-cubic, which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found within any of the other regular convex 4-polytopes, except in the largest and most complex one, the [[120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). However, we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the 16-cell is their very starting point, and the most frequently used tool in the box.

== 5-cells and hemi-icosahedra in the 11-cell ==
[[File:Symmetrical_5-set_Venn_diagram.svg|thumb|The 5-point (10-face) regular 5-cell (the regular 4-simplex). Grünbaum's rotationally symmetrical 5-set Venn diagram{{Sfn|Grünbaum|1975|loc=''Rotationally symmetrical 5-set Venn diagram'', Fig 1 (e)|ps=; partitions the individual elements of the 5-cell.}} is an illustration of the 5-cell labeling each of its &lt;math&gt;2^5&lt;/math&gt; elements.{{Sfn|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled; image includes the Python code to render it, optimising for maximum area of the smallest regions.}}]]
[[File:Hemi-icosahedron.png|thumb|The 6-point (10 face) [[W:hemi-icosahedron|hemi-icosahedron]], an abstraction of the regular icosahedron, has half as many faces, edges and vertices. Each element of the abstract polyhedron represents two real elements found in different places in the 11-cell's realization.{{Sfn|Ruen: hemi-icosahedron|2007}}]]
The most apparent relationship between the pentad 5-cell and the hexad hemi-icosahedron is that they both have 10 triangular faces. When we see a facet congruence between a 4-polytope and a 3-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 6-point 5-simplex. It is also the same size as the 6-point 3-orthoplex; thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no chord congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals, pentad and hexad, who are famously stand-offish with each other. Interesting!

The cell of the 11-cell is an abstract hexad hemi-icosahedron containing pentads in its interior, most handsomely illustrated by Séquin.{{Sfn|Séquin &amp; Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing rings of polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The 11-cell's 11 hemi-icosahedral cells have 10 triangle faces each, and we shall assume a 5-cell is behind each face.{{Efn|A 5-cell of radius {{radic|2}} has an edge length of {{radic|5}}.}} Only 3 of the 5-cell's 5 vertices are visible on the surface of the hemi-icosahedron as the exposed face, but its other two vertices are visible on the opposite side of the hemi-icosahedron as an exposed edge.{{Sfn|Coxeter|1984|p=110|loc=§6. The Petrie polygon [of the 11-cell]|ps=; &quot;We may reasonably call this edge and face ''opposites''. It is easy to find the face opposite to a given edge by looking at the faces to which a given edge belongs. ... Conversely, given a face, we can find the opposite edge by seeing which vertices belong to neither of the hemi-icosahedra which share that face. The ten edges opposite to the ten faces of one hemi-icosahedron are the edges of the complementary &lt;math&gt;a_4&lt;/math&gt; [4-simplex], that is, the joins of all pairs of the five vertices [of the 11-cell] not belonging to the given hemi-icosahedron.&quot;}} The other parts of the 5-cell (its other 6 edges, its other 9 faces and its 5 tetrahedral cells) lie outside the hyperplane occupied by the hemi-icosahedron.

There are 11 disjoint 5-cells in each 11-cell, which is comprised of 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. The real 11-cell is harder to see than the abstract 11-cell that represents it, because the real hemi-icosahedron that the abstract hemi-icosahedron represents is hard to see, but seeing these objects will be easier once we have identified the real hemi-icosahedron and exactly where the 11-cell's real elements reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each hemi-icosahedron's 10 faces come from 10 disjoint 5-cells, and there is just one 5-cell with which it does not share a face.{{Efn|As Coxeter observes (in the previous citation), that unrepresented 5-point (5-cell) is the other 5 vertices of the 11-point (11-cell) that are not vertices of this 6-point (hemi-icosahedron), the hemi-icosahedron's disjoint complement.}} Each 5-cell has 10 faces from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share a face.

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in two different planes which are completely orthogonal to each other. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must be sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness: 8 concentric hulls|2022|loc=Hull #8 (lower right)|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell of the 11-cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Compounds in the 120-cell|§Compounds in the 120-cell]]'' that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software which uses Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s to render the polyhedra which are found in the interior of ''n''-dimensional polytopes.{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the tetrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is a real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (Hull #8) is a concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this {12} irregular great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing {6} irregular great hexagons (truncated triangles) of alternating 120-cell edges and 5-cell edges. The great dodecagon is a great circle of the hemi-icosahedral cell, and also a great circle of the 120-cell, because unlike the cells of most 4-polytopes, the 11-cell's cells are embedded in 4-space at the exact center of the 4-polytope, making them inscribed 4-polytopes themselves.{{Efn|name=Embedding point}} There are 25 great dodecagons and 120 5-cell edges in Moxness's 60-point (Hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (Hull #8) polyhedron altogether.

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point (Hull #8) is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] similar to the one from the catalog shown here,{{Sfn|Piesk: Rhombicosidodecahedron|2018}} but a slightly shallower truncation with smaller red pentagons (120-cell faces) and narrower rhombs. Rhombicosidodecahedra are made by truncating the [[W:Rhombic triacontahedron|rhombic triacontahedron]], which is the 30-sided polyhedron with only one kind of face, the dual of the 30-point icosidodecahedron. The 120-cell contains 120 of these 11-cell cells.]]
Moxness's 60-point (Hull #8) is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (Hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|The 5-point (regular 5-cell) can be another abstraction of Moxness's 60-point (Hull #8), 12-vertices-into-1, but it is not strictly the result of a contraction operation as described by Boole Stott, since she did not apply her expansion and contraction operations to uniform polytopes with more than one edge length.}} In fact Moxness was not the first to find a rhombicosidodecahedron in the 120-cell; Alicia Boole Stott was. It is the 6th section of {5,3,3} beginning with a cell, which Boole Stott identified as the ''e&lt;sub&gt;2&lt;/sub&gt; expansion'' of the icosahedron (or equivalently of its dual polyhedron the dodecahedron).{{Sfn|Boole Stott|1910|loc=§Examples of the e&lt;sub&gt;2&lt;/sub&gt; expansion|p=7}} But that rhombicosidodecahedron identified by Boole Stott is not Moxness's Hull #8: rather, it is the semi-regular Archimedean solid, with a single edge length and square faces. Moxness's Hull #8 with its two distinct edge lengths and rectangular faces is Coxeter's 8th section of {5,3,3} beginning with a cell, which is missing from the sections illustrated by Boole Stott.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than tetrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. The 120-cell contains 120 of Moxness's 60-point (11-cell cells). They are non-disjoint, with 12 60-points sharing each vertex.

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (Hull #8) is the real hemi-icosahedron cell, with 120 5-cell edges in it as chords lurking under its surface. The rhombicosadodecahedron has enough of those 5-cell edges in its interior to make 20 disjoint tetrahedral cells which belong to 20 distinct 5-cells.

The 600-point (120-cell) may be constructed as a semi-regular honeycomb of two kinds of cells, the 120 non-disjoint 60-point (Moxness's rhombicosadodecahedra) and their neighbors the 120 disjoint 5-point (5-cells), cell-bonded together by shared tetrahedral cells of edge-length {{radic|5}}. Each rhombicosadodecahedron is bonded to 20 islanded 5-cells, sharing with each a 5-cell tetrahedron (4 of its interior 5-cell faces, and 6 of its interior 5-cell edges). The rhombicosadodecahedron is also bonded directly to 10 other rhombicosadodecahedra (since each 11-cell has 11 rhombicosadodecahedron cells), again not by sharing one of its exterior faces, but this time by sharing an interior face, a {{radic|6}} triangle which lies in a central plane.{{Efn|The {{radic|6}} chord is another hidden chord lurking beneath the 60-point rhombicosadodecahedron's surface, like the {{radic|5}} 5-cell edge. Both these hidden chords lie on the same great circle with the 120-cell edges (the pentagon face edges). Three {{radic|6}} chords make a triangle which lies in a central plane. This {{radic|6}} great triangle must not be confused with the {{radic|5}} face triangle of the 5-cell, which does not lie in a central plane.}} The 120 rhombicosadodecahedra are also bonded directly together at ''all'' their exterior faces: at their pentagon faces (which are 120-cell faces of dodecahedral cells); at their small triangular faces (which are 600-cell faces of small tetrahedral cells, 10 of which are inscribed in each dodecahedral cell); and at their rectangular faces (which are faces of 10-point [[w:Pentagonal_prism|pentagonal prism]] cells, where the pentagonal prism is a column inscribed in a dodecahedral cell, joining its opposite pentagon faces). The 120 rhombicosadodecahedra are bonded directly to each other at their own exterior faces; no dodecahedra, small tetrahedra, or pentagonal prisms lie between them, and those objects are not cells of this honeycomb; we mention them here only to help locate the rhombicosadodecahedra within the 120-cell. The only objects which fill holes between the rhombicosadodecahedra are the 120 islanded 5-cells.{{Efn|This honeycomb of the 120-cell resembles the [[600-cell#Icosahedra|honeycomb of the 600-cell consisting of 24 face-bonded regular icosahedra]] with 24 clusters of 5 tetrahedra filling the holes between them. In the 120-cell there are 5 times as many of everything as in the 600-cell, so there are 120 (rhombicosadodecahedron) icosahedral cells with 600 tetrahedra filling the holes between them.}}

It is possible for rhombicosadodecahedra to meet each other at more than one kind of exterior face because they are not meeting there as cells of the same 11-cell; they belong to different 11-cells. The 11 rhombicosadodecahedra in the same 11-cell do not meet at their own exterior faces, they meet at their interior {{radic|6}} triangle faces. As a cellular 4-polytope, the 11-cell is inside-out. Its cells meet each other at interior planes, sharing interior {{radic|6}} triangle faces, instead of in the usual way that 4-polytopes' cells share an exterior face. It is a deeper kind of intertwining to be sure, but the 11 rhombicosadodecahedra in the same 11-cell are still volumetrically disjoint cells, and the 11-cell is a perfectly real 4-polytope with real cells. It is a semi-regular 4-polytope, because it has two different kinds of cells: 11 rhombicosadodecahedra and 11 5-cells. The 11 5-cells are islanded (not bound to each other), and the 11 rhombicosadodecahedra are bound both to 5-cells (at {{radic|5}} 5-cell faces) and to each other (at {{radic|6}} central triangle faces). In the section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#What's in the box|§What's in the box]]'', below, we will look closely at the relation between the {{radic|5}} chord and the {{radic|6}} chord, which is the relation between the pentad and the hexad, and at the general phenomenon of hemi-polytopes which meet each other at a central plane. Then in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Cell rings of the 11-cells|§Cell rings of the 11-cells]]'' we will look at the two-cell cellular structure of the 11-cell in more detail.

Considered as a single complex of overlapping cells rather than as distinct honeycombs of volumetrically disjoint cells, the 120-cell contains 60-point (rhombicosadodecahedral cells), islanded 5-point (5-cells), 20-point (dodecahedral cells), and 10-point ([[w:Pentagonal_prism|pentagonal prism]] cells). The 60-points meet each other in pairs at their pentagon faces, meet regular dodecahedra at those pentagon faces also, meet each other in quads at their 600-cell faces where they share an interior 4-point (600-cell tetrahedral cell) and its four faces (just as they share a larger interior 5-cell tetrahedron and its faces), and meet each other and the 10-point (pentagonal prisms) at their rectangle faces. We might try to imagine such a complex with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with parts truncated from 30 neighboring truncated dodecahedra pentagonal prisms). There are 120 of each of these 3 kinds of dodecahedron. 

Moxness's 60-point (rhombicosadodecahedron cell) is a central object of the 120-cell, and contains many wonders. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra by expansion and contraction operations, as Alicia Boole Stott discovered before 1910,{{Sfn|Polo-Blanco: ''Theory and history of geometric models of Alicia Boole Stott''|2007|loc=§5.3.2 1910 paper on semi-regular polytopes|pp=152-158|ps=; summarizes Boole Stott's method and results from {{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}}, including her tables of expansion-contraction dimensional analogies and a few of her illustrations.}} and those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, rediscovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter &amp; Petrie|1938|p=4|loc=''[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]''}}|name=Concentric polyhedra}} Before we move on from the 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice one more thing about the hidden {{radic|5}} chords lurking below its surface. The 12 little pentagon faces are spanned by sets of 6 {{radic|5}} chords (5-cell edges). The 6 chords of each set are disjoint (they don't touch or form 5-cell faces), but they are symmetrically arranged in 3 parallel pairs, {{radic|3}} apart, which lie in 3 orthogonal central planes. It turns out they are the 6 reflex edges of a 12-point non-convex polyhedron called the [[w:Jessen's_icosahedron|Jessen's icosahedron]], inscribed in the 60-point cell. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again and work our way outwards, before eventually in ''[[#The perfection of Fuller's cyclic design|§The perfection of Fuller's cyclic design]]'' we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cell)s ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks, showing its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Pentahemidemicube.png|120px]]
|[[File:Tetrahemihexahedron_rotation.gif|120px]]
|[[File:Pentahemicosahedron.png|120px]]
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts needed to build the 120-cell (the picture on the cover of the box), and everything else that's invisible in that picture which fits inside the 120-cell: all the astonishing regular 4-polytopes. The pentad building block is half a pair of orthogonal 5-point (regular 5-cells), the hexad building block is half an 8-point (16-cell), and the heptad building block is a quarter of an 11-point (11-cell). Everything can be built from these blocks by embedding them at the center of the 3-sphere, inscribing them as 4-polytopes to make compounds as with the 16-cell building block, rather than by assembling them face-to-face as 3-polytopes to make honeycombs in which they are cells or vertex figures.{{Efn|name=Embedding point}}

The pentad building block is the 5-point '''pentahemidemicube''', an abstract polyhedron with 10 equilateral triangle faces, 10 {{radic|5}} edges, and 5 vertices.{{Sfn|Christie: Pentahemidemicube|2024|loc=File:Pentahemidemicube.png|ps=; &quot;The 5-point ''pentahemidemicube'' is an abstract polyhedron with 10 equilateral triangle faces, 10 edges, and 5 vertices. It has a real presentation as the regular 5-cell.&quot;}} It has a real presentation as the 5-point (regular 5-cell 4-polytope), a regular tetrahedron expanded by addition of a fifth vertex equidistant from all the others in a fourth dimension. The pentad block is a tetrahedral pyramid (a 5-cell) inscribed in a cube: a tetrahedron inscribed in a cube (a demicube) expanded by addition of a fifth vertex, one of the cube's other 4 vertices. Each pentad (5-cell) building block lies in the 120-cell orthogonal to three other pentad (5-cell) building blocks. They are four completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges. Each regular 5-cell in the 120-cell is a pentad (pentahemidemicube) building block, in one of 30 orthogonal sets of 4. There are 120 pentad building blocks in the box.

Every vertex of the 16-cell, and therefore every vertex of the 8-cell, 24-cell, 600-cell and 120-cell, is surrounded by a 6-point (octahedral vertex figure). The hexad building block is that 6-point octahedron embedded at the center of the 4-polytope instead of at a vertex, that is, inscribed in the 3-sphere as a 4-polytope, the 6-point '''[[w:Tetrahemihexahedron|tetrahemihexahedron]]''' or '''hemi-cuboctahedron'''.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; &quot;The tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U&lt;sub&gt;4&lt;/sub&gt;. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices.&quot;}} The hexads have 4 red triangle 5-cell faces, bounded by 12 {{radic|5}} edges. Their 3 yellow square faces lie in central planes, because the building block is an abstract 4-polytope. The hexad is an irregular abstract polyhedron that also has 6 {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

Two orthogonal 5-point (pentad) building blocks intersect in a 4-point (inscribed tetrahedron) to form a 6-point (hexad) building block. Two orthogonal 6-point (hexad) building blocks intersect in a 4-point (central square) to form an 8-point (16-cell) building block, the most usable building block, from which everything larger can be built by compounding.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 120 pentads to pair with a completely orthogonal hexad, and although there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed) they only result in 120 pairs two different ways, so there are 120 11-cells. The 11-cell is 6 pentads' and 5 hexads' smallest joint compound, and the 120-cell is their largest, 120 pentads and 100 hexads.

The heptad{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''|pages=22–31}} building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|4}} edges and 9 {{radic|5}} edges), and 7 vertices.{{Sfn|Christie: Pentahemicosahedron|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;}} It is a compound of a 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]) and a 6-point (hexad [[W:hemi-icosahedron|hemi-icosahedron]]) intersecting in a 4-point (tetrahedron), that is, of three orthogonal 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]&lt;nowiki/&gt;s) intersecting in their common 4-point (tetrahedron). It is also the abstract cube contracted by removal of one vertex, which reduces one of the cube's two inscribed tetrahedra (demicubes) to a single remaining triangular face. As with all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Four &lt;small&gt;&lt;math&gt;(11-7)&lt;/math&gt;&lt;/small&gt; orthogonal 7-point (pentahemicosahedron) blocks can be joined at a common Petrie polygon (a skew hexagon which contains their orthogonal 6-edge paths) to construct a real 11-point ([[W:11-cell|11-cell]] semi-regular 4-polytope), which magically contains six &lt;small&gt;&lt;math&gt;(11-5)&lt;/math&gt;&lt;/small&gt; 5-point (1-pentad regular decahedra) as cells and five &lt;small&gt;&lt;math&gt;(11-6)&lt;/math&gt;&lt;/small&gt; 12-point (2-hexad semi-regular cuboctahedra) as cells. There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like [[W:Euclid's postulates|Euclid's postulates]], and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as honeycombs of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles. It possesses the complete symmetry of the polytope without any redundancy, because it contains one of each of the polytope's characteristic root elements.{{Efn|A [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be partitioned radially by its planes of symmetry (Coxeter's &quot;reflecting circles&quot;) into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated rotationally to generate its regular 4-polytope because it is the complete [[W:gene|gene]] for it, containing all of its elements and capturing all of its symmetry without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring that triggers the reflections.|name=Characteristic orthoscheme}} It is the gene for the polytope, which can be replicated to construct the polytope.

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its &lt;math&gt;A_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the regular compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;small&gt;&lt;math&gt;120\times 120 = 14400&lt;/math&gt;&lt;/small&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[16--cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its &lt;math&gt;B_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 384. The 120-cell is the convex hull of the regular compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 24-point (24-cell) is subdivided into 1152 instances of its [[24-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;F_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 1152. The 120-cell is the convex hull of the regular compound of 25 disjoint 24-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;25\times 1152 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has three equivalent origins in distinct root systems, though most are obviously constructed from one root, and not so obviously from the others. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything larger than the pentad can be constructed by compounding hexads, and we shall see that everything as large as a 24-cell has a construction from heptads which are a product of a pentad and a hexad. Surely every uniform polytope can be constructed by some function of expansions and contractions from any of three root systems, however indirect the recipe. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes, related by an equals sign between them, then give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics by [[W:Noether's theorem|Noether's theorem]].

=== Pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 octahedral cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell contains the 600 octahedral cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It also contains the 600 cubical cells of its 75 8-cells. It is a 600-cell in three ways, although the octahedra and cubes are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell and the 8-cell, respectively.|name=600 cells three ways}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (1-pentad 5-cell), out of the 8-point (2-hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (1-pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope that contains 5-point (1-pentad 5-cell) building blocks beside the largest one, it just isn't a ''regular'' 4-polytope. It is semi-regular with two kinds of cells, the 4-point (regular tetrahedron) and the 12-point (truncated cuboctahedron), which have just one kind of regular face between them, a 5-cell face. That semi-regular 4-polytope is the 11-cell.

We said earlier that the 11-cell had 11 disjoint 5-cells and 11 hemi-icosahedral cells in it, and that its construction from them might be hard to see, but now we are going to account numerically, at least, for how it can have 6 pentad cells plus 5 hexad cells magically adding to 11 of the same kind of elevenad cell. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, 11. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (5-cell regular decahedron) cells, and the 12 6-point (hexad) building blocks into 100 12-point (regular icosahedron) cells, and then magically adding one whole 5-point (5-cell regular decahedron) cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 12-point (regular icosahedron) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such (cell bundles of 6 + 5 = 11 cells), with each 11-cell bundle containing 11 of both kinds of cell. Very, very strange!{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden'': §II To Add is to Multiply|p=38|ps=; the sequence of regular polygons with harmonious proportions beginning {5}, {7}, {9}, {11} as a family of natural numbers; &quot;The numbers in this optimal family have remarkable properties [that allow] similar figures to be arranged easily or allow a figure to be dissected into a set of similar figures. This is the origin of the Greek idea of geometric progression — that ''multiplication can be accomplished instead by addition''.... For instance, it is well known that the square of the golden ratio is one more than itself (written above as 𝜙&lt;sup&gt;2&lt;/sup&gt; = 𝜙 + 1) and one less than 𝜙 is its reciprocal (1/𝜙 = 𝜙 - 1). The first relation equates multiplication with addition, while the second accomplishes division through subtraction. [These are pentagon {5} bisectional golden ratios, but] the heptagonal [{7} tri-sectional] ratios also behave in surprising ways — adding to multiply, subtracting to divide.... There are analogous constructions for [nonagon {9}] quadrisectional and [hendecagon {11}] pentasectional systems.&quot;}}

The 120-cell has three interchangeable cyclic expansion-contraction cycles, the pentad, hexad and heptad cyclic designs, rooted in the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems, respectively. Each of them constructs the whole sequence of regular 4-polytopes from a single building block. The key to all three cycles is that 5 + 6 = 11, and there is an 11-point quasi-regular 4-polytope in the sequence of convex 4-polytopes. It lies between the 8-point (16-cell) and the 16-point (8-cell), and is itself a &lt;math&gt;B_4&lt;/math&gt; polytope. Its role is to be a unified single building block common to all three cyclic expansion-contraction cycles. It is involved with the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems and expresses in one building block their relations to each other.

We have already seen how the hexad building cycle works. It is plain compounding of 4-orthoplexes at the beginning, increasing to densely intricate non-disjoint symmetry toward the end, making it difficult to see why it works once past the 24-cell. This is how the build works, in terms of sixes. The 16-cell is a 2-hexad (3 pairs of completely orthogonal great square planes). The 24-cell is 3 16-cells and a 4-hexad, 4 different ways: 16 non-disjoint hexads (great hexagon planes). The 120-point (600-cell) is 25 24-cells and a 20-hexad, 10 different ways: 200 non-disjoint hexads (great hexagon planes). The 600-point (120-cell) is 225 (9 times 25) 24-cells. ....

The pentad building cycle doesn't do plain compounding to regular 4-polytopes until the last one, the 120-cell. Fives don't even appear as pentagons until after the 24-cell. We have seen how there are 11 5-cells in the quasi-regular 11-cell, which occurs before the 8-cell, and also how everything can be compounded from irregular 5-cells (4-orthoschemes specialized for each 4-polytope). It is still not evident how a pentad cycle can work to build all the regular 4-polytopes from the regular 4-simplex. This is how is how the build works, in terms of fives. ....

This is how the build works, in terms of elevens. The 11-cell is 4 orthogonal heptads (intersecting in their skew hexagon Petrie polygon). The 24-cell is also 4 heptads (intersecting in 4 orthogonal digon axes, so each heptad contributes 6 unique points to a single fibration of 24 points). Heptads are quarter elevenads, so the 11-cell and 24-cell are both 1 elevenad. ....

See the tabular sequence of convex 4-polytopes in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Build with the blocks|§Build with the blocks]]'', below: table rows ''Inscribed, Pentad, Hexad, Heptad''.

=== 11 of them ===
There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of eleven 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum eleven of them. There is only a single Hopf fibration of the 11 great circles of an 11-cell. 

The 5-cell and the 11-cell are limit cases. The 5-cells are completely disjoint, but the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 10 other instances of itself.

== Cell rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer &amp; Segerman|2013|loc=''Puzzling the 120-cell''}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his section of the Wikipedia article on the 120-cell entitled ''[[120-cell#Visualization|§Visualization]]''.{{Sfn|Ruen &amp; Goucher et al. eds. 120-cell|2024|loc=Goucher's ''[[120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious place to look for the 11-cell's Hopf map is in Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other place it could be hiding. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The quasi-regular convex 11-cell|§The quasi-regular convex 11-cell]]'', and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen: triacontagon|2011|loc=''Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them except his illustrations. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Four orthogonal projections of the 120-cell to a central plane, selected from the table of 3 regular and 11 compound 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|100 icosahedra&lt;br&gt;120 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|5 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 icosahedra&lt;br&gt;6 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|120
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|4
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|50
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|7.2°
|align=center|36°
|align=center|6°
|align=center|7.2°
|}

....

Here we summarize our findings on the 11-cells so far, including in advance some results to be obtained in the next section two sections.

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section....

== The 15 major chords ring everything ==
The natural numbers are each a distinct flavor.

Each integer has its own distinct taste. The first six integers {0..5} are the six strongest, most familiar tastes in nature, instantly distinguished and utterly unique.

The seventh flavor {6} is the first ratio of two of the others. Thereafter follows in strange sequence all the ratios of all the flavors, each marriage of every two flavors, most of which are fractions &lt;math&gt;\{k/d\}&lt;/math&gt;, but some of which are integers &lt;math&gt;\{k\}&lt;/math&gt;.

Some of these ratios are successful flavors in their own right, as distinct and inimitable as the marriage of chocolate and oranges, but the marriage of many flavor pairs is mud. It is a rule of thumb among flavor specialists that nearly any taste can be successfully alloyed with either onions or chocolate, but the direct marriage of these two flavors is notably unsuccessful. Some flavors make famous pairings or infamous ones, and subtle new pairings are always to be discovered that work surprisingly well for some purpose, but the majority of flavor marriages are undistinguished compounds.

In polytope geometry, each chord of a polytope is both is a distinct 1-dimensional object, a chord of the sphere of a distinct length &lt;math&gt;l&lt;/math&gt;, and a distinct natural number &lt;math&gt;h&lt;/math&gt;, a unique flavor. If the polytope is regular, it is a noteworthy and successful flavor. The chord's length &lt;math&gt;l&lt;/math&gt; is a square root, related to the natural number &lt;math&gt;h = k/d&lt;/math&gt; and to a set of polytopes they both represent, by a formula discovered by Steinbach.{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden''; Figure 5. Optimal sections and proportions|p=37|ps=; the regular polygons {5}, {7}, {9} and {11} with their diagonals define respectively: {5} the golden bisection proportional to 𝜙; {7} an analogous trisection; {9} an analogous quadrasection; {11} an analogous pentasection.}} The chord length &lt;math&gt;l&lt;/math&gt; is related to the number of sides of the regular polygon formed by the chord (its numerator &lt;math&gt;k&lt;/math&gt;), and to the winding number or density of its regular skew Petrie polygon (its denominator &lt;math&gt;d&lt;/math&gt;).

The regular 4-polytopes do not contain an infinite variety of these chord flavors, but only 30 of them: only 15 unique pairs of 180° degree complementary chords, which combine to make every regular ''n''-dimensional thing found in nature.{{Efn|The 15 ''major chords'' suffice to construct all the regular ''n''-polytopes. Their 180° complements include 15 ''minor chords'' which occur only in the 120-cell, not in any smaller regular polytope.{{Efn|name=Major and minor chords}} Including the complement of the 180° {{radic|4}} digon {2} chord (the {0} chord {{radic|0}}), and the skew variant of the 90° {{radic|2}} square {4} chord (the {30/7} chord), there are 32 distinct natural numbers which are all the chords of all the regular polytopes in all dimensions.|name=32 chords}} All the regular polytopes (including the largest one on the cover of the box) can be built not only from a box containing a number of blocks of 3 shapes glued together by gravity, they can be also be built from a box containing a number of sticks of 15 lengths glued together by rubber joints.

[[File:15 major chords.png|500px|thumb|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the others occur only in the interior of 120-cell, not as edges of any of the regular 4-polytopes. There are 30 distinct 4-space chordal distances between vertices within the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to identify any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=Major and minor chords}} chords are properly labelled #1 — #15 in the order they fan out by size.{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''; Figure 3|ps=; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with a &quot;fan of chords&quot; diagram like the one here.|p=23}} They join vertex pairs which are 1 — 15 edges apart on a &lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon. A #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; chord forms &lt;small&gt;&lt;math&gt;f+1&lt;/math&gt;&lt;/small&gt; regular geodesic polygons which include a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; disjoint regular {{nowrap|&lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt;-gons}} (e.g. &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; a compound of 5 disjoint hexagons) plus a single [[W:Triacontagon#Triacontagram|regular skew {{nowrap|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram}}]] that wraps around the sphere &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times before closing its circle. Notice the maverick #11 chord, with its non-conforming origin point for its {{nowrap|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;}} polygram.]]

These 15 chords (natural numbers) include the integers 2, 3, 4, 5, 6, 10, 11, 15 and 30. The others are the fractions 5/2, 10/3, 13/7, 15/2, 15/4, and 15/7. Represented among the chord lengths (the square roots) measured in ''unit radii'' (not in {{radic|2}} radii, as metrics are often given elsewhere in this article) are the {{background color|paleturquoise|roots of the first 4 positive integers {{radic|1}}, {{radic|2}}, {{radic|3}} and {{radic|4}}}}, the {{background color|#FFCCCC|root of the very distinguished rational fraction {{radic|5/2}}}}, and the roots of various distinguished irrational fractions as Steinbach discovered: for example, 4 of them are {{background color|yellow|roots of golden numbers (a small integer ± 𝜙)}}. The 15 chords each form their regular polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, which is just &lt;small&gt;&lt;math&gt;\{k\}&lt;/math&gt;&lt;/small&gt; in 8 cases. Arranged in order #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; from shortest edge-length square root to longest, the #1 — #15 chords bridge vertex pairs which are &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; vertices apart on a [[W:Triacontagon#Petrie polygons|&lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon]]. Each chord makes a [[W:Triacontagon#Triacontagram|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; polygram]] of its edge length, a compound of the smaller &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. The 15 distinct polygons form a sequence you would not anticipate exactly unless you have Steinbach's formula:

{| style=&quot;background-color:transparent; white-space:nowrap;&quot;
|-
| {30/1} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|.073}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{30} 
| 120-cell edges
|-
| {30/2}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.191}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15}
|- style=&quot;background: yellow;&quot;|
| {30/3}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.382}} 
| &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi}&lt;/math&gt;&lt;/small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{10}''' 
| 600-cell edges
|-
| {30/4} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.573}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/2}
|- style=&quot;background: paleturquoise;&quot;|
| {30/5}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|1}}'''
| &lt;math&gt;1&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{6}''' 
| 24-cell, 8-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/6}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|1.382}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{5}'''
|- style=&quot;background: paleturquoise;&quot;|
| {30/7}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|2}}'''
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{4}''' 
| 16-cell edges
|- style=&quot;background: #FFCCCC;&quot;|
| {30/8}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.5}} 
| &lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;&lt;small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/4} 
| 5-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/9}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.618}} 
| &lt;math&gt;\phi&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{10/3}
|- style=&quot;background: paleturquoise;&quot;|
| {30/10}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|3}}''' 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{3}'''
|-
| {30/11}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.426}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{11/11}
|- style=&quot;background: yellow;&quot;|
| {30/12}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.618}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{5/2}
|-
| {30/13}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.810}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{13/7}
|-
| {30/14}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.928}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/7}
|- style=&quot;background: paleturquoise;&quot;|
| {30/15}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|4}}''' 
| &lt;math&gt;2&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{2}'''
|}

{| class=wikitable style=&quot;white-space:nowrap; float:right;&quot;
!colspan=6|Natural number &lt;small&gt;&lt;math&gt;h=\{k/d\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Fibers&lt;br&gt;&lt;small&gt;&lt;math&gt;f\{h\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Cells&lt;br&gt;&lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|4-tope&lt;br&gt;&lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Verts&lt;br&gt;&lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|[[W:Triacontagon#Triacontagram|30-gram]]&lt;br&gt;&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;
|-
!Chord
!Arc
!colspan=2|&lt;small&gt;&lt;math&gt;l/\sqrt{1}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;l/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;h&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#1
|15.5~°
|{{radic|0.073}}
|0.270
|{{radic|0.146}}
|&lt;small&gt;&lt;math&gt;30&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1\{30\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;5\{3,3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/1\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#2
|25.2~°
|{{radic|0.191}}
|0.437
|{{radic|0.382}}
|&lt;small&gt;&lt;math&gt;15&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;..\{3,3\}..&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/2\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#3
|36°
|{{radic|0.382}}
|0.618
|{{radic|0.764}}
|&lt;small&gt;&lt;math&gt;10&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;30\{3,3\}20&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/3\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#4
|44.5~°
|{{radic|0.573}}
|0.757
|{{radic|1.146}}
|&lt;small&gt;&lt;math&gt;15/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/4\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5
|60°
|{{radic|1}}
|1
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;6&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/5\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#6
|72°
|{{radic|1.382}}
|1.175
|{{radic|2.624}}
|&lt;small&gt;&lt;math&gt;5&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}} 
|&lt;small&gt;&lt;math&gt;10\{5,3\}12&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/6\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#7
|90°
|{{radic|2}}
|1.414
|{{radic|4}}
|&lt;small&gt;&lt;math&gt;4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7\{4\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;4\{4,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;9\{3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/7\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: #FFCCCC;&quot;|
|#8
|104.5~°
|{{radic|2.5}}
|1.581
|{{radic|5}}
|&lt;small&gt;&lt;math&gt;15/4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/4\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}} 
|&lt;small&gt;&lt;math&gt;15\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/8\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#9
|108°
|{{radic|2.618}}
|1.618
|{{radic|5.236}}
|&lt;small&gt;&lt;math&gt;10/3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10/3\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/9\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#10
|120°
|{{radic|3}}
|1.732
|{{radic|6}}
|&lt;small&gt;&lt;math&gt;3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;10\{3\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/10\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#11
|135.5~°
|{{radic|3.426}}
|1.851~
|{{radic|6.852}}
|&lt;small&gt;&lt;math&gt;11/11&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;11\{11/11\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;11\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#12
|144°
|{{radic|3.618}}
|1.902
|{{radic|7.235}}
|&lt;small&gt;&lt;math&gt;5/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2},3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/12\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#13
|154.8~°
|{{radic|3.810}}
|1.952
|{{radic|7.621}}
|&lt;small&gt;&lt;math&gt;13/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;13\{13/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;7\{5,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/13\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#14
|164.5~°
|{{radic|3.928}}
|1.982
|{{radic|7.857}}
|&lt;small&gt;&lt;math&gt;15/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;137\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/14\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{radic|8}}
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15\{2\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐𝜙&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;8\{3,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/15\}&lt;/math&gt;&lt;/small&gt;
|}

The polygons '''{10}''', '''{6}''', '''{5}''', '''{4}''', '''{3}''', and '''{2}''' are flat polygons that each lie on a great circle in a central plane of the 3-sphere. The polygons {30}, {15}, {15/2}, {15/4}, {10/3}, {11/11}, {5/2}, {13/7}, {15/7} and all the {30/''n''} polygons are skew polygrams spiraling through curved 3-space, each successive edge in a different central plane of the 3-sphere. Projected to the 1-sphere (a circle in the plane), they are [[W:Star polygon#Regular star polygon|regular star polygrams]], the simplest being the pentagram {5/2}. In 4-space they wind around the 3-sphere more than once ''without self-intersecting'', before they close their circuit. They lie on helical geodesic circles or ''isoclines'' of circumference greater than 2𝝅''r''.

Even the 6 integer chords which form flat polygons also form skew polygrams. Each of the 15 chords occurs as a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, a fiber bundle of Clifford parallel polygons that totals 30 vertices per bundle. Each chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; also forms a single skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram that winds through another, shorter chord's bundle of parallel polygons, intersecting them in succession multiple times each, and weaving that whole bundle of parallel fibers together as the cross-thread does on a loom. The polygram cross-thread has to wind multiple times through the whole bundle (&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times around the [[W:Triacontagon#Triacontagram|30-gram]] in which each edge connects every &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt;th vertex) to complete an enumeration of all 30 vertices in the bundle. For example, the #5 chord forms hexagon &lt;small&gt;&lt;math&gt;\{6\}&lt;/math&gt;&lt;/small&gt; flat planes in parallel bundles of 5, written &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt;, and also a skew &lt;small&gt;&lt;math&gt;\{15/2\}&lt;/math&gt;&lt;/small&gt;-gram of #5 edges which winds 2 times through the #3 chord's bundle of 3 decagon &lt;small&gt;&lt;math&gt;\{10\}&lt;/math&gt;&lt;/small&gt; flat planes. Every chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; forms both its &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram and its fiber bundle of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, and the edges of one chord's parallel polygons are also the edges of its skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram weaving through some shorter chord's bundle of parallel polygons. Each numbered chord is a natural number, a ratio of two integers and their polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, either flat or skew. Each chord number defines a &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; skew polygon that is a compound bundle of the &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. All these flat or skew polygons are regular polygons made of their numerator instances of a single chord #, whose length is that square root, Steinbach's function of the natural number.

The relationship between each bundle of flat Clifford parallel great circle polygons of a distinct length, and its corresponding bundle of skew polygrams of a longer length, is precisely that of a discrete simple rotation and its corresponding discrete isoclinic rotation. For example, the flat great circle hexagons of edge length {{radic|1}} lie in the invariant rotation planes of the 24-cell's characteristic 60° isoclinic rotation, in which the vertices move on skew hexagram rotation paths of edge length {{radic|3}}. The 24-cell contains 4 bundles of 4 Clifford parallel (disjoint) hexagon central planes each. The 24 vertices of the 24-cell in each bundle are linked by rotation in the 4 hexagon invariant planes in two different ways: simply and isoclinically. In a simple rotation the vertices of a single invariant hexagon exchange places on that great circle hexagon of edge {{radic|1}}. In an isoclinic rotation in which any one of these 4 disjoint hexagon central planes is invariant, all 4 of them are necessarily invariant planes in the rotation, because they rotate in parallel. In that isoclinic rotation the vertices of all 4 hexagon invariant planes (all 24 vertices of the 24-cell) exchange places concurrently, moving along 4 disjoint helical hexagrams of edge {{radic|3}} in parallel. It is a characteristic only of this distinct rotation in hexagon invariant planes that the helical vertex path or ''isocline'' is a hexagram of period 6, edge length {{radic|3}}, and circumference 720°. Rotations in other kinds of polygon invariant planes have different characteristic isoclines. The period of their isoclinic rotation must be ''the same or greater'' than the period of their simple rotation; in most cases (unlike this one) it is greater.

The 15 chords form not only distinct fiber bundles of polygons, but also distinct bundles of polyhedra: cell rings &lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt; and compound vertex figures &lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;. These combine to form their characteristic regular 4-polytope &lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Coxeter|1973|p=269|loc=Compounds|ps=; &quot;It is remarkable that the vertices of {5, 3, 3} include the vertices of all the other fifteen regular polytopes in four dimensions.&quot;}} All these elements of a regular 4-polytope constructed from one chord are themselves regular polytopes, and each of the 16 regular 4-polytopes is a function of one characteristic chord. If the regular 4-polytope is convex, its characteristic chord is its unit-radius edge. For a non-convex regular 4-polytope, the characteristic chord is its unit-radius ''mid-edge diameter'', the distance from the midpoint of each edge to the midpoint of the completely orthogonal edge. Each of the characteristic chords (each distinct natural number) has its expression as one of the 16 regular 4-polytopes and all the products of the distinct regular 1-polytopes, 2-polytopes and 3-polytopes that construct it, and the whole distinct construction from the natural number to the 4-polytope is an [[W:SO(4)|SO(4) symmetry group]] expression of that 4-polytope's characteristic discrete [[W:SO(4)#Isoclinic rotations|isoclinic rotation]], in the invariant central planes of its characteristic chord, by the characteristic chord arc.

...demonstrate this theory by proper enumeration of the chord lengths in the table...

...color-code the △ ☐ 𝜙 ✩ symbols ... chord colors in diagram should match them ... some of the diagram colors are wrong and must be changed, but otherwise diagram is pretty good ... i think in a colored diagram all chords of the same length should have the same size line dash, the polygons are not obvious enough and the 15 fanning-out chords are readily identified by their labels...

... add back the counts of chords in the box (very small font number under each table entry), and present these two images as the front and back a box of building sticks and rubber joints...What's in the other box...

...say the rational-root major chords {{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|5/2}} are the ''n''-edges: their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge...

...abbreviated text of the [[120-cell#Relationships among interior polytopes|''§Relationships among interior polytopes'']] section from the 120-cell article...to explain the color-coding

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius=1}}

See also the elements, properties and metrics of the sequence of regular 4-polytopes summarized in tabular form in the section ''[[#Build with the blocks|§Build with the blocks]]'', below.

== Alicia Boole Stott's original formulation of dimensional analogy ==
&lt;blockquote&gt;&quot;...a method by which bodies having a certain kind of semi-regularity may be derived from regular bodies in an Euclidean space of any number of dimensions...&quot;{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''; presents cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues|pp=12-45}}&lt;/blockquote&gt;

Alica Boole Stott's 1910 paper reads like an exceptionally well-written patent application. It is a comprehensive stepwise description of her original invention, a general method of dimensional analogy between regular and semiregular polytopes of the same edge length.

[[File:The Roots of Three.png|thumb|An illustration of Boole Stott's expansion of unit-edge triangle to hexagon captures the geometric basis in 3 of the &lt;math&gt;A_n&lt;/math&gt;root system.]]

The engineering-like articulation of her new principles is entirely discrete and group-theoretic. Boole Stott repeatedly refers to sets of polytope elements as ''groups''. All her transformations are discrete instances of one or another special case of her &quot;two inverse operations which may be called ''expansion'' and ''contraction''.&quot; Hers is an exploration of ''n''-dimensional space at its most basic: space defined by the discrete objects which fill it, themselves defined directly as groups of elements with a few operations on them, which transform them into each other. This is geometry which seems to precede Euclid's postulates, and does not depend on them. Instead it corresponds directly to the deeper group theory mathematics which underlies geometry, even though Boole Stott, like the ancients, did not have that mathematics.

Boole Stott enumerated her expansion and contraction operations as transformations between regular and semi-regular polytopes of a single edge length. In her 1910 paper she did not consider any polytopes of multiple edge lengths, or expansions of the subject polytope by a distance other than that exactly required to produce new edges of the original edge length, so of course she did not enumerate Moxness's non-uniform rhombicosidodecahedron (Hull #8). This may be why section 8&lt;sub&gt;3&lt;/sub&gt; is missing from her earlier drawings of the sections of {5,3,3}; she may not have had a construction for it.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The Boole Stott expansion (or contraction) operation is as precisely defined a transformation as Coxeter's reflection, rotation and translation operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions). Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;
where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;
where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;
For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} An instance of the expansion (or contraction) operation is an application of the general operation to a particular subject polytope, operating on all of its elements symmetrically at once. The general operation is restricted, such that one may only expand (or contract) the subject polytope by pushing all its edges, faces or cells directly away from (or toward) its center, without changing the size of the moved elements. This splits each vertex into several (or coalesces several vertices into one), and has corresponding effects on the other elements, producing new edges ''of the same size'' between pairs of separated vertices (or reducing some edges to single vertices). The distance all the elements are moved away from (or toward) the center must be such that the vertices separate producing new edges of only the original edge length (or come together reducing edges to single vertices). It is permitted to combine these three operations, e.g. by pushing the edges ''and'' the faces, or by pushing all three element sets, provided the combination results in a semi-regular polytope of only one edge length. The three operations are all commutative with one another, so combined operations can be considered either as a sequence of expansions (or contractions) linking several related polytopes, or as their concurrent movements, a single transformation of the subject polytope. Since there are 8 combinations of one to three operations there are 8 polytope variants (including the original regular polytope), and all are adjacent to one another by a single transformation. The variants of the original regular polytope are named for which element set(s) are moved. By enumerating all 8 variants of expansion (and contraction) for each of the six regular convex 4-polytopes, Boole Stott produced a table of regular and semi-regular 4-polytopes divided into 12 sets of 8: six expansion sets and six contraction sets.{{Sfn|Boole Stott|1910|loc=''Table of Polytopes in S&lt;sub&gt;4&lt;/sub&gt;''; note that Boole Stott's usage of ''S&lt;sub&gt;4&lt;/sub&gt;'' denotes a Euclidean 4-space ''E&lt;sub&gt;4&lt;/sub&gt;'', or a 3-sphere space embedded in it, ''not'' a 4-sphere space embedded in a Euclidean 5-space|p=30}} Each set includes just one of the six regular convex 4-polytopes, with its distinct unit-radius edge length, and the 7 possible semi-regular expansions (or contractions) of it, which all preserve that characteristic edge length. The polytopes in each set of 8 all have the same total number of elements as their original regular polytope, but each polytope in the set has a distinct combination of cell, face, and vertex figures. The 12 sets are not completely disjoint, and even within one set the 8 operations do not necessarily produce 8 unique polytopes. For example, the regular 5-cell (the 4-simplex polytope) has 5 cells, 10 faces, 10 edges, and 5 vertices (30 elements). Expanding the 5-cell produces 7 related semi-regular 4-polytopes of 30 elements each (two of which happen to be the same polytope). In each expansion, the 5 cells have expanded into semi-regular or regular polyhedra, the 10 faces and 10 edges ''may'' have expanded into prisms, and the 5 original vertices have expanded into regular or semi-regular cells.

== The regular 11-cell 4-polytope ==

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosadodecahedron). Each non-convex Jessen's icosahedron is a set of 6 reflex edges that are {{radic|5}} 5-cell edges, each set symmetrically arranged in 3 parallel pairs, not as far apart ({{radic|3}}) as they are long, in 3 orthogonal central planes. The 12-point (Jessen's sets of 6 reflex 5-cell edges) are inscribed in the 60-point with their endpoints as a vertex of a pentagon face. The 12 pentagon faces are joined to 5 other pentagon faces by the 5-cell edges; not to their nearest neighbor pentagons but to their next-to-nearest neighbor pentagons; and since there are 5 vertices to each pentagon face, there are 10 12-point (Jessen's sets of 6 reflex 5-cell edges). Each pentagon face is joined to another pentagon face by 5 reflex edges of five distinct Jessen's. The 5 reflex edges are Clifford parallel, the 5 edges of a pentagonal prism column that has a twist to it. The contraction of those 5 edges into 1, and those 10 vertices (two pentagons) into 2, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing all their edges as a single Jessen's. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (octahedron) by conflating its parallel pairs of edges, leaving the 11-cell's abstract 6-point (hexad cell). The hexad's situation is that it occurs in bundles of 5, disjoint but close together in the sense of adjacent, like the great circle rings in a Hopf fiber bundle. 

To summarize, the 12-point (2-hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell). We shall now be able to see how Fuller's perfect hexads combine with Todd's perfect pentads to form the perfect heptads of the 11-cell and 137-cell.  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=alias the hemi-cuboctahedron}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces. (The ''heptad'' arrives at the party, anonymously; they are our distinguished anti-celebrity, who most of the other guests have never even heard of.) The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. This 12-point (truncated tetrahedron) is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 100 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp: Truncated tetrahedron|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp: Truncated tetrahedron|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}} Of course Moxness has been there, seen that, already with his graphics software;{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} see his parameterized illustration of it exactly as it occurs in the 11-cell, below in his bestiary of A3, B3, and H3 Archimedian and Catalan polyhedra in the 120-cell.]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:Archimedean_and_Catalan_solid_hulls_with_their_Weyl_orbit_definitions.svg|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of the 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. In this catalog of Archimedean and Catalan polyhedra sighted in the 120-cell by Moxness's software,{{Sfn|Moxness: Archimedean and Catalan hulls|2023|loc=Hull #1 Archimedean Name A3 110 Truncated Tetrahedron A (upper left)}} it is the first entry (upper left), Moxness's Hull # = 1 with 12 vertices of 3D norm {{radic|11}}/2. On a {{radic|2}}-radius 3-sphere, 5 of these with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell. 20 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges are cells in the [[#The rings of the 11-cells|cell rings of the quasi-regular 11-cell]], which also has 12 regular 5-cells as cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

...

Both kinds of cells in the 11-cell have expressions in two different dimensions (on the 2-sphere and 3-sphere), because they are all centered on the center of the 3-sphere, unlike the cells of most 4-polytopes, which are centered on various off-center points within the 4-polytope's interior. 

One kind of 11-cell cell appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (semi-regular truncated cuboctahedron), a.k.a. the truncated tetrahedron. The other kind of cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now, except in the 120-cell. But this is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron, and a regular tetrahedron behind each face. The 11-cell is thus a honeycomb of mixed regular tetrahedra and truncated cuboctahedra (a.k.a. truncated tetrahedra) that meet at their shared 5-cell faces. 

Both the 5-cells (decahedra) and the truncated cuboctahedra (truncated tetrahedra) are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks in their 4-polytopes (off-center).{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} Therefore all the 11-cell's cells are concentric, which indeed makes a different kind of honeycomb, an inside-out cellular honeycomb with all the cells piled on top of each other overlapping in 4-space. They do not overlap in 3-sphere space however; their 3-volumes (but not their 4-contents) are disjoint. They make a proper 4D honeycomb like the cells of any 4-polytope, with volumetrically disjoint 3D cells that meet each other in the proper way at their shared 2D faces.

An 11-cell contains 6 completely disjoint 5-point (1-pentad 5-cell decahedron cells), and 5 volumetrically disjoint 12-point (2-hexad truncated cuboctahedron cells), 11 volumetrically disjoint cells altogether. Of course it is supposed to have 11 identical 11-point (elevenad hemi-icosahedral cells), and that is also true in its abstract sense, which is not the same sense as disjoint cellular containment: each pentad or hexad cell is inscribed in one of 11 larger 60-point (Moxness's rhombicosadodecahedron cells). The abstract pentads and hexads are only hemi-polyhedra, so the 11-cell's pentad cells are 12 (out of 120) disjoint instances of a pentad 5-point, and its hexad cells are 10 (out of 100) disjoint instances of a hexad 6-point, 1/10th of the 120-cell's cellular volume in both cases. The 11-cell's hemi-icosahedral cells are 11 (out of 120) non-disjoint instances of an elevenad 60-point. Each 11-cell shares each of its hexad cells with 9 other 11-cells, and each of its 11 vertices (out of 600) occurs in 9 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.|name=Tetrahedral vertex figures}}

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

...insert image thumb of 11-cell...

...Construct the 11-cell from the heptad....

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...Four concentric 4-polytopes may be mutually orthogonal. In any set of 4 mutually orthogonal regular 5-cells, one pair of faces in each orthogonal pair are completely orthogonal to each other. Thus three edges in one 5-cell in an orthogonal pair are opposite (completely orthogonal) to a face of the orthogonal 5-cell. .... A 5-cell has 5 vertices in its abstract hemi-icosahedral cells, but only 4 vertices in one concrete icosahedral cell; the 5th lies in an icosahedral cell of another 11-cell (not in another icosahedral cell of the same 11-cell).

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

....

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 11-cell ===

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon.... (wrong, 11-cell's Hopf map is the truncated icosahedron)

....

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk: Truncated icosahedron|2018}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

...

Since the rhombicosadodecahedron has 120 5-cell edges, the 120-cell has 120 completely disjoint 5-cells, and the rhombicosadodecahedron shares each vertex with 11 other rhombicosadodecahedra, how a rhombicosadodecahedron is bound to its neighbor 5-cells is constrained. A further constraint is that the abstract hemi-icosahedron has only 15 5-cell edges, so the 120 actual 5-cell edges of the rhombicosadodecahedron are conflated 8-to-1 in the hemi-icosahedron. To be conflated by contraction into a single edge, those 8 edges must lie parallel to each other in the 120-cell; either actually parallel in the same plane, or Clifford parallel in separate planes, as two edges in completely orthogonal planes may be parallel and orthogonal at the same time. Each 5-cell inscribed in the 120-cell contains 5 such tetrahedra occur in parallel pairs in 4 orthogonal planes (none of them completely orthogonal planes). 

.... 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
Each conflated vertex is the 10 vertices of two orthogonal regular 5-cells...
The 60-point contains 6 pairs of these orthogonal regular 5-cell cells...
...its 20 hexad cells are ..., a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere.

== The regular 137-cell 4-polytope ==

(description originally written for the 11-cell, which applies only to a fibration of 11 11-cells)

It is quasi-regular with two kinds of cells, the 12-point (regular icosahedron) and the 5-point (regular 5-cell), which have just one kind of regular face between them. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of regular facets (it is semi-regular with regular facets), it also means that its facets have expressions of two different dimensionalities because they are concentric in the 3-sphere. The 11-cell has two kinds of regular cells. One appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (regular icosahedron), but since it is embedded at the center of the 3-sphere (unlike the cells of most 4-polytopes) it also acts as an inscribed 4-polytope.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} The other cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now except in the 120-cell. This is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron. The 11-cell is a honeycomb of mixed icosahedron and half-icosahedron cells. Like the icosahedra, the decahedra are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks, off-center in their 4-polytopes. All the 11-cell's cells are concentric, which indeed makes an unusual honeycomb of cells, an inside-out honeycomb. It is a proper 4D honeycomb however, with volumetrically disjoint 3D cells that meet each other in the proper way at shared 2D faces. The embedded 12-point (regular icosahedron) cell is an icosahedral pyramid (as a 4-polytope), which is filled by 20 4-point (irregular tetrahedra) that meet at its center; the center point of the 11-cell acts as a 12th vertex, just as the center point of the 24-cell acts as a 25th vertex in some honeycombs. The 5-point (regular 5-cell) cell contains five 4-point (regular tetrahedra) which meet the 12-point (regular icosahedra)'s irregular tetrahedral cells face-to-face, so the 11-cell is quasi-regular in the usual sense of having two varieties of regular 3-polytope cells, but additionally both varieties of of its cells are bi-dimensional: another clue that the 11-cell has something to do with dimensional analogy

....

The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

....

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur to make an 11-cell. 11 of them occur in a 137-cell, all 120 of them occur in the 120-cell, and they occur nowhere else.

How can this regular 4-polytope possibly exist? By Schläfli's criterion {3, 3, ..} exists, but not as a finite polytope.{{Sfn|Coxeter|1973|loc=§7.7 Schläfli's criterion}} Coxeter found {3, 5, 3} as a finite hyperbolic honeycomb, but not in Euclidean 4-space.{{Sfn|Coxeter|1968|title=The Beauty of Geometry: Twelve Geometric Essays|loc=''Regular Honeycombs in Hyperbolic Space''|ps=; &quot;The honeycombs considered by the above authors [Schlegel, Schläfli, Klein, Sommerville] have finite cells and finite vertex figures. ... A further extension allows a cell or vertex figure to be a ''star''-polytope, so that the honeycomb covers the space several times. .... We shall find that there are four regular star-honeycombs in hyperbolic 4-space, as well as two infinite families of them in the hyperbolic plane.&quot;}} We find that {3, 3, ..} is a central expression of the &lt;math&gt;H_4&lt;/math&gt; root system, being the regular convex hull in 4 dimensions of 137 points, 96 pentads and 600 hexads. The minimal expression of &lt;math&gt;H_4&lt;/math&gt; is the regular convex hull {3, 3, 5} of 120 points, 600 tetrads and 75 hexads, and its maximal expression is the regular convex hull {5, 3, 3} of 600 points, 120 pentads and 675 hexads. {3, 3, ..} is a third regular &lt;math&gt;H_4&lt;/math&gt; polytope which fits between them. The properties and metrics of this regular sequence are presented in tabular form below in ''[[#Build with the blocks|§Build with the blocks]]''.

{|class=&quot;wikitable&quot;
!colspan=3|Regular &lt;math&gt;H_4&lt;/math&gt; polytopes
|-
|[[File:600-cell.gif|200px]]
|[[File:5-cell.gif|200px]]
|[[File:120-cell.gif|200px]]
|-
!120-point 600-cell
!137-point 137-cell
!600-point 120-cell
|}

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 137-cell ===
....

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=Apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|2022|loc=''[[Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]''}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie: On Fuller's use of language|2024|loc=''[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry'']]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center, 1948-1949|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with their mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the gender neutral shad can open their six beaks all the way, until they become flat squares and they becomes a cuboctahedron, or they can shut them all tight like a turtle retracting into their octahedron shell. The six mouths always move in unison. This is [[Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do their ''really'' odd trick -- where they flip their 6 jaws 90 degrees in their 6 faces and shut their 6 beaks on the opposite axis of their squares than the one they opened them on -- or whether they will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than their normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make their edge length exactly the same size as their radius, when they open their mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing their mouths in spherical synchrony, their 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron (the hexad), then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=''[[W:SO(4)|SO(4)]]''}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=Isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its orthogonal hexad's planes (wz, wx, wy).{{Efn|name=Six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of (expanded octahederon 16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh &amp; Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== Legendre's 12-point hexad binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a 12-point double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's perfectly realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, this 2-hexad expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre hexad) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction.{{Efn|name=Concentric polyhedra}} In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== ''n''-simplexes ===
....

The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== ''n''-orthoplexes ===
....
[[File:Great 5-cell √5 digons rectangle.png|thumb|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} ''regular'' great hexagons. These are the same 200 dodecagon central planes illustrated above, which also contain 6 120-cell edges and two opposing ''irregular'' great hexagons (truncated {{radic|6}} triangles). The 1200 {{radic|5}} 5-cell face triangles occupy 1200 other, non-central planes.]]

...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's Hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[16--cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[16--cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=Concentric polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[16--cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== ''n''-taliesins ===
The 4-simplex pentad building block combines with the 4-orthoplex hexad building block to make the 4-taliesin heptad building block. The heptad is the essential building block of the ''taliesin'' family of polytopes. The 11-cell and 57-cell hemi-polytopes and the 137-cell and 120-cell regular 4-polytopes are nuclear members of this trans-dimensional family. The 24-cell and 600-cell are extended members of it, because they also contain instances of the heptad building block.

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its spherical symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* [[W:Taliesin Myrddin Namkai-Meche|Taliesin Myrddin Namkai-Meche]], a young Reed College scholar who was stabbed to death by a crazy man in 2017 on a Portland, Oregon light rail train, when he rose to the nonviolent defense of two black teenagers who were being violently assaulted in racist and anti-Muslim language by a right-wing extremist.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in orthogonal [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].
* '''Taliesan polytope''', an (''n+1)''-polytope whose ''n''-polytope dimensional analogue inscribed concentrically in the ''n''-sphere forms a convex honeycomb of itself which is the (''n+1)''-polytope.{{Efn|name=Embedding point}}
* The taliesan polytope's honeycomb, a Hopf fibration of its great circles.
* The ''n''-dimensional family of dimensionally analogous polytopes related to a taliesan polytope by expansion and contraction operations.
* The characteristic [[w:Orthoscheme|orthoscheme]] of a teliesan polytope, an irregular simplex that is the complete gene for the entire expression of trans-dimensional symmetry relationships among the polytopes of the ''n''-dimensional family, as well as the complete gene for the teliesan polytope's rotational symmetry.{{Efn|name=Characteristic orthoscheme}}

=== ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== ''n''-equilaterals ===
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed at four different [[#n-leviathon|chord]] lengths from each other. In the {{radic|2}}-radius polytopes we have using as examples in this paper they are {{radic|2}}, {{radic|4}}, {{radic|6}} and {{radic|8}}, but in a ''unit-radius'' polytope they are: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}.

Each vertex is joined to 8 others by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}. The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== ''n''-pentagonals ===

[[File:600-cell.gif|thumb|The 120-point (600-cell), &lt;math&gt;H^4&lt;/math&gt;.]]
...

[[File:600-cell vertex geometry.png|thumb|Vertex geometry of the 600-cell, showing the 5 regular great circle polygons and the 8 vertex-to-vertex chord lengths with angles of arc. The golden ratio{{Efn|name=golden chords|group=}} governs the fractional roots of every other chord,{{Efn|name=fractional root chords}} and the radial golden triangles which meet at the center.|alt=|400x400px]]
The 120 vertices of the 600-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏&lt;sup&gt;−1&lt;/sup&gt;) beginning with a vertex; see column ''a''}} at eight different [[#n-leviathon|chord]] lengths from each other. These edges and chords of the 600-cell are simply the edges and chords of its five great circle polygons. In ascending order of length in a ''unit-radius'' polytope (not in a {{radic|2}} radius polytopes like our usual examples), they are {{radic|0.𝚫}}, {{radic|1}}, {{radic|1.𝚫}}, {{radic|2}}, {{radic|2.𝚽}}, {{radic|3}}, {{radic|3.𝚽}}, and {{radic|4}}.{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=Decimal fractions of square roots|group=}}

Notice that the four [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#''n-''equilaterals|hypercubic chords]] of the 24-cell ({{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|4}}) alternate with the four new chords of the 600-cell's additional great circles, the decagons and pentagons. The new ''golden chord'' lengths are necessarily square roots of fractions, but very special fractions related to the golden ratio{{Efn|1=The fractional-root ''golden chords'' are irrational fractions that are functions of {{radic|5}}. They exemplify that the [[W:golden ratio|golden ratio]] &lt;big&gt;𝜙&lt;/big&gt; {{=}} {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618 is a circle ratio related to &lt;big&gt;𝜋&lt;/big&gt;:&lt;br&gt;
: {{sfrac|𝜋|5}} = arccos (𝜙/2)
is one decagon edge, the 𝚽 = {{radic|0.𝚫}} = {{radic|0.382~}} ≈ 0.618 chord.
Reciprocally, in this function discovered by Robert Everest expressing &lt;big&gt;𝜙&lt;/big&gt; as a function of &lt;big&gt;𝜋&lt;/big&gt; and the numbers 1, 2, 3 and 5 of the Fibonacci series:&lt;br&gt;
: &lt;big&gt;𝜙&lt;/big&gt; = 1 – 2 cos ({{sfrac|3𝜋|5}})
{{sfrac|3𝜋|5}} is the arc length of the &lt;big&gt;𝜙&lt;/big&gt; = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618 chord.|name=golden chords|group=}} including the two [[W:golden section|golden section]]s of {{radic|5}}, as shown in the diagram.{{Efn|The 600-cell edges are decagon edges of length {{radic|0.𝚫}}, which is 𝚽, the ''smaller'' golden section of {{radic|5}}; the edges are in the inverse [[W:golden ratio|golden ratio]] {{sfrac|1|φ}} to the {{radic|1}} hexagon chords (the 24-cell edges).
The other fractional-root chords exhibit golden relationships as well. The chord of length {{radic|1.𝚫}} is a pentagon edge.
The next fractional-root chord is a decagon diagonal of length {{radic|2.𝚽}} which is &lt;big&gt;𝜙&lt;/big&gt;, the ''larger'' golden section of {{radic|5}}; it is in the golden ratio{{Efn|name=golden chords|group=}} to the {{radic|1}} chord (and the radius).{{Efn|Notice in the diagram how the &lt;big&gt;𝜙&lt;/big&gt; chord (the ''larger'' golden section) sums with the adjacent 𝚽 edge (the ''smaller'' golden section) to {{radic|5}}, as if together they were a {{radic|5}} chord bent to fit inside the {{radic|4}} diameter.}}
The last fractional-root chord is the pentagon diagonal of length {{radic|3.𝚽}}.
The [[W:Pentagon#Side length is given|diagonal of a regular pentagon]] is always in the golden ratio to its edge, and indeed &lt;big&gt;𝜙&lt;/big&gt;{{radic|1.𝚫}} is {{radic|3.𝚽}}.|name=fractional root chords|group=}}

....


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== ''n''-leviathon ===
...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of its &lt;math&gt;10^2&lt;/math&gt; disjoint instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction, before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]].{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}} .... It is only because she did not have the 11-cell to start with that she did not discover the regular 137-cell {3, 3, ..} by her method.

Between Alicia Boole Stott and her identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and her identification of symmetry groups with conservation laws (in her 1st theorem), we owe the deepest mathematics of geometry and physics originally to two pioneering women, colleagues of their contemporaries the greatest men of the academy, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just an [[W:abstract polytope|abstract 4-polytope]], not just a singleton convex 4-polytope, and not just a honeycomb.{{Sfn|Coxeter|1970|loc=''Twisted Honeycombs''}} The 11-point (11-cell) has a concrete ...regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}&lt;nowiki/&gt;} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, 3, 3} as the 137-point (137-cell).

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 11-cell, 137-cell and 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=''Clifford Spinors and Root System Induction: H4 and the Grand Antiprism''}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=Apology}}

== Build with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Duveneck|1978|loc=Carl Jung, quoted in ''Life on Two Levels''|p=ii|ps=.{{Sfn|Jung|1961|ps=: &quot;The best of truths is of no use unless it has become one's most personal inner experience. It is the duty of everyone who takes a solitary path to share with society what he finds on his journey of discovery.&quot;}}}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1954|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius={{radic|2}}}}

== Acknowledgements ==
...

== Notes ==
{{Regular convex 4-polytopes Notelist}}

== Citations ==
{{Reflist}}

== References ==
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      <text bytes="208171" sha1="iqs9habvw2zvay480pjqubal71owfcw" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|June 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a regular 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the &lt;s&gt;120-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 5, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 11 [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] cells. Eleven 11-cells (plural) are the 137-cell, the &lt;s&gt;600-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 3, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 137 [[W:Rhombic triacontahedron|triacontahedron]] cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=''Regularity of Graphs, Complexes and Designs''}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=''A Symmetrical Arrangement of Eleven Hemi-Icosahedra''}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]]
The 4-dimensional regular polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:Simplex|simplex]] (called the [[5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:Orthoplex|orthoplex]] (called the [[16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular convex 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the [[w:Tesseract|16-point (8-cell) tesseract]], the 4-cubic, which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found within any of the other regular convex 4-polytopes, except in the largest and most complex one, the [[120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). However, we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the 16-cell is their very starting point, and the most frequently used tool in the box.

== 5-cells and hemi-icosahedra in the 11-cell ==
[[File:Symmetrical_5-set_Venn_diagram.svg|thumb|The 5-point (10-face) regular 5-cell (the regular 4-simplex). Grünbaum's rotationally symmetrical 5-set Venn diagram{{Sfn|Grünbaum|1975|loc=''Rotationally symmetrical 5-set Venn diagram'', Fig 1 (e)|ps=; partitions the individual elements of the 5-cell.}} is an illustration of the 5-cell labeling each of its &lt;math&gt;2^5&lt;/math&gt; elements.{{Sfn|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled; image includes the Python code to render it, optimising for maximum area of the smallest regions.}}]]
[[File:Hemi-icosahedron.png|thumb|The 6-point (10 face) [[W:hemi-icosahedron|hemi-icosahedron]], an abstraction of the regular icosahedron, has half as many faces, edges and vertices. Each element of the abstract polyhedron represents two real elements found in different places in the 11-cell's realization.{{Sfn|Ruen: hemi-icosahedron|2007}}]]
The most apparent relationship between the pentad 5-cell and the hexad hemi-icosahedron is that they both have 10 triangular faces. When we see a facet congruence between a 4-polytope and a 3-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 6-point 5-simplex. It is also the same size as the 6-point 3-orthoplex; thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no chord congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals, pentad and hexad, who are famously stand-offish with each other. Interesting!

The cell of the 11-cell is an abstract hexad hemi-icosahedron containing pentads in its interior, most handsomely illustrated by Séquin.{{Sfn|Séquin &amp; Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing rings of polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The 11-cell's 11 hemi-icosahedral cells have 10 triangle faces each. If we assume a 5-cell is behind each face, only 3 of the 5-cell's 5 vertices are visible on the surface of the hemi-icosahedron as the exposed face. The 5-cell's other two vertices must define an edge on the opposite side of the 11-cell. Coxeter determined that the 11-cell does indeed have an edge opposite each face, but that it does not belong to the same hemi-icosahedral cell as its opposing face.{{Sfn|Coxeter|1984|p=110|loc=§6. The Petrie polygon [of the 11-cell]|ps=; &quot;We may reasonably call this edge and face ''opposites''. It is easy to find the face opposite to a given edge by looking at the faces to which a given edge belongs. ... Conversely, given a face, we can find the opposite edge by seeing which vertices belong to neither of the hemi-icosahedra which share that face. The ten edges opposite to the ten faces of one hemi-icosahedron are the edges of the complementary &lt;math&gt;a_4&lt;/math&gt; [4-simplex], that is, the joins of all pairs of the five vertices [of the 11-cell] not belonging to the given hemi-icosahedron.&quot;}}

There are 11 disjoint 5-cells in each 11-cell, which is comprised of 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. The real 11-cell is harder to see than the abstract 11-cell that represents it, because the real hemi-icosahedron that the abstract hemi-icosahedron represents is hard to see, but seeing these objects will be easier once we have identified the real hemi-icosahedron and exactly where the 11-cell's real elements reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each hemi-icosahedron's 10 faces come from 10 disjoint 5-cells, and there is just one 5-cell with which it does not share a face.{{Efn|As Coxeter observes (in the previous citation), that unrepresented 5-point (5-cell) is the other 5 vertices of the 11-point (11-cell) that are not vertices of this 6-point (hemi-icosahedron), the hemi-icosahedron's disjoint complement.}} Each 5-cell has 10 faces from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share a face.

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in two different planes which are completely orthogonal to each other. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must be sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness: 8 concentric hulls|2022|loc=Hull #8 (lower right)|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell of the 11-cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Compounds in the 120-cell|§Compounds in the 120-cell]]'' that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software which uses Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s to render the polyhedra which are found in the interior of ''n''-dimensional polytopes.{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the tetrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is a real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (Hull #8) is a concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this {12} irregular great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing {6} irregular great hexagons (truncated triangles) of alternating 120-cell edges and 5-cell edges. The great dodecagon is a great circle of the hemi-icosahedral cell, and also a great circle of the 120-cell, because unlike the cells of most 4-polytopes, the 11-cell's cells are embedded in 4-space at the exact center of the 4-polytope, making them inscribed 4-polytopes themselves.{{Efn|name=Embedding point}} There are 25 great dodecagons and 120 5-cell edges in Moxness's 60-point (Hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (Hull #8) polyhedron altogether.

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point (Hull #8) is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] similar to the one from the catalog shown here,{{Sfn|Piesk: Rhombicosidodecahedron|2018}} but a slightly shallower truncation with smaller red pentagons (120-cell faces) and narrower rhombs. Rhombicosidodecahedra are made by truncating the [[W:Rhombic triacontahedron|rhombic triacontahedron]], which is the 30-sided polyhedron with only one kind of face, the dual of the 30-point icosidodecahedron. The 120-cell contains 120 of these 11-cell cells.]]
Moxness's 60-point (Hull #8) is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (Hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|The 5-point (regular 5-cell) can be another abstraction of Moxness's 60-point (Hull #8), 12-vertices-into-1, but it is not strictly the result of a contraction operation as described by Boole Stott, since she did not apply her expansion and contraction operations to uniform polytopes with more than one edge length.}} In fact Moxness was not the first to find a rhombicosidodecahedron in the 120-cell; Alicia Boole Stott was. It is the 6th section of {5,3,3} beginning with a cell, which Boole Stott identified as the ''e&lt;sub&gt;2&lt;/sub&gt; expansion'' of the icosahedron (or equivalently of its dual polyhedron the dodecahedron).{{Sfn|Boole Stott|1910|loc=§Examples of the e&lt;sub&gt;2&lt;/sub&gt; expansion|p=7}} But that rhombicosidodecahedron identified by Boole Stott is not Moxness's Hull #8: rather, it is the semi-regular Archimedean solid, with a single edge length and square faces. Moxness's Hull #8 with its two distinct edge lengths and rectangular faces is Coxeter's 8th section of {5,3,3} beginning with a cell, which is missing from the sections illustrated by Boole Stott.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than tetrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. The 120-cell contains 120 of Moxness's 60-point (11-cell cells). They are non-disjoint, with 12 60-points sharing each vertex.

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (Hull #8) is the real hemi-icosahedron cell, with 120 5-cell edges in it as chords lurking under its surface. The rhombicosadodecahedron has enough of those 5-cell edges in its interior to make 20 disjoint tetrahedral cells which belong to 20 distinct 5-cells.

The 600-point (120-cell) may be constructed as a semi-regular honeycomb of two kinds of cells, the 120 non-disjoint 60-point (Moxness's rhombicosadodecahedra) and their neighbors the 120 disjoint 5-point (5-cells), cell-bonded together by shared tetrahedral cells of edge-length {{radic|5}}. Each rhombicosadodecahedron is bonded to 20 islanded 5-cells, sharing with each a 5-cell tetrahedron (4 of its interior 5-cell faces, and 6 of its interior 5-cell edges). The rhombicosadodecahedron is also bonded directly to 10 other rhombicosadodecahedra (since each 11-cell has 11 rhombicosadodecahedron cells), again not by sharing one of its exterior faces, but this time by sharing an interior face, a {{radic|6}} triangle which lies in a central plane.{{Efn|The {{radic|6}} chord is another hidden chord lurking beneath the 60-point rhombicosadodecahedron's surface, like the {{radic|5}} 5-cell edge. Both these hidden chords lie on the same great circle with the 120-cell edges (the pentagon face edges). Three {{radic|6}} chords make a triangle which lies in a central plane. This {{radic|6}} great triangle must not be confused with the {{radic|5}} face triangle of the 5-cell, which does not lie in a central plane.}} The 120 rhombicosadodecahedra are also bonded directly together at ''all'' their exterior faces: at their pentagon faces (which are 120-cell faces of dodecahedral cells); at their small triangular faces (which are 600-cell faces of small tetrahedral cells, 10 of which are inscribed in each dodecahedral cell); and at their rectangular faces (which are faces of 10-point [[w:Pentagonal_prism|pentagonal prism]] cells, where the pentagonal prism is a column inscribed in a dodecahedral cell, joining its opposite pentagon faces). The 120 rhombicosadodecahedra are bonded directly to each other at their own exterior faces; no dodecahedra, small tetrahedra, or pentagonal prisms lie between them, and those objects are not cells of this honeycomb; we mention them here only to help locate the rhombicosadodecahedra within the 120-cell. The only objects which fill holes between the rhombicosadodecahedra are the 120 islanded 5-cells.{{Efn|This honeycomb of the 120-cell resembles the [[600-cell#Icosahedra|honeycomb of the 600-cell consisting of 24 face-bonded regular icosahedra]] with 24 clusters of 5 tetrahedra filling the holes between them. In the 120-cell there are 5 times as many of everything as in the 600-cell, so there are 120 (rhombicosadodecahedron) icosahedral cells with 600 tetrahedra filling the holes between them.}}

It is possible for rhombicosadodecahedra to meet each other at more than one kind of exterior face because they are not meeting there as cells of the same 11-cell; they belong to different 11-cells. The 11 rhombicosadodecahedra in the same 11-cell do not meet at their own exterior faces, they meet at their interior {{radic|6}} triangle faces. As a cellular 4-polytope, the 11-cell is inside-out. Its cells meet each other at interior planes, sharing interior {{radic|6}} triangle faces, instead of in the usual way that 4-polytopes' cells share an exterior face. It is a deeper kind of intertwining to be sure, but the 11 rhombicosadodecahedra in the same 11-cell are still volumetrically disjoint cells, and the 11-cell is a perfectly real 4-polytope with real cells. It is a semi-regular 4-polytope, because it has two different kinds of cells: 11 rhombicosadodecahedra and 11 5-cells. The 11 5-cells are islanded (not bound to each other), and the 11 rhombicosadodecahedra are bound both to 5-cells (at {{radic|5}} 5-cell faces) and to each other (at {{radic|6}} central triangle faces). In the section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#What's in the box|§What's in the box]]'', below, we will look closely at the relation between the {{radic|5}} chord and the {{radic|6}} chord, which is the relation between the pentad and the hexad, and at the general phenomenon of hemi-polytopes which meet each other at a central plane. Then in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Cell rings of the 11-cells|§Cell rings of the 11-cells]]'' we will look at the two-cell cellular structure of the 11-cell in more detail.

Considered as a single complex of overlapping cells rather than as distinct honeycombs of volumetrically disjoint cells, the 120-cell contains 60-point (rhombicosadodecahedral cells), islanded 5-point (5-cells), 20-point (dodecahedral cells), and 10-point ([[w:Pentagonal_prism|pentagonal prism]] cells). The 60-points meet each other in pairs at their pentagon faces, meet regular dodecahedra at those pentagon faces also, meet each other in quads at their 600-cell faces where they share an interior 4-point (600-cell tetrahedral cell) and its four faces (just as they share a larger interior 5-cell tetrahedron and its faces), and meet each other and the 10-point (pentagonal prisms) at their rectangle faces. We might try to imagine such a complex with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with parts truncated from 30 neighboring truncated dodecahedra pentagonal prisms). There are 120 of each of these 3 kinds of dodecahedron. 

Moxness's 60-point (rhombicosadodecahedron cell) is a central object of the 120-cell, and contains many wonders. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra by expansion and contraction operations, as Alicia Boole Stott discovered before 1910,{{Sfn|Polo-Blanco: ''Theory and history of geometric models of Alicia Boole Stott''|2007|loc=§5.3.2 1910 paper on semi-regular polytopes|pp=152-158|ps=; summarizes Boole Stott's method and results from {{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}}, including her tables of expansion-contraction dimensional analogies and a few of her illustrations.}} and those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, rediscovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter &amp; Petrie|1938|p=4|loc=''[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]''}}|name=Concentric polyhedra}} Before we move on from the 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice one more thing about the hidden {{radic|5}} chords lurking below its surface. The 12 little pentagon faces are spanned by sets of 6 {{radic|5}} chords (5-cell edges). The 6 chords of each set are disjoint (they don't touch or form 5-cell faces), but they are symmetrically arranged in 3 parallel pairs, {{radic|3}} apart, which lie in 3 orthogonal central planes. It turns out they are the 6 reflex edges of a 12-point non-convex polyhedron called the [[w:Jessen's_icosahedron|Jessen's icosahedron]], inscribed in the 60-point cell. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again and work our way outwards, before eventually in ''[[#The perfection of Fuller's cyclic design|§The perfection of Fuller's cyclic design]]'' we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cell)s ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks, showing its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Pentahemidemicube.png|120px]]
|[[File:Tetrahemihexahedron_rotation.gif|120px]]
|[[File:Pentahemicosahedron.png|120px]]
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts needed to build the 120-cell (the picture on the cover of the box), and everything else that's invisible in that picture which fits inside the 120-cell: all the astonishing regular 4-polytopes. The pentad building block is half a pair of orthogonal 5-point (regular 5-cells), the hexad building block is half an 8-point (16-cell), and the heptad building block is a quarter of an 11-point (11-cell). Everything can be built from these blocks by embedding them at the center of the 3-sphere, inscribing them as 4-polytopes to make compounds as with the 16-cell building block, rather than by assembling them face-to-face as 3-polytopes to make honeycombs in which they are cells or vertex figures.{{Efn|name=Embedding point}}

The pentad building block is the 5-point '''pentahemidemicube''', an abstract polyhedron with 10 equilateral triangle faces, 10 {{radic|5}} edges, and 5 vertices.{{Sfn|Christie: Pentahemidemicube|2024|loc=File:Pentahemidemicube.png|ps=; &quot;The 5-point ''pentahemidemicube'' is an abstract polyhedron with 10 equilateral triangle faces, 10 edges, and 5 vertices. It has a real presentation as the regular 5-cell.&quot;}} It has a real presentation as the 5-point (regular 5-cell 4-polytope), a regular tetrahedron expanded by addition of a fifth vertex equidistant from all the others in a fourth dimension. The pentad block is a tetrahedral pyramid (a 5-cell) inscribed in a cube: a tetrahedron inscribed in a cube (a demicube) expanded by addition of a fifth vertex, one of the cube's other 4 vertices. Each pentad (5-cell) building block lies in the 120-cell orthogonal to three other pentad (5-cell) building blocks. They are four completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges. Each regular 5-cell in the 120-cell is a pentad (pentahemidemicube) building block, in one of 30 orthogonal sets of 4. There are 120 pentad building blocks in the box.

Every vertex of the 16-cell, and therefore every vertex of the 8-cell, 24-cell, 600-cell and 120-cell, is surrounded by a 6-point (octahedral vertex figure). The hexad building block is that 6-point octahedron embedded at the center of the 4-polytope instead of at a vertex, that is, inscribed in the 3-sphere as a 4-polytope, the 6-point '''[[w:Tetrahemihexahedron|tetrahemihexahedron]]''' or '''hemi-cuboctahedron'''.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; &quot;The tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U&lt;sub&gt;4&lt;/sub&gt;. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices.&quot;}} The hexads have 4 red triangle 5-cell faces, bounded by 12 {{radic|5}} edges. Their 3 yellow square faces lie in central planes, because the building block is an abstract 4-polytope. The hexad is an irregular abstract polyhedron that also has 6 {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

Two orthogonal 5-point (pentad) building blocks intersect in a 4-point (inscribed tetrahedron) to form a 6-point (hexad) building block. Two orthogonal 6-point (hexad) building blocks intersect in a 4-point (central square) to form an 8-point (16-cell) building block, the most usable building block, from which everything larger can be built by compounding.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 120 pentads to pair with a completely orthogonal hexad, and although there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed) they only result in 120 pairs two different ways, so there are 120 11-cells. The 11-cell is 6 pentads' and 5 hexads' smallest joint compound, and the 120-cell is their largest, 120 pentads and 100 hexads.

The heptad{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''|pages=22–31}} building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|4}} edges and 9 {{radic|5}} edges), and 7 vertices.{{Sfn|Christie: Pentahemicosahedron|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;}} It is a compound of a 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]) and a 6-point (hexad [[W:hemi-icosahedron|hemi-icosahedron]]) intersecting in a 4-point (tetrahedron), that is, of three orthogonal 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]&lt;nowiki/&gt;s) intersecting in their common 4-point (tetrahedron). It is also the abstract cube contracted by removal of one vertex, which reduces one of the cube's two inscribed tetrahedra (demicubes) to a single remaining triangular face. As with all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Four &lt;small&gt;&lt;math&gt;(11-7)&lt;/math&gt;&lt;/small&gt; orthogonal 7-point (pentahemicosahedron) blocks can be joined at a common Petrie polygon (a skew hexagon which contains their orthogonal 6-edge paths) to construct a real 11-point ([[W:11-cell|11-cell]] semi-regular 4-polytope), which magically contains six &lt;small&gt;&lt;math&gt;(11-5)&lt;/math&gt;&lt;/small&gt; 5-point (1-pentad regular decahedra) as cells and five &lt;small&gt;&lt;math&gt;(11-6)&lt;/math&gt;&lt;/small&gt; 12-point (2-hexad semi-regular cuboctahedra) as cells. There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like [[W:Euclid's postulates|Euclid's postulates]], and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as honeycombs of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles. It possesses the complete symmetry of the polytope without any redundancy, because it contains one of each of the polytope's characteristic root elements.{{Efn|A [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be partitioned radially by its planes of symmetry (Coxeter's &quot;reflecting circles&quot;) into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated rotationally to generate its regular 4-polytope because it is the complete [[W:gene|gene]] for it, containing all of its elements and capturing all of its symmetry without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring that triggers the reflections.|name=Characteristic orthoscheme}} It is the gene for the polytope, which can be replicated to construct the polytope.

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its &lt;math&gt;A_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the regular compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;small&gt;&lt;math&gt;120\times 120 = 14400&lt;/math&gt;&lt;/small&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[16--cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its &lt;math&gt;B_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 384. The 120-cell is the convex hull of the regular compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 24-point (24-cell) is subdivided into 1152 instances of its [[24-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;F_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 1152. The 120-cell is the convex hull of the regular compound of 25 disjoint 24-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;25\times 1152 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has three equivalent origins in distinct root systems, though most are obviously constructed from one root, and not so obviously from the others. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything larger than the pentad can be constructed by compounding hexads, and we shall see that everything as large as a 24-cell has a construction from heptads which are a product of a pentad and a hexad. Surely every uniform polytope can be constructed by some function of expansions and contractions from any of three root systems, however indirect the recipe. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes, related by an equals sign between them, then give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics by [[W:Noether's theorem|Noether's theorem]].

=== Pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 octahedral cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell contains the 600 octahedral cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It also contains the 600 cubical cells of its 75 8-cells. It is a 600-cell in three ways, although the octahedra and cubes are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell and the 8-cell, respectively.|name=600 cells three ways}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (1-pentad 5-cell), out of the 8-point (2-hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (1-pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope that contains 5-point (1-pentad 5-cell) building blocks beside the largest one, it just isn't a ''regular'' 4-polytope. It is semi-regular with two kinds of cells, the 4-point (regular tetrahedron) and the 12-point (truncated cuboctahedron), which have just one kind of regular face between them, a 5-cell face. That semi-regular 4-polytope is the 11-cell.

We said earlier that the 11-cell had 11 disjoint 5-cells and 11 hemi-icosahedral cells in it, and that its construction from them might be hard to see, but now we are going to account numerically, at least, for how it can have 6 pentad cells plus 5 hexad cells magically adding to 11 of the same kind of elevenad cell. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, 11. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (5-cell regular decahedron) cells, and the 12 6-point (hexad) building blocks into 100 12-point (regular icosahedron) cells, and then magically adding one whole 5-point (5-cell regular decahedron) cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 12-point (regular icosahedron) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such (cell bundles of 6 + 5 = 11 cells), with each 11-cell bundle containing 11 of both kinds of cell. Very, very strange!{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden'': §II To Add is to Multiply|p=38|ps=; the sequence of regular polygons with harmonious proportions beginning {5}, {7}, {9}, {11} as a family of natural numbers; &quot;The numbers in this optimal family have remarkable properties [that allow] similar figures to be arranged easily or allow a figure to be dissected into a set of similar figures. This is the origin of the Greek idea of geometric progression — that ''multiplication can be accomplished instead by addition''.... For instance, it is well known that the square of the golden ratio is one more than itself (written above as 𝜙&lt;sup&gt;2&lt;/sup&gt; = 𝜙 + 1) and one less than 𝜙 is its reciprocal (1/𝜙 = 𝜙 - 1). The first relation equates multiplication with addition, while the second accomplishes division through subtraction. [These are pentagon {5} bisectional golden ratios, but] the heptagonal [{7} tri-sectional] ratios also behave in surprising ways — adding to multiply, subtracting to divide.... There are analogous constructions for [nonagon {9}] quadrisectional and [hendecagon {11}] pentasectional systems.&quot;}}

The 120-cell has three interchangeable cyclic expansion-contraction cycles, the pentad, hexad and heptad cyclic designs, rooted in the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems, respectively. Each of them constructs the whole sequence of regular 4-polytopes from a single building block. The key to all three cycles is that 5 + 6 = 11, and there is an 11-point quasi-regular 4-polytope in the sequence of convex 4-polytopes. It lies between the 8-point (16-cell) and the 16-point (8-cell), and is itself a &lt;math&gt;B_4&lt;/math&gt; polytope. Its role is to be a unified single building block common to all three cyclic expansion-contraction cycles. It is involved with the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems and expresses in one building block their relations to each other.

We have already seen how the hexad building cycle works. It is plain compounding of 4-orthoplexes at the beginning, increasing to densely intricate non-disjoint symmetry toward the end, making it difficult to see why it works once past the 24-cell. This is how the build works, in terms of sixes. The 16-cell is a 2-hexad (3 pairs of completely orthogonal great square planes). The 24-cell is 3 16-cells and a 4-hexad, 4 different ways: 16 non-disjoint hexads (great hexagon planes). The 120-point (600-cell) is 25 24-cells and a 20-hexad, 10 different ways: 200 non-disjoint hexads (great hexagon planes). The 600-point (120-cell) is 225 (9 times 25) 24-cells. ....

The pentad building cycle doesn't do plain compounding to regular 4-polytopes until the last one, the 120-cell. Fives don't even appear as pentagons until after the 24-cell. We have seen how there are 11 5-cells in the quasi-regular 11-cell, which occurs before the 8-cell, and also how everything can be compounded from irregular 5-cells (4-orthoschemes specialized for each 4-polytope). It is still not evident how a pentad cycle can work to build all the regular 4-polytopes from the regular 4-simplex. This is how is how the build works, in terms of fives. ....

This is how the build works, in terms of elevens. The 11-cell is 4 orthogonal heptads (intersecting in their skew hexagon Petrie polygon). The 24-cell is also 4 heptads (intersecting in 4 orthogonal digon axes, so each heptad contributes 6 unique points to a single fibration of 24 points). Heptads are quarter elevenads, so the 11-cell and 24-cell are both 1 elevenad. ....

See the tabular sequence of convex 4-polytopes in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Build with the blocks|§Build with the blocks]]'', below: table rows ''Inscribed, Pentad, Hexad, Heptad''.

=== 11 of them ===
There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of eleven 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum eleven of them. There is only a single Hopf fibration of the 11 great circles of an 11-cell. 

The 5-cell and the 11-cell are limit cases. The 5-cells are completely disjoint, but the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 10 other instances of itself.

== Cell rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer &amp; Segerman|2013|loc=''Puzzling the 120-cell''}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his section of the Wikipedia article on the 120-cell entitled ''[[120-cell#Visualization|§Visualization]]''.{{Sfn|Ruen &amp; Goucher et al. eds. 120-cell|2024|loc=Goucher's ''[[120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious place to look for the 11-cell's Hopf map is in Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other place it could be hiding. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The quasi-regular convex 11-cell|§The quasi-regular convex 11-cell]]'', and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen: triacontagon|2011|loc=''Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them except his illustrations. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Four orthogonal projections of the 120-cell to a central plane, selected from the table of 3 regular and 11 compound 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|100 icosahedra&lt;br&gt;120 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|5 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 icosahedra&lt;br&gt;6 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|120
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|4
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|50
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|7.2°
|align=center|36°
|align=center|6°
|align=center|7.2°
|}

....

Here we summarize our findings on the 11-cells so far, including in advance some results to be obtained in the next section two sections.

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section....

== The 15 major chords ring everything ==
The natural numbers are each a distinct flavor.

Each integer has its own distinct taste. The first six integers {0..5} are the six strongest, most familiar tastes in nature, instantly distinguished and utterly unique.

The seventh flavor {6} is the first ratio of two of the others. Thereafter follows in strange sequence all the ratios of all the flavors, each marriage of every two flavors, most of which are fractions &lt;math&gt;\{k/d\}&lt;/math&gt;, but some of which are integers &lt;math&gt;\{k\}&lt;/math&gt;.

Some of these ratios are successful flavors in their own right, as distinct and inimitable as the marriage of chocolate and oranges, but the marriage of many flavor pairs is mud. It is a rule of thumb among flavor specialists that nearly any taste can be successfully alloyed with either onions or chocolate, but the direct marriage of these two flavors is notably unsuccessful. Some flavors make famous pairings or infamous ones, and subtle new pairings are always to be discovered that work surprisingly well for some purpose, but the majority of flavor marriages are undistinguished compounds.

In polytope geometry, each chord of a polytope is both is a distinct 1-dimensional object, a chord of the sphere of a distinct length &lt;math&gt;l&lt;/math&gt;, and a distinct natural number &lt;math&gt;h&lt;/math&gt;, a unique flavor. If the polytope is regular, it is a noteworthy and successful flavor. The chord's length &lt;math&gt;l&lt;/math&gt; is a square root, related to the natural number &lt;math&gt;h = k/d&lt;/math&gt; and to a set of polytopes they both represent, by a formula discovered by Steinbach.{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden''; Figure 5. Optimal sections and proportions|p=37|ps=; the regular polygons {5}, {7}, {9} and {11} with their diagonals define respectively: {5} the golden bisection proportional to 𝜙; {7} an analogous trisection; {9} an analogous quadrasection; {11} an analogous pentasection.}} The chord length &lt;math&gt;l&lt;/math&gt; is related to the number of sides of the regular polygon formed by the chord (its numerator &lt;math&gt;k&lt;/math&gt;), and to the winding number or density of its regular skew Petrie polygon (its denominator &lt;math&gt;d&lt;/math&gt;).

The regular 4-polytopes do not contain an infinite variety of these chord flavors, but only 30 of them: only 15 unique pairs of 180° degree complementary chords, which combine to make every regular ''n''-dimensional thing found in nature.{{Efn|The 15 ''major chords'' suffice to construct all the regular ''n''-polytopes. Their 180° complements include 15 ''minor chords'' which occur only in the 120-cell, not in any smaller regular polytope.{{Efn|name=Major and minor chords}} Including the complement of the 180° {{radic|4}} digon {2} chord (the {0} chord {{radic|0}}), and the skew variant of the 90° {{radic|2}} square {4} chord (the {30/7} chord), there are 32 distinct natural numbers which are all the chords of all the regular polytopes in all dimensions.|name=32 chords}} All the regular polytopes (including the largest one on the cover of the box) can be built not only from a box containing a number of blocks of 3 shapes glued together by gravity, they can be also be built from a box containing a number of sticks of 15 lengths glued together by rubber joints.

[[File:15 major chords.png|500px|thumb|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the others occur only in the interior of 120-cell, not as edges of any of the regular 4-polytopes. There are 30 distinct 4-space chordal distances between vertices within the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to identify any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=Major and minor chords}} chords are properly labelled #1 — #15 in the order they fan out by size.{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''; Figure 3|ps=; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with a &quot;fan of chords&quot; diagram like the one here.|p=23}} They join vertex pairs which are 1 — 15 edges apart on a &lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon. A #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; chord forms &lt;small&gt;&lt;math&gt;f+1&lt;/math&gt;&lt;/small&gt; regular geodesic polygons which include a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; disjoint regular {{nowrap|&lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt;-gons}} (e.g. &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; a compound of 5 disjoint hexagons) plus a single [[W:Triacontagon#Triacontagram|regular skew {{nowrap|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram}}]] that wraps around the sphere &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times before closing its circle. Notice the maverick #11 chord, with its non-conforming origin point for its {{nowrap|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;}} polygram.]]

These 15 chords (natural numbers) include the integers 2, 3, 4, 5, 6, 10, 11, 15 and 30. The others are the fractions 5/2, 10/3, 13/7, 15/2, 15/4, and 15/7. Represented among the chord lengths (the square roots) measured in ''unit radii'' (not in {{radic|2}} radii, as metrics are often given elsewhere in this article) are the {{background color|paleturquoise|roots of the first 4 positive integers {{radic|1}}, {{radic|2}}, {{radic|3}} and {{radic|4}}}}, the {{background color|#FFCCCC|root of the very distinguished rational fraction {{radic|5/2}}}}, and the roots of various distinguished irrational fractions as Steinbach discovered: for example, 4 of them are {{background color|yellow|roots of golden numbers (a small integer ± 𝜙)}}. The 15 chords each form their regular polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, which is just &lt;small&gt;&lt;math&gt;\{k\}&lt;/math&gt;&lt;/small&gt; in 8 cases. Arranged in order #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; from shortest edge-length square root to longest, the #1 — #15 chords bridge vertex pairs which are &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; vertices apart on a [[W:Triacontagon#Petrie polygons|&lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon]]. Each chord makes a [[W:Triacontagon#Triacontagram|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; polygram]] of its edge length, a compound of the smaller &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. The 15 distinct polygons form a sequence you would not anticipate exactly unless you have Steinbach's formula:

{| style=&quot;background-color:transparent; white-space:nowrap;&quot;
|-
| {30/1} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|.073}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{30} 
| 120-cell edges
|-
| {30/2}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.191}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15}
|- style=&quot;background: yellow;&quot;|
| {30/3}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.382}} 
| &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi}&lt;/math&gt;&lt;/small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{10}''' 
| 600-cell edges
|-
| {30/4} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.573}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/2}
|- style=&quot;background: paleturquoise;&quot;|
| {30/5}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|1}}'''
| &lt;math&gt;1&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{6}''' 
| 24-cell, 8-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/6}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|1.382}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{5}'''
|- style=&quot;background: paleturquoise;&quot;|
| {30/7}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|2}}'''
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{4}''' 
| 16-cell edges
|- style=&quot;background: #FFCCCC;&quot;|
| {30/8}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.5}} 
| &lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;&lt;small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/4} 
| 5-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/9}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.618}} 
| &lt;math&gt;\phi&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{10/3}
|- style=&quot;background: paleturquoise;&quot;|
| {30/10}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|3}}''' 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{3}'''
|-
| {30/11}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.426}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{11/11}
|- style=&quot;background: yellow;&quot;|
| {30/12}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.618}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{5/2}
|-
| {30/13}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.810}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{13/7}
|-
| {30/14}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.928}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/7}
|- style=&quot;background: paleturquoise;&quot;|
| {30/15}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|4}}''' 
| &lt;math&gt;2&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{2}'''
|}

{| class=wikitable style=&quot;white-space:nowrap; float:right;&quot;
!colspan=6|Natural number &lt;small&gt;&lt;math&gt;h=\{k/d\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Fibers&lt;br&gt;&lt;small&gt;&lt;math&gt;f\{h\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Cells&lt;br&gt;&lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|4-tope&lt;br&gt;&lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Verts&lt;br&gt;&lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|[[W:Triacontagon#Triacontagram|30-gram]]&lt;br&gt;&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;
|-
!Chord
!Arc
!colspan=2|&lt;small&gt;&lt;math&gt;l/\sqrt{1}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;l/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;h&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#1
|15.5~°
|{{radic|0.073}}
|0.270
|{{radic|0.146}}
|&lt;small&gt;&lt;math&gt;30&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1\{30\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;5\{3,3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/1\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#2
|25.2~°
|{{radic|0.191}}
|0.437
|{{radic|0.382}}
|&lt;small&gt;&lt;math&gt;15&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;..\{3,3\}..&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/2\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#3
|36°
|{{radic|0.382}}
|0.618
|{{radic|0.764}}
|&lt;small&gt;&lt;math&gt;10&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;30\{3,3\}20&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/3\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#4
|44.5~°
|{{radic|0.573}}
|0.757
|{{radic|1.146}}
|&lt;small&gt;&lt;math&gt;15/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/4\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5
|60°
|{{radic|1}}
|1
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;6&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/5\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#6
|72°
|{{radic|1.382}}
|1.175
|{{radic|2.624}}
|&lt;small&gt;&lt;math&gt;5&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}} 
|&lt;small&gt;&lt;math&gt;10\{5,3\}12&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/6\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#7
|90°
|{{radic|2}}
|1.414
|{{radic|4}}
|&lt;small&gt;&lt;math&gt;4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7\{4\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;4\{4,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;9\{3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/7\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: #FFCCCC;&quot;|
|#8
|104.5~°
|{{radic|2.5}}
|1.581
|{{radic|5}}
|&lt;small&gt;&lt;math&gt;15/4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/4\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}} 
|&lt;small&gt;&lt;math&gt;15\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/8\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#9
|108°
|{{radic|2.618}}
|1.618
|{{radic|5.236}}
|&lt;small&gt;&lt;math&gt;10/3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10/3\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/9\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#10
|120°
|{{radic|3}}
|1.732
|{{radic|6}}
|&lt;small&gt;&lt;math&gt;3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;10\{3\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/10\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#11
|135.5~°
|{{radic|3.426}}
|1.851~
|{{radic|6.852}}
|&lt;small&gt;&lt;math&gt;11/11&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;11\{11/11\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;11\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#12
|144°
|{{radic|3.618}}
|1.902
|{{radic|7.235}}
|&lt;small&gt;&lt;math&gt;5/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2},3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/12\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#13
|154.8~°
|{{radic|3.810}}
|1.952
|{{radic|7.621}}
|&lt;small&gt;&lt;math&gt;13/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;13\{13/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;7\{5,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/13\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#14
|164.5~°
|{{radic|3.928}}
|1.982
|{{radic|7.857}}
|&lt;small&gt;&lt;math&gt;15/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;137\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/14\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{radic|8}}
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15\{2\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐𝜙&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;8\{3,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/15\}&lt;/math&gt;&lt;/small&gt;
|}

The polygons '''{10}''', '''{6}''', '''{5}''', '''{4}''', '''{3}''', and '''{2}''' are flat polygons that each lie on a great circle in a central plane of the 3-sphere. The polygons {30}, {15}, {15/2}, {15/4}, {10/3}, {11/11}, {5/2}, {13/7}, {15/7} and all the {30/''n''} polygons are skew polygrams spiraling through curved 3-space, each successive edge in a different central plane of the 3-sphere. Projected to the 1-sphere (a circle in the plane), they are [[W:Star polygon#Regular star polygon|regular star polygrams]], the simplest being the pentagram {5/2}. In 4-space they wind around the 3-sphere more than once ''without self-intersecting'', before they close their circuit. They lie on helical geodesic circles or ''isoclines'' of circumference greater than 2𝝅''r''.

Even the 6 integer chords which form flat polygons also form skew polygrams. Each of the 15 chords occurs as a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, a fiber bundle of Clifford parallel polygons that totals 30 vertices per bundle. Each chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; also forms a single skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram that winds through another, shorter chord's bundle of parallel polygons, intersecting them in succession multiple times each, and weaving that whole bundle of parallel fibers together as the cross-thread does on a loom. The polygram cross-thread has to wind multiple times through the whole bundle (&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times around the [[W:Triacontagon#Triacontagram|30-gram]] in which each edge connects every &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt;th vertex) to complete an enumeration of all 30 vertices in the bundle. For example, the #5 chord forms hexagon &lt;small&gt;&lt;math&gt;\{6\}&lt;/math&gt;&lt;/small&gt; flat planes in parallel bundles of 5, written &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt;, and also a skew &lt;small&gt;&lt;math&gt;\{15/2\}&lt;/math&gt;&lt;/small&gt;-gram of #5 edges which winds 2 times through the #3 chord's bundle of 3 decagon &lt;small&gt;&lt;math&gt;\{10\}&lt;/math&gt;&lt;/small&gt; flat planes. Every chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; forms both its &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram and its fiber bundle of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, and the edges of one chord's parallel polygons are also the edges of its skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram weaving through some shorter chord's bundle of parallel polygons. Each numbered chord is a natural number, a ratio of two integers and their polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, either flat or skew. Each chord number defines a &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; skew polygon that is a compound bundle of the &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. All these flat or skew polygons are regular polygons made of their numerator instances of a single chord #, whose length is that square root, Steinbach's function of the natural number.

The relationship between each bundle of flat Clifford parallel great circle polygons of a distinct length, and its corresponding bundle of skew polygrams of a longer length, is precisely that of a discrete simple rotation and its corresponding discrete isoclinic rotation. For example, the flat great circle hexagons of edge length {{radic|1}} lie in the invariant rotation planes of the 24-cell's characteristic 60° isoclinic rotation, in which the vertices move on skew hexagram rotation paths of edge length {{radic|3}}. The 24-cell contains 4 bundles of 4 Clifford parallel (disjoint) hexagon central planes each. The 24 vertices of the 24-cell in each bundle are linked by rotation in the 4 hexagon invariant planes in two different ways: simply and isoclinically. In a simple rotation the vertices of a single invariant hexagon exchange places on that great circle hexagon of edge {{radic|1}}. In an isoclinic rotation in which any one of these 4 disjoint hexagon central planes is invariant, all 4 of them are necessarily invariant planes in the rotation, because they rotate in parallel. In that isoclinic rotation the vertices of all 4 hexagon invariant planes (all 24 vertices of the 24-cell) exchange places concurrently, moving along 4 disjoint helical hexagrams of edge {{radic|3}} in parallel. It is a characteristic only of this distinct rotation in hexagon invariant planes that the helical vertex path or ''isocline'' is a hexagram of period 6, edge length {{radic|3}}, and circumference 720°. Rotations in other kinds of polygon invariant planes have different characteristic isoclines. The period of their isoclinic rotation must be ''the same or greater'' than the period of their simple rotation; in most cases (unlike this one) it is greater.

The 15 chords form not only distinct fiber bundles of polygons, but also distinct bundles of polyhedra: cell rings &lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt; and compound vertex figures &lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;. These combine to form their characteristic regular 4-polytope &lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Coxeter|1973|p=269|loc=Compounds|ps=; &quot;It is remarkable that the vertices of {5, 3, 3} include the vertices of all the other fifteen regular polytopes in four dimensions.&quot;}} All these elements of a regular 4-polytope constructed from one chord are themselves regular polytopes, and each of the 16 regular 4-polytopes is a function of one characteristic chord. If the regular 4-polytope is convex, its characteristic chord is its unit-radius edge. For a non-convex regular 4-polytope, the characteristic chord is its unit-radius ''mid-edge diameter'', the distance from the midpoint of each edge to the midpoint of the completely orthogonal edge. Each of the characteristic chords (each distinct natural number) has its expression as one of the 16 regular 4-polytopes and all the products of the distinct regular 1-polytopes, 2-polytopes and 3-polytopes that construct it, and the whole distinct construction from the natural number to the 4-polytope is an [[W:SO(4)|SO(4) symmetry group]] expression of that 4-polytope's characteristic discrete [[W:SO(4)#Isoclinic rotations|isoclinic rotation]], in the invariant central planes of its characteristic chord, by the characteristic chord arc.

...demonstrate this theory by proper enumeration of the chord lengths in the table...

...color-code the △ ☐ 𝜙 ✩ symbols ... chord colors in diagram should match them ... some of the diagram colors are wrong and must be changed, but otherwise diagram is pretty good ... i think in a colored diagram all chords of the same length should have the same size line dash, the polygons are not obvious enough and the 15 fanning-out chords are readily identified by their labels...

... add back the counts of chords in the box (very small font number under each table entry), and present these two images as the front and back a box of building sticks and rubber joints...What's in the other box...

...say the rational-root major chords {{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|5/2}} are the ''n''-edges: their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge...

...abbreviated text of the [[120-cell#Relationships among interior polytopes|''§Relationships among interior polytopes'']] section from the 120-cell article...to explain the color-coding

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius=1}}

See also the elements, properties and metrics of the sequence of regular 4-polytopes summarized in tabular form in the section ''[[#Build with the blocks|§Build with the blocks]]'', below.

== Alicia Boole Stott's original formulation of dimensional analogy ==
&lt;blockquote&gt;&quot;...a method by which bodies having a certain kind of semi-regularity may be derived from regular bodies in an Euclidean space of any number of dimensions...&quot;{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''; presents cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues|pp=12-45}}&lt;/blockquote&gt;

Alica Boole Stott's 1910 paper reads like an exceptionally well-written patent application. It is a comprehensive stepwise description of her original invention, a general method of dimensional analogy between regular and semiregular polytopes of the same edge length.

[[File:The Roots of Three.png|thumb|An illustration of Boole Stott's expansion of unit-edge triangle to hexagon captures the geometric basis in 3 of the &lt;math&gt;A_n&lt;/math&gt;root system.]]

The engineering-like articulation of her new principles is entirely discrete and group-theoretic. Boole Stott repeatedly refers to sets of polytope elements as ''groups''. All her transformations are discrete instances of one or another special case of her &quot;two inverse operations which may be called ''expansion'' and ''contraction''.&quot; Hers is an exploration of ''n''-dimensional space at its most basic: space defined by the discrete objects which fill it, themselves defined directly as groups of elements with a few operations on them, which transform them into each other. This is geometry which seems to precede Euclid's postulates, and does not depend on them. Instead it corresponds directly to the deeper group theory mathematics which underlies geometry, even though Boole Stott, like the ancients, did not have that mathematics.

Boole Stott enumerated her expansion and contraction operations as transformations between regular and semi-regular polytopes of a single edge length. In her 1910 paper she did not consider any polytopes of multiple edge lengths, or expansions of the subject polytope by a distance other than that exactly required to produce new edges of the original edge length, so of course she did not enumerate Moxness's non-uniform rhombicosidodecahedron (Hull #8). This may be why section 8&lt;sub&gt;3&lt;/sub&gt; is missing from her earlier drawings of the sections of {5,3,3}; she may not have had a construction for it.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The Boole Stott expansion (or contraction) operation is as precisely defined a transformation as Coxeter's reflection, rotation and translation operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions). Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;
where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;
where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;
For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} An instance of the expansion (or contraction) operation is an application of the general operation to a particular subject polytope, operating on all of its elements symmetrically at once. The general operation is restricted, such that one may only expand (or contract) the subject polytope by pushing all its edges, faces or cells directly away from (or toward) its center, without changing the size of the moved elements. This splits each vertex into several (or coalesces several vertices into one), and has corresponding effects on the other elements, producing new edges ''of the same size'' between pairs of separated vertices (or reducing some edges to single vertices). The distance all the elements are moved away from (or toward) the center must be such that the vertices separate producing new edges of only the original edge length (or come together reducing edges to single vertices). It is permitted to combine these three operations, e.g. by pushing the edges ''and'' the faces, or by pushing all three element sets, provided the combination results in a semi-regular polytope of only one edge length. The three operations are all commutative with one another, so combined operations can be considered either as a sequence of expansions (or contractions) linking several related polytopes, or as their concurrent movements, a single transformation of the subject polytope. Since there are 8 combinations of one to three operations there are 8 polytope variants (including the original regular polytope), and all are adjacent to one another by a single transformation. The variants of the original regular polytope are named for which element set(s) are moved. By enumerating all 8 variants of expansion (and contraction) for each of the six regular convex 4-polytopes, Boole Stott produced a table of regular and semi-regular 4-polytopes divided into 12 sets of 8: six expansion sets and six contraction sets.{{Sfn|Boole Stott|1910|loc=''Table of Polytopes in S&lt;sub&gt;4&lt;/sub&gt;''; note that Boole Stott's usage of ''S&lt;sub&gt;4&lt;/sub&gt;'' denotes a Euclidean 4-space ''E&lt;sub&gt;4&lt;/sub&gt;'', or a 3-sphere space embedded in it, ''not'' a 4-sphere space embedded in a Euclidean 5-space|p=30}} Each set includes just one of the six regular convex 4-polytopes, with its distinct unit-radius edge length, and the 7 possible semi-regular expansions (or contractions) of it, which all preserve that characteristic edge length. The polytopes in each set of 8 all have the same total number of elements as their original regular polytope, but each polytope in the set has a distinct combination of cell, face, and vertex figures. The 12 sets are not completely disjoint, and even within one set the 8 operations do not necessarily produce 8 unique polytopes. For example, the regular 5-cell (the 4-simplex polytope) has 5 cells, 10 faces, 10 edges, and 5 vertices (30 elements). Expanding the 5-cell produces 7 related semi-regular 4-polytopes of 30 elements each (two of which happen to be the same polytope). In each expansion, the 5 cells have expanded into semi-regular or regular polyhedra, the 10 faces and 10 edges ''may'' have expanded into prisms, and the 5 original vertices have expanded into regular or semi-regular cells.

== The regular 11-cell 4-polytope ==

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosadodecahedron). Each non-convex Jessen's icosahedron is a set of 6 reflex edges that are {{radic|5}} 5-cell edges, each set symmetrically arranged in 3 parallel pairs, not as far apart ({{radic|3}}) as they are long, in 3 orthogonal central planes. The 12-point (Jessen's sets of 6 reflex 5-cell edges) are inscribed in the 60-point with their endpoints as a vertex of a pentagon face. The 12 pentagon faces are joined to 5 other pentagon faces by the 5-cell edges; not to their nearest neighbor pentagons but to their next-to-nearest neighbor pentagons; and since there are 5 vertices to each pentagon face, there are 10 12-point (Jessen's sets of 6 reflex 5-cell edges). Each pentagon face is joined to another pentagon face by 5 reflex edges of five distinct Jessen's. The 5 reflex edges are Clifford parallel, the 5 edges of a pentagonal prism column that has a twist to it. The contraction of those 5 edges into 1, and those 10 vertices (two pentagons) into 2, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing all their edges as a single Jessen's. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (octahedron) by conflating its parallel pairs of edges, leaving the 11-cell's abstract 6-point (hexad cell). The hexad's situation is that it occurs in bundles of 5, disjoint but close together in the sense of adjacent, like the great circle rings in a Hopf fiber bundle. 

To summarize, the 12-point (2-hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell). We shall now be able to see how Fuller's perfect hexads combine with Todd's perfect pentads to form the perfect heptads of the 11-cell and 137-cell.  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=alias the hemi-cuboctahedron}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces. (The ''heptad'' arrives at the party, anonymously; they are our distinguished anti-celebrity, who most of the other guests have never even heard of.) The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. This 12-point (truncated tetrahedron) is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 100 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp: Truncated tetrahedron|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp: Truncated tetrahedron|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}} Of course Moxness has been there, seen that, already with his graphics software;{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} see his parameterized illustration of it exactly as it occurs in the 11-cell, below in his bestiary of A3, B3, and H3 Archimedian and Catalan polyhedra in the 120-cell.]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:Archimedean_and_Catalan_solid_hulls_with_their_Weyl_orbit_definitions.svg|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of the 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. In this catalog of Archimedean and Catalan polyhedra sighted in the 120-cell by Moxness's software,{{Sfn|Moxness: Archimedean and Catalan hulls|2023|loc=Hull #1 Archimedean Name A3 110 Truncated Tetrahedron A (upper left)}} it is the first entry (upper left), Moxness's Hull # = 1 with 12 vertices of 3D norm {{radic|11}}/2. On a {{radic|2}}-radius 3-sphere, 5 of these with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell. 20 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges are cells in the [[#The rings of the 11-cells|cell rings of the quasi-regular 11-cell]], which also has 12 regular 5-cells as cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

...

Both kinds of cells in the 11-cell have expressions in two different dimensions (on the 2-sphere and 3-sphere), because they are all centered on the center of the 3-sphere, unlike the cells of most 4-polytopes, which are centered on various off-center points within the 4-polytope's interior. 

One kind of 11-cell cell appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (semi-regular truncated cuboctahedron), a.k.a. the truncated tetrahedron. The other kind of cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now, except in the 120-cell. But this is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron, and a regular tetrahedron behind each face. The 11-cell is thus a honeycomb of mixed regular tetrahedra and truncated cuboctahedra (a.k.a. truncated tetrahedra) that meet at their shared 5-cell faces. 

Both the 5-cells (decahedra) and the truncated cuboctahedra (truncated tetrahedra) are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks in their 4-polytopes (off-center).{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} Therefore all the 11-cell's cells are concentric, which indeed makes a different kind of honeycomb, an inside-out cellular honeycomb with all the cells piled on top of each other overlapping in 4-space. They do not overlap in 3-sphere space however; their 3-volumes (but not their 4-contents) are disjoint. They make a proper 4D honeycomb like the cells of any 4-polytope, with volumetrically disjoint 3D cells that meet each other in the proper way at their shared 2D faces.

An 11-cell contains 6 completely disjoint 5-point (1-pentad 5-cell decahedron cells), and 5 volumetrically disjoint 12-point (2-hexad truncated cuboctahedron cells), 11 volumetrically disjoint cells altogether. Of course it is supposed to have 11 identical 11-point (elevenad hemi-icosahedral cells), and that is also true in its abstract sense, which is not the same sense as disjoint cellular containment: each pentad or hexad cell is inscribed in one of 11 larger 60-point (Moxness's rhombicosadodecahedron cells). The abstract pentads and hexads are only hemi-polyhedra, so the 11-cell's pentad cells are 12 (out of 120) disjoint instances of a pentad 5-point, and its hexad cells are 10 (out of 100) disjoint instances of a hexad 6-point, 1/10th of the 120-cell's cellular volume in both cases. The 11-cell's hemi-icosahedral cells are 11 (out of 120) non-disjoint instances of an elevenad 60-point. Each 11-cell shares each of its hexad cells with 9 other 11-cells, and each of its 11 vertices (out of 600) occurs in 9 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.|name=Tetrahedral vertex figures}}

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

...insert image thumb of 11-cell...

...Construct the 11-cell from the heptad....

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...Four concentric 4-polytopes may be mutually orthogonal. In any set of 4 mutually orthogonal regular 5-cells, one pair of faces in each orthogonal pair are completely orthogonal to each other. Thus three edges in one 5-cell in an orthogonal pair are opposite (completely orthogonal) to a face of the orthogonal 5-cell. .... A 5-cell has 5 vertices in its abstract hemi-icosahedral cells, but only 4 vertices in one concrete icosahedral cell; the 5th lies in an icosahedral cell of another 11-cell (not in another icosahedral cell of the same 11-cell).

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

....

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 11-cell ===

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon.... (wrong, 11-cell's Hopf map is the truncated icosahedron)

....

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk: Truncated icosahedron|2018}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

...

Since the rhombicosadodecahedron has 120 5-cell edges, the 120-cell has 120 completely disjoint 5-cells, and the rhombicosadodecahedron shares each vertex with 11 other rhombicosadodecahedra, how a rhombicosadodecahedron is bound to its neighbor 5-cells is constrained. A further constraint is that the abstract hemi-icosahedron has only 15 5-cell edges, so the 120 actual 5-cell edges of the rhombicosadodecahedron are conflated 8-to-1 in the hemi-icosahedron. To be conflated by contraction into a single edge, those 8 edges must lie parallel to each other in the 120-cell; either actually parallel in the same plane, or Clifford parallel in separate planes, as two edges in completely orthogonal planes may be parallel and orthogonal at the same time. Each 5-cell inscribed in the 120-cell contains 5 such tetrahedra occur in parallel pairs in 4 orthogonal planes (none of them completely orthogonal planes). 

.... 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
Each conflated vertex is the 10 vertices of two orthogonal regular 5-cells...
The 60-point contains 6 pairs of these orthogonal regular 5-cell cells...
...its 20 hexad cells are ..., a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere.

== The regular 137-cell 4-polytope ==

(description originally written for the 11-cell, which applies only to a fibration of 11 11-cells)

It is quasi-regular with two kinds of cells, the 12-point (regular icosahedron) and the 5-point (regular 5-cell), which have just one kind of regular face between them. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of regular facets (it is semi-regular with regular facets), it also means that its facets have expressions of two different dimensionalities because they are concentric in the 3-sphere. The 11-cell has two kinds of regular cells. One appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (regular icosahedron), but since it is embedded at the center of the 3-sphere (unlike the cells of most 4-polytopes) it also acts as an inscribed 4-polytope.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} The other cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now except in the 120-cell. This is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron. The 11-cell is a honeycomb of mixed icosahedron and half-icosahedron cells. Like the icosahedra, the decahedra are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks, off-center in their 4-polytopes. All the 11-cell's cells are concentric, which indeed makes an unusual honeycomb of cells, an inside-out honeycomb. It is a proper 4D honeycomb however, with volumetrically disjoint 3D cells that meet each other in the proper way at shared 2D faces. The embedded 12-point (regular icosahedron) cell is an icosahedral pyramid (as a 4-polytope), which is filled by 20 4-point (irregular tetrahedra) that meet at its center; the center point of the 11-cell acts as a 12th vertex, just as the center point of the 24-cell acts as a 25th vertex in some honeycombs. The 5-point (regular 5-cell) cell contains five 4-point (regular tetrahedra) which meet the 12-point (regular icosahedra)'s irregular tetrahedral cells face-to-face, so the 11-cell is quasi-regular in the usual sense of having two varieties of regular 3-polytope cells, but additionally both varieties of of its cells are bi-dimensional: another clue that the 11-cell has something to do with dimensional analogy

....

The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

....

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur to make an 11-cell. 11 of them occur in a 137-cell, all 120 of them occur in the 120-cell, and they occur nowhere else.

How can this regular 4-polytope possibly exist? By Schläfli's criterion {3, 3, ..} exists, but not as a finite polytope.{{Sfn|Coxeter|1973|loc=§7.7 Schläfli's criterion}} Coxeter found {3, 5, 3} as a finite hyperbolic honeycomb, but not in Euclidean 4-space.{{Sfn|Coxeter|1968|title=The Beauty of Geometry: Twelve Geometric Essays|loc=''Regular Honeycombs in Hyperbolic Space''|ps=; &quot;The honeycombs considered by the above authors [Schlegel, Schläfli, Klein, Sommerville] have finite cells and finite vertex figures. ... A further extension allows a cell or vertex figure to be a ''star''-polytope, so that the honeycomb covers the space several times. .... We shall find that there are four regular star-honeycombs in hyperbolic 4-space, as well as two infinite families of them in the hyperbolic plane.&quot;}} We find that {3, 3, ..} is a central expression of the &lt;math&gt;H_4&lt;/math&gt; root system, being the regular convex hull in 4 dimensions of 137 points, 96 pentads and 600 hexads. The minimal expression of &lt;math&gt;H_4&lt;/math&gt; is the regular convex hull {3, 3, 5} of 120 points, 600 tetrads and 75 hexads, and its maximal expression is the regular convex hull {5, 3, 3} of 600 points, 120 pentads and 675 hexads. {3, 3, ..} is a third regular &lt;math&gt;H_4&lt;/math&gt; polytope which fits between them. The properties and metrics of this regular sequence are presented in tabular form below in ''[[#Build with the blocks|§Build with the blocks]]''.

{|class=&quot;wikitable&quot;
!colspan=3|Regular &lt;math&gt;H_4&lt;/math&gt; polytopes
|-
|[[File:600-cell.gif|200px]]
|[[File:5-cell.gif|200px]]
|[[File:120-cell.gif|200px]]
|-
!120-point 600-cell
!137-point 137-cell
!600-point 120-cell
|}

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 137-cell ===
....

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=Apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|2022|loc=''[[Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]''}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie: On Fuller's use of language|2024|loc=''[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry'']]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center, 1948-1949|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with their mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the gender neutral shad can open their six beaks all the way, until they become flat squares and they becomes a cuboctahedron, or they can shut them all tight like a turtle retracting into their octahedron shell. The six mouths always move in unison. This is [[Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do their ''really'' odd trick -- where they flip their 6 jaws 90 degrees in their 6 faces and shut their 6 beaks on the opposite axis of their squares than the one they opened them on -- or whether they will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than their normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make their edge length exactly the same size as their radius, when they open their mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing their mouths in spherical synchrony, their 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron (the hexad), then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=''[[W:SO(4)|SO(4)]]''}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=Isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its orthogonal hexad's planes (wz, wx, wy).{{Efn|name=Six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of (expanded octahederon 16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh &amp; Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== Legendre's 12-point hexad binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a 12-point double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's perfectly realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, this 2-hexad expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre hexad) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction.{{Efn|name=Concentric polyhedra}} In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== ''n''-simplexes ===
....

The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== ''n''-orthoplexes ===
....
[[File:Great 5-cell √5 digons rectangle.png|thumb|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} ''regular'' great hexagons. These are the same 200 dodecagon central planes illustrated above, which also contain 6 120-cell edges and two opposing ''irregular'' great hexagons (truncated {{radic|6}} triangles). The 1200 {{radic|5}} 5-cell face triangles occupy 1200 other, non-central planes.]]

...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's Hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[16--cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[16--cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=Concentric polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[16--cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== ''n''-taliesins ===
The 4-simplex pentad building block combines with the 4-orthoplex hexad building block to make the 4-taliesin heptad building block. The heptad is the essential building block of the ''taliesin'' family of polytopes. The 11-cell and 57-cell hemi-polytopes and the 137-cell and 120-cell regular 4-polytopes are nuclear members of this trans-dimensional family. The 24-cell and 600-cell are extended members of it, because they also contain instances of the heptad building block.

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its spherical symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* [[W:Taliesin Myrddin Namkai-Meche|Taliesin Myrddin Namkai-Meche]], a young Reed College scholar who was stabbed to death by a crazy man in 2017 on a Portland, Oregon light rail train, when he rose to the nonviolent defense of two black teenagers who were being violently assaulted in racist and anti-Muslim language by a right-wing extremist.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in orthogonal [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].
* '''Taliesan polytope''', an (''n+1)''-polytope whose ''n''-polytope dimensional analogue inscribed concentrically in the ''n''-sphere forms a convex honeycomb of itself which is the (''n+1)''-polytope.{{Efn|name=Embedding point}}
* The taliesan polytope's honeycomb, a Hopf fibration of its great circles.
* The ''n''-dimensional family of dimensionally analogous polytopes related to a taliesan polytope by expansion and contraction operations.
* The characteristic [[w:Orthoscheme|orthoscheme]] of a teliesan polytope, an irregular simplex that is the complete gene for the entire expression of trans-dimensional symmetry relationships among the polytopes of the ''n''-dimensional family, as well as the complete gene for the teliesan polytope's rotational symmetry.{{Efn|name=Characteristic orthoscheme}}

=== ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== ''n''-equilaterals ===
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed at four different [[#n-leviathon|chord]] lengths from each other. In the {{radic|2}}-radius polytopes we have using as examples in this paper they are {{radic|2}}, {{radic|4}}, {{radic|6}} and {{radic|8}}, but in a ''unit-radius'' polytope they are: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}.

Each vertex is joined to 8 others by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}. The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== ''n''-pentagonals ===

[[File:600-cell.gif|thumb|The 120-point (600-cell), &lt;math&gt;H^4&lt;/math&gt;.]]
...

[[File:600-cell vertex geometry.png|thumb|Vertex geometry of the 600-cell, showing the 5 regular great circle polygons and the 8 vertex-to-vertex chord lengths with angles of arc. The golden ratio{{Efn|name=golden chords|group=}} governs the fractional roots of every other chord,{{Efn|name=fractional root chords}} and the radial golden triangles which meet at the center.|alt=|400x400px]]
The 120 vertices of the 600-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏&lt;sup&gt;−1&lt;/sup&gt;) beginning with a vertex; see column ''a''}} at eight different [[#n-leviathon|chord]] lengths from each other. These edges and chords of the 600-cell are simply the edges and chords of its five great circle polygons. In ascending order of length in a ''unit-radius'' polytope (not in a {{radic|2}} radius polytopes like our usual examples), they are {{radic|0.𝚫}}, {{radic|1}}, {{radic|1.𝚫}}, {{radic|2}}, {{radic|2.𝚽}}, {{radic|3}}, {{radic|3.𝚽}}, and {{radic|4}}.{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=Decimal fractions of square roots|group=}}

Notice that the four [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#''n-''equilaterals|hypercubic chords]] of the 24-cell ({{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|4}}) alternate with the four new chords of the 600-cell's additional great circles, the decagons and pentagons. The new ''golden chord'' lengths are necessarily square roots of fractions, but very special fractions related to the golden ratio{{Efn|1=The fractional-root ''golden chords'' are irrational fractions that are functions of {{radic|5}}. They exemplify that the [[W:golden ratio|golden ratio]] &lt;big&gt;𝜙&lt;/big&gt; {{=}} {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618 is a circle ratio related to &lt;big&gt;𝜋&lt;/big&gt;:&lt;br&gt;
: {{sfrac|𝜋|5}} = arccos (𝜙/2)
is one decagon edge, the 𝚽 = {{radic|0.𝚫}} = {{radic|0.382~}} ≈ 0.618 chord.
Reciprocally, in this function discovered by Robert Everest expressing &lt;big&gt;𝜙&lt;/big&gt; as a function of &lt;big&gt;𝜋&lt;/big&gt; and the numbers 1, 2, 3 and 5 of the Fibonacci series:&lt;br&gt;
: &lt;big&gt;𝜙&lt;/big&gt; = 1 – 2 cos ({{sfrac|3𝜋|5}})
{{sfrac|3𝜋|5}} is the arc length of the &lt;big&gt;𝜙&lt;/big&gt; = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618 chord.|name=golden chords|group=}} including the two [[W:golden section|golden section]]s of {{radic|5}}, as shown in the diagram.{{Efn|The 600-cell edges are decagon edges of length {{radic|0.𝚫}}, which is 𝚽, the ''smaller'' golden section of {{radic|5}}; the edges are in the inverse [[W:golden ratio|golden ratio]] {{sfrac|1|φ}} to the {{radic|1}} hexagon chords (the 24-cell edges).
The other fractional-root chords exhibit golden relationships as well. The chord of length {{radic|1.𝚫}} is a pentagon edge.
The next fractional-root chord is a decagon diagonal of length {{radic|2.𝚽}} which is &lt;big&gt;𝜙&lt;/big&gt;, the ''larger'' golden section of {{radic|5}}; it is in the golden ratio{{Efn|name=golden chords|group=}} to the {{radic|1}} chord (and the radius).{{Efn|Notice in the diagram how the &lt;big&gt;𝜙&lt;/big&gt; chord (the ''larger'' golden section) sums with the adjacent 𝚽 edge (the ''smaller'' golden section) to {{radic|5}}, as if together they were a {{radic|5}} chord bent to fit inside the {{radic|4}} diameter.}}
The last fractional-root chord is the pentagon diagonal of length {{radic|3.𝚽}}.
The [[W:Pentagon#Side length is given|diagonal of a regular pentagon]] is always in the golden ratio to its edge, and indeed &lt;big&gt;𝜙&lt;/big&gt;{{radic|1.𝚫}} is {{radic|3.𝚽}}.|name=fractional root chords|group=}}

....


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== ''n''-leviathon ===
...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of its &lt;math&gt;10^2&lt;/math&gt; disjoint instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction, before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]].{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}} .... It is only because she did not have the 11-cell to start with that she did not discover the regular 137-cell {3, 3, ..} by her method.

Between Alicia Boole Stott and her identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and her identification of symmetry groups with conservation laws (in her 1st theorem), we owe the deepest mathematics of geometry and physics originally to two pioneering women, colleagues of their contemporaries the greatest men of the academy, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just an [[W:abstract polytope|abstract 4-polytope]], not just a singleton convex 4-polytope, and not just a honeycomb.{{Sfn|Coxeter|1970|loc=''Twisted Honeycombs''}} The 11-point (11-cell) has a concrete ...regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}&lt;nowiki/&gt;} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, 3, 3} as the 137-point (137-cell).

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 11-cell, 137-cell and 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=''Clifford Spinors and Root System Induction: H4 and the Grand Antiprism''}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=Apology}}

== Build with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Duveneck|1978|loc=Carl Jung, quoted in ''Life on Two Levels''|p=ii|ps=.{{Sfn|Jung|1961|ps=: &quot;The best of truths is of no use unless it has become one's most personal inner experience. It is the duty of everyone who takes a solitary path to share with society what he finds on his journey of discovery.&quot;}}}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1954|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius={{radic|2}}}}

== Acknowledgements ==
...

== Notes ==
{{Regular convex 4-polytopes Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Citation|last=Boole Stott|first=Alicia|author-link=W:Alicia Boole Stott|title=Geometrical deduction of semiregular from regular polytopes and space fillings|date=1910|pages=12-45|publisher=Johannes Muller|place=Amsterdam|url=https://dwc.knaw.nl/DL/publications/PU00011492.pdf}}
* {{Citation|last=Schoute|first=Pieter Hendrik|author-link=W:Pieter Hendrik Schoute|title=Analytical treatment of the polytopes regularly derived from the regular  polytopes|date=1911|pages=114-197|publisher=Johannes Muller|place=Amsterdam|url=https://www.google.com/books/edition/Analytical_Treatment_of_the_Polytopes_Re/G_oxAQAAMAAJ?hl=en&amp;gbpv=0}}
* {{Citation | last=Grünbaum | first=Branko | author-link=W:Branko Grünbaum | year=1976 | title=Regularity of Graphs, Complexes and Designs | journal=Colloques Internationaux C.N.R.S. | publisher=Orsay | volume=260 | pages=191-197 | url=https://faculty.washington.edu/moishe/branko/BG111.Regularity%20of%20graphs,etc.pdf | ref=}}
* {{Citation | last=Grünbaum | first=Branko | author-link=W:Branko Grünbaum | year=1975 | title=Venn Diagrams and Independent Families of Sets | journal=Mathematics Magazine | volume=48 | issue=1 | url=https://maa.org/sites/default/files/pdf/upload_library/22/Ford/BrankoGrunbaum.pdf | doi=10.1080/0025570X.1975.11976431 }}
* {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1984 | title=A Symmetrical Arrangement of Eleven Hemi-Icosahedra | journal=Annals of Discrete Mathematics (20): Convexity and graph theory | series=North-Holland Mathematics Studies | publisher=North-Holland | volume=87 | pages=103-114 | doi=10.1016/S0304-0208(08)72814-7 | url=https://www.sciencedirect.com/science/article/pii/S0304020808728147
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      <text bytes="208171" sha1="3x605h27pfanu109fo3k77ott72ybd5" xml:space="preserve">{{align|center|David Brooks Christie}}
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{{align|center|June 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a regular 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the &lt;s&gt;120-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 5, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 11 [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] cells. Eleven 11-cells (plural) are the 137-cell, the &lt;s&gt;600-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 3, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 137 [[W:Rhombic triacontahedron|triacontahedron]] cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=''Regularity of Graphs, Complexes and Designs''}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=''A Symmetrical Arrangement of Eleven Hemi-Icosahedra''}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]]
The 4-dimensional regular polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:Simplex|simplex]] (called the [[5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:Orthoplex|orthoplex]] (called the [[16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular convex 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the [[w:Tesseract|16-point (8-cell) tesseract]], the 4-cubic, which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found within any of the other regular convex 4-polytopes, except in the largest and most complex one, the [[120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). However, we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the 16-cell is their very starting point, and the most frequently used tool in the box.

== 5-cells and hemi-icosahedra in the 11-cell ==
[[File:Symmetrical_5-set_Venn_diagram.svg|thumb|The 5-point (10-face) regular 5-cell (the regular 4-simplex). Grünbaum's rotationally symmetrical 5-set Venn diagram{{Sfn|Grünbaum|1975|loc=''Rotationally symmetrical 5-set Venn diagram'', Fig 1 (e)|ps=; partitions the individual elements of the 5-cell.}} is an illustration of the 5-cell labeling each of its &lt;math&gt;2^5&lt;/math&gt; elements.{{Sfn|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled; image includes the Python code to render it, optimising for maximum area of the smallest regions.}}]]
[[File:Hemi-icosahedron.png|thumb|The 6-point (10 face) [[W:hemi-icosahedron|hemi-icosahedron]], an abstraction of the regular icosahedron, has half as many faces, edges and vertices. Each element of the abstract polyhedron represents two real elements found in different places in the 11-cell's realization.{{Sfn|Ruen: hemi-icosahedron|2007}}]]
The most apparent relationship between the pentad 5-cell and the hexad hemi-icosahedron is that they both have 10 triangular faces. When we see a facet congruence between a 4-polytope and a 3-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 6-point 5-simplex. It is also the same size as the 6-point 3-orthoplex; thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no chord congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals, pentad and hexad, who are famously stand-offish with each other. Interesting!

The cell of the 11-cell is an abstract hexad hemi-icosahedron containing pentads in its interior, most handsomely illustrated by Séquin.{{Sfn|Séquin &amp; Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing rings of polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The 11-cell's 11 hemi-icosahedral cells have 10 triangle faces each. If we assume a 5-cell is behind each face, only 3 of the 5-cell's 5 vertices are visible on the surface of the hemi-icosahedron as the exposed face. The 5-cell's other two vertices must define an edge on the opposite side of the 11-cell. Coxeter determined that the 11-cell does indeed have an edge opposite each face, and that it does not belong to the same hemi-icosahedral cell as its opposing face.{{Sfn|Coxeter|1984|p=110|loc=§6. The Petrie polygon [of the 11-cell]|ps=; &quot;We may reasonably call this edge and face ''opposites''. It is easy to find the face opposite to a given edge by looking at the faces to which a given edge belongs. ... Conversely, given a face, we can find the opposite edge by seeing which vertices belong to neither of the hemi-icosahedra which share that face. The ten edges opposite to the ten faces of one hemi-icosahedron are the edges of the complementary &lt;math&gt;a_4&lt;/math&gt; [4-simplex], that is, the joins of all pairs of the five vertices [of the 11-cell] not belonging to the given hemi-icosahedron.&quot;}}

There are 11 disjoint 5-cells in each 11-cell, which is comprised of 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. The real 11-cell is harder to see than the abstract 11-cell that represents it, because the real hemi-icosahedron that the abstract hemi-icosahedron represents is hard to see, but seeing these objects will be easier once we have identified the real hemi-icosahedron and exactly where the 11-cell's real elements reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each hemi-icosahedron's 10 faces come from 10 disjoint 5-cells, and there is just one 5-cell with which it does not share a face.{{Efn|As Coxeter observes (in the previous citation), that unrepresented 5-point (5-cell) is the other 5 vertices of the 11-point (11-cell) that are not vertices of this 6-point (hemi-icosahedron), the hemi-icosahedron's disjoint complement.}} Each 5-cell has 10 faces from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share a face.

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in two different planes which are completely orthogonal to each other. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must be sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness: 8 concentric hulls|2022|loc=Hull #8 (lower right)|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell of the 11-cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Compounds in the 120-cell|§Compounds in the 120-cell]]'' that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software which uses Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s to render the polyhedra which are found in the interior of ''n''-dimensional polytopes.{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the tetrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is a real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (Hull #8) is a concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this {12} irregular great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing {6} irregular great hexagons (truncated triangles) of alternating 120-cell edges and 5-cell edges. The great dodecagon is a great circle of the hemi-icosahedral cell, and also a great circle of the 120-cell, because unlike the cells of most 4-polytopes, the 11-cell's cells are embedded in 4-space at the exact center of the 4-polytope, making them inscribed 4-polytopes themselves.{{Efn|name=Embedding point}} There are 25 great dodecagons and 120 5-cell edges in Moxness's 60-point (Hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (Hull #8) polyhedron altogether.

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point (Hull #8) is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] similar to the one from the catalog shown here,{{Sfn|Piesk: Rhombicosidodecahedron|2018}} but a slightly shallower truncation with smaller red pentagons (120-cell faces) and narrower rhombs. Rhombicosidodecahedra are made by truncating the [[W:Rhombic triacontahedron|rhombic triacontahedron]], which is the 30-sided polyhedron with only one kind of face, the dual of the 30-point icosidodecahedron. The 120-cell contains 120 of these 11-cell cells.]]
Moxness's 60-point (Hull #8) is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (Hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|The 5-point (regular 5-cell) can be another abstraction of Moxness's 60-point (Hull #8), 12-vertices-into-1, but it is not strictly the result of a contraction operation as described by Boole Stott, since she did not apply her expansion and contraction operations to uniform polytopes with more than one edge length.}} In fact Moxness was not the first to find a rhombicosidodecahedron in the 120-cell; Alicia Boole Stott was. It is the 6th section of {5,3,3} beginning with a cell, which Boole Stott identified as the ''e&lt;sub&gt;2&lt;/sub&gt; expansion'' of the icosahedron (or equivalently of its dual polyhedron the dodecahedron).{{Sfn|Boole Stott|1910|loc=§Examples of the e&lt;sub&gt;2&lt;/sub&gt; expansion|p=7}} But that rhombicosidodecahedron identified by Boole Stott is not Moxness's Hull #8: rather, it is the semi-regular Archimedean solid, with a single edge length and square faces. Moxness's Hull #8 with its two distinct edge lengths and rectangular faces is Coxeter's 8th section of {5,3,3} beginning with a cell, which is missing from the sections illustrated by Boole Stott.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than tetrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. The 120-cell contains 120 of Moxness's 60-point (11-cell cells). They are non-disjoint, with 12 60-points sharing each vertex.

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (Hull #8) is the real hemi-icosahedron cell, with 120 5-cell edges in it as chords lurking under its surface. The rhombicosadodecahedron has enough of those 5-cell edges in its interior to make 20 disjoint tetrahedral cells which belong to 20 distinct 5-cells.

The 600-point (120-cell) may be constructed as a semi-regular honeycomb of two kinds of cells, the 120 non-disjoint 60-point (Moxness's rhombicosadodecahedra) and their neighbors the 120 disjoint 5-point (5-cells), cell-bonded together by shared tetrahedral cells of edge-length {{radic|5}}. Each rhombicosadodecahedron is bonded to 20 islanded 5-cells, sharing with each a 5-cell tetrahedron (4 of its interior 5-cell faces, and 6 of its interior 5-cell edges). The rhombicosadodecahedron is also bonded directly to 10 other rhombicosadodecahedra (since each 11-cell has 11 rhombicosadodecahedron cells), again not by sharing one of its exterior faces, but this time by sharing an interior face, a {{radic|6}} triangle which lies in a central plane.{{Efn|The {{radic|6}} chord is another hidden chord lurking beneath the 60-point rhombicosadodecahedron's surface, like the {{radic|5}} 5-cell edge. Both these hidden chords lie on the same great circle with the 120-cell edges (the pentagon face edges). Three {{radic|6}} chords make a triangle which lies in a central plane. This {{radic|6}} great triangle must not be confused with the {{radic|5}} face triangle of the 5-cell, which does not lie in a central plane.}} The 120 rhombicosadodecahedra are also bonded directly together at ''all'' their exterior faces: at their pentagon faces (which are 120-cell faces of dodecahedral cells); at their small triangular faces (which are 600-cell faces of small tetrahedral cells, 10 of which are inscribed in each dodecahedral cell); and at their rectangular faces (which are faces of 10-point [[w:Pentagonal_prism|pentagonal prism]] cells, where the pentagonal prism is a column inscribed in a dodecahedral cell, joining its opposite pentagon faces). The 120 rhombicosadodecahedra are bonded directly to each other at their own exterior faces; no dodecahedra, small tetrahedra, or pentagonal prisms lie between them, and those objects are not cells of this honeycomb; we mention them here only to help locate the rhombicosadodecahedra within the 120-cell. The only objects which fill holes between the rhombicosadodecahedra are the 120 islanded 5-cells.{{Efn|This honeycomb of the 120-cell resembles the [[600-cell#Icosahedra|honeycomb of the 600-cell consisting of 24 face-bonded regular icosahedra]] with 24 clusters of 5 tetrahedra filling the holes between them. In the 120-cell there are 5 times as many of everything as in the 600-cell, so there are 120 (rhombicosadodecahedron) icosahedral cells with 600 tetrahedra filling the holes between them.}}

It is possible for rhombicosadodecahedra to meet each other at more than one kind of exterior face because they are not meeting there as cells of the same 11-cell; they belong to different 11-cells. The 11 rhombicosadodecahedra in the same 11-cell do not meet at their own exterior faces, they meet at their interior {{radic|6}} triangle faces. As a cellular 4-polytope, the 11-cell is inside-out. Its cells meet each other at interior planes, sharing interior {{radic|6}} triangle faces, instead of in the usual way that 4-polytopes' cells share an exterior face. It is a deeper kind of intertwining to be sure, but the 11 rhombicosadodecahedra in the same 11-cell are still volumetrically disjoint cells, and the 11-cell is a perfectly real 4-polytope with real cells. It is a semi-regular 4-polytope, because it has two different kinds of cells: 11 rhombicosadodecahedra and 11 5-cells. The 11 5-cells are islanded (not bound to each other), and the 11 rhombicosadodecahedra are bound both to 5-cells (at {{radic|5}} 5-cell faces) and to each other (at {{radic|6}} central triangle faces). In the section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#What's in the box|§What's in the box]]'', below, we will look closely at the relation between the {{radic|5}} chord and the {{radic|6}} chord, which is the relation between the pentad and the hexad, and at the general phenomenon of hemi-polytopes which meet each other at a central plane. Then in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Cell rings of the 11-cells|§Cell rings of the 11-cells]]'' we will look at the two-cell cellular structure of the 11-cell in more detail.

Considered as a single complex of overlapping cells rather than as distinct honeycombs of volumetrically disjoint cells, the 120-cell contains 60-point (rhombicosadodecahedral cells), islanded 5-point (5-cells), 20-point (dodecahedral cells), and 10-point ([[w:Pentagonal_prism|pentagonal prism]] cells). The 60-points meet each other in pairs at their pentagon faces, meet regular dodecahedra at those pentagon faces also, meet each other in quads at their 600-cell faces where they share an interior 4-point (600-cell tetrahedral cell) and its four faces (just as they share a larger interior 5-cell tetrahedron and its faces), and meet each other and the 10-point (pentagonal prisms) at their rectangle faces. We might try to imagine such a complex with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with parts truncated from 30 neighboring truncated dodecahedra pentagonal prisms). There are 120 of each of these 3 kinds of dodecahedron. 

Moxness's 60-point (rhombicosadodecahedron cell) is a central object of the 120-cell, and contains many wonders. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra by expansion and contraction operations, as Alicia Boole Stott discovered before 1910,{{Sfn|Polo-Blanco: ''Theory and history of geometric models of Alicia Boole Stott''|2007|loc=§5.3.2 1910 paper on semi-regular polytopes|pp=152-158|ps=; summarizes Boole Stott's method and results from {{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}}, including her tables of expansion-contraction dimensional analogies and a few of her illustrations.}} and those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, rediscovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter &amp; Petrie|1938|p=4|loc=''[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]''}}|name=Concentric polyhedra}} Before we move on from the 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice one more thing about the hidden {{radic|5}} chords lurking below its surface. The 12 little pentagon faces are spanned by sets of 6 {{radic|5}} chords (5-cell edges). The 6 chords of each set are disjoint (they don't touch or form 5-cell faces), but they are symmetrically arranged in 3 parallel pairs, {{radic|3}} apart, which lie in 3 orthogonal central planes. It turns out they are the 6 reflex edges of a 12-point non-convex polyhedron called the [[w:Jessen's_icosahedron|Jessen's icosahedron]], inscribed in the 60-point cell. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again and work our way outwards, before eventually in ''[[#The perfection of Fuller's cyclic design|§The perfection of Fuller's cyclic design]]'' we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cell)s ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks, showing its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Pentahemidemicube.png|120px]]
|[[File:Tetrahemihexahedron_rotation.gif|120px]]
|[[File:Pentahemicosahedron.png|120px]]
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts needed to build the 120-cell (the picture on the cover of the box), and everything else that's invisible in that picture which fits inside the 120-cell: all the astonishing regular 4-polytopes. The pentad building block is half a pair of orthogonal 5-point (regular 5-cells), the hexad building block is half an 8-point (16-cell), and the heptad building block is a quarter of an 11-point (11-cell). Everything can be built from these blocks by embedding them at the center of the 3-sphere, inscribing them as 4-polytopes to make compounds as with the 16-cell building block, rather than by assembling them face-to-face as 3-polytopes to make honeycombs in which they are cells or vertex figures.{{Efn|name=Embedding point}}

The pentad building block is the 5-point '''pentahemidemicube''', an abstract polyhedron with 10 equilateral triangle faces, 10 {{radic|5}} edges, and 5 vertices.{{Sfn|Christie: Pentahemidemicube|2024|loc=File:Pentahemidemicube.png|ps=; &quot;The 5-point ''pentahemidemicube'' is an abstract polyhedron with 10 equilateral triangle faces, 10 edges, and 5 vertices. It has a real presentation as the regular 5-cell.&quot;}} It has a real presentation as the 5-point (regular 5-cell 4-polytope), a regular tetrahedron expanded by addition of a fifth vertex equidistant from all the others in a fourth dimension. The pentad block is a tetrahedral pyramid (a 5-cell) inscribed in a cube: a tetrahedron inscribed in a cube (a demicube) expanded by addition of a fifth vertex, one of the cube's other 4 vertices. Each pentad (5-cell) building block lies in the 120-cell orthogonal to three other pentad (5-cell) building blocks. They are four completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges. Each regular 5-cell in the 120-cell is a pentad (pentahemidemicube) building block, in one of 30 orthogonal sets of 4. There are 120 pentad building blocks in the box.

Every vertex of the 16-cell, and therefore every vertex of the 8-cell, 24-cell, 600-cell and 120-cell, is surrounded by a 6-point (octahedral vertex figure). The hexad building block is that 6-point octahedron embedded at the center of the 4-polytope instead of at a vertex, that is, inscribed in the 3-sphere as a 4-polytope, the 6-point '''[[w:Tetrahemihexahedron|tetrahemihexahedron]]''' or '''hemi-cuboctahedron'''.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; &quot;The tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U&lt;sub&gt;4&lt;/sub&gt;. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices.&quot;}} The hexads have 4 red triangle 5-cell faces, bounded by 12 {{radic|5}} edges. Their 3 yellow square faces lie in central planes, because the building block is an abstract 4-polytope. The hexad is an irregular abstract polyhedron that also has 6 {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

Two orthogonal 5-point (pentad) building blocks intersect in a 4-point (inscribed tetrahedron) to form a 6-point (hexad) building block. Two orthogonal 6-point (hexad) building blocks intersect in a 4-point (central square) to form an 8-point (16-cell) building block, the most usable building block, from which everything larger can be built by compounding.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 120 pentads to pair with a completely orthogonal hexad, and although there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed) they only result in 120 pairs two different ways, so there are 120 11-cells. The 11-cell is 6 pentads' and 5 hexads' smallest joint compound, and the 120-cell is their largest, 120 pentads and 100 hexads.

The heptad{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''|pages=22–31}} building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|4}} edges and 9 {{radic|5}} edges), and 7 vertices.{{Sfn|Christie: Pentahemicosahedron|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;}} It is a compound of a 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]) and a 6-point (hexad [[W:hemi-icosahedron|hemi-icosahedron]]) intersecting in a 4-point (tetrahedron), that is, of three orthogonal 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]&lt;nowiki/&gt;s) intersecting in their common 4-point (tetrahedron). It is also the abstract cube contracted by removal of one vertex, which reduces one of the cube's two inscribed tetrahedra (demicubes) to a single remaining triangular face. As with all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Four &lt;small&gt;&lt;math&gt;(11-7)&lt;/math&gt;&lt;/small&gt; orthogonal 7-point (pentahemicosahedron) blocks can be joined at a common Petrie polygon (a skew hexagon which contains their orthogonal 6-edge paths) to construct a real 11-point ([[W:11-cell|11-cell]] semi-regular 4-polytope), which magically contains six &lt;small&gt;&lt;math&gt;(11-5)&lt;/math&gt;&lt;/small&gt; 5-point (1-pentad regular decahedra) as cells and five &lt;small&gt;&lt;math&gt;(11-6)&lt;/math&gt;&lt;/small&gt; 12-point (2-hexad semi-regular cuboctahedra) as cells. There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like [[W:Euclid's postulates|Euclid's postulates]], and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as honeycombs of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles. It possesses the complete symmetry of the polytope without any redundancy, because it contains one of each of the polytope's characteristic root elements.{{Efn|A [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be partitioned radially by its planes of symmetry (Coxeter's &quot;reflecting circles&quot;) into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated rotationally to generate its regular 4-polytope because it is the complete [[W:gene|gene]] for it, containing all of its elements and capturing all of its symmetry without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring that triggers the reflections.|name=Characteristic orthoscheme}} It is the gene for the polytope, which can be replicated to construct the polytope.

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its &lt;math&gt;A_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the regular compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;small&gt;&lt;math&gt;120\times 120 = 14400&lt;/math&gt;&lt;/small&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[16--cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its &lt;math&gt;B_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 384. The 120-cell is the convex hull of the regular compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 24-point (24-cell) is subdivided into 1152 instances of its [[24-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;F_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 1152. The 120-cell is the convex hull of the regular compound of 25 disjoint 24-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;25\times 1152 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has three equivalent origins in distinct root systems, though most are obviously constructed from one root, and not so obviously from the others. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything larger than the pentad can be constructed by compounding hexads, and we shall see that everything as large as a 24-cell has a construction from heptads which are a product of a pentad and a hexad. Surely every uniform polytope can be constructed by some function of expansions and contractions from any of three root systems, however indirect the recipe. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes, related by an equals sign between them, then give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics by [[W:Noether's theorem|Noether's theorem]].

=== Pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 octahedral cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell contains the 600 octahedral cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It also contains the 600 cubical cells of its 75 8-cells. It is a 600-cell in three ways, although the octahedra and cubes are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell and the 8-cell, respectively.|name=600 cells three ways}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (1-pentad 5-cell), out of the 8-point (2-hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (1-pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope that contains 5-point (1-pentad 5-cell) building blocks beside the largest one, it just isn't a ''regular'' 4-polytope. It is semi-regular with two kinds of cells, the 4-point (regular tetrahedron) and the 12-point (truncated cuboctahedron), which have just one kind of regular face between them, a 5-cell face. That semi-regular 4-polytope is the 11-cell.

We said earlier that the 11-cell had 11 disjoint 5-cells and 11 hemi-icosahedral cells in it, and that its construction from them might be hard to see, but now we are going to account numerically, at least, for how it can have 6 pentad cells plus 5 hexad cells magically adding to 11 of the same kind of elevenad cell. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, 11. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (5-cell regular decahedron) cells, and the 12 6-point (hexad) building blocks into 100 12-point (regular icosahedron) cells, and then magically adding one whole 5-point (5-cell regular decahedron) cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 12-point (regular icosahedron) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such (cell bundles of 6 + 5 = 11 cells), with each 11-cell bundle containing 11 of both kinds of cell. Very, very strange!{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden'': §II To Add is to Multiply|p=38|ps=; the sequence of regular polygons with harmonious proportions beginning {5}, {7}, {9}, {11} as a family of natural numbers; &quot;The numbers in this optimal family have remarkable properties [that allow] similar figures to be arranged easily or allow a figure to be dissected into a set of similar figures. This is the origin of the Greek idea of geometric progression — that ''multiplication can be accomplished instead by addition''.... For instance, it is well known that the square of the golden ratio is one more than itself (written above as 𝜙&lt;sup&gt;2&lt;/sup&gt; = 𝜙 + 1) and one less than 𝜙 is its reciprocal (1/𝜙 = 𝜙 - 1). The first relation equates multiplication with addition, while the second accomplishes division through subtraction. [These are pentagon {5} bisectional golden ratios, but] the heptagonal [{7} tri-sectional] ratios also behave in surprising ways — adding to multiply, subtracting to divide.... There are analogous constructions for [nonagon {9}] quadrisectional and [hendecagon {11}] pentasectional systems.&quot;}}

The 120-cell has three interchangeable cyclic expansion-contraction cycles, the pentad, hexad and heptad cyclic designs, rooted in the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems, respectively. Each of them constructs the whole sequence of regular 4-polytopes from a single building block. The key to all three cycles is that 5 + 6 = 11, and there is an 11-point quasi-regular 4-polytope in the sequence of convex 4-polytopes. It lies between the 8-point (16-cell) and the 16-point (8-cell), and is itself a &lt;math&gt;B_4&lt;/math&gt; polytope. Its role is to be a unified single building block common to all three cyclic expansion-contraction cycles. It is involved with the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems and expresses in one building block their relations to each other.

We have already seen how the hexad building cycle works. It is plain compounding of 4-orthoplexes at the beginning, increasing to densely intricate non-disjoint symmetry toward the end, making it difficult to see why it works once past the 24-cell. This is how the build works, in terms of sixes. The 16-cell is a 2-hexad (3 pairs of completely orthogonal great square planes). The 24-cell is 3 16-cells and a 4-hexad, 4 different ways: 16 non-disjoint hexads (great hexagon planes). The 120-point (600-cell) is 25 24-cells and a 20-hexad, 10 different ways: 200 non-disjoint hexads (great hexagon planes). The 600-point (120-cell) is 225 (9 times 25) 24-cells. ....

The pentad building cycle doesn't do plain compounding to regular 4-polytopes until the last one, the 120-cell. Fives don't even appear as pentagons until after the 24-cell. We have seen how there are 11 5-cells in the quasi-regular 11-cell, which occurs before the 8-cell, and also how everything can be compounded from irregular 5-cells (4-orthoschemes specialized for each 4-polytope). It is still not evident how a pentad cycle can work to build all the regular 4-polytopes from the regular 4-simplex. This is how is how the build works, in terms of fives. ....

This is how the build works, in terms of elevens. The 11-cell is 4 orthogonal heptads (intersecting in their skew hexagon Petrie polygon). The 24-cell is also 4 heptads (intersecting in 4 orthogonal digon axes, so each heptad contributes 6 unique points to a single fibration of 24 points). Heptads are quarter elevenads, so the 11-cell and 24-cell are both 1 elevenad. ....

See the tabular sequence of convex 4-polytopes in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Build with the blocks|§Build with the blocks]]'', below: table rows ''Inscribed, Pentad, Hexad, Heptad''.

=== 11 of them ===
There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of eleven 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum eleven of them. There is only a single Hopf fibration of the 11 great circles of an 11-cell. 

The 5-cell and the 11-cell are limit cases. The 5-cells are completely disjoint, but the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 10 other instances of itself.

== Cell rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer &amp; Segerman|2013|loc=''Puzzling the 120-cell''}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his section of the Wikipedia article on the 120-cell entitled ''[[120-cell#Visualization|§Visualization]]''.{{Sfn|Ruen &amp; Goucher et al. eds. 120-cell|2024|loc=Goucher's ''[[120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious place to look for the 11-cell's Hopf map is in Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other place it could be hiding. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The quasi-regular convex 11-cell|§The quasi-regular convex 11-cell]]'', and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen: triacontagon|2011|loc=''Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them except his illustrations. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Four orthogonal projections of the 120-cell to a central plane, selected from the table of 3 regular and 11 compound 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|100 icosahedra&lt;br&gt;120 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|5 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 icosahedra&lt;br&gt;6 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|120
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|4
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|50
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|7.2°
|align=center|36°
|align=center|6°
|align=center|7.2°
|}

....

Here we summarize our findings on the 11-cells so far, including in advance some results to be obtained in the next section two sections.

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section....

== The 15 major chords ring everything ==
The natural numbers are each a distinct flavor.

Each integer has its own distinct taste. The first six integers {0..5} are the six strongest, most familiar tastes in nature, instantly distinguished and utterly unique.

The seventh flavor {6} is the first ratio of two of the others. Thereafter follows in strange sequence all the ratios of all the flavors, each marriage of every two flavors, most of which are fractions &lt;math&gt;\{k/d\}&lt;/math&gt;, but some of which are integers &lt;math&gt;\{k\}&lt;/math&gt;.

Some of these ratios are successful flavors in their own right, as distinct and inimitable as the marriage of chocolate and oranges, but the marriage of many flavor pairs is mud. It is a rule of thumb among flavor specialists that nearly any taste can be successfully alloyed with either onions or chocolate, but the direct marriage of these two flavors is notably unsuccessful. Some flavors make famous pairings or infamous ones, and subtle new pairings are always to be discovered that work surprisingly well for some purpose, but the majority of flavor marriages are undistinguished compounds.

In polytope geometry, each chord of a polytope is both is a distinct 1-dimensional object, a chord of the sphere of a distinct length &lt;math&gt;l&lt;/math&gt;, and a distinct natural number &lt;math&gt;h&lt;/math&gt;, a unique flavor. If the polytope is regular, it is a noteworthy and successful flavor. The chord's length &lt;math&gt;l&lt;/math&gt; is a square root, related to the natural number &lt;math&gt;h = k/d&lt;/math&gt; and to a set of polytopes they both represent, by a formula discovered by Steinbach.{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden''; Figure 5. Optimal sections and proportions|p=37|ps=; the regular polygons {5}, {7}, {9} and {11} with their diagonals define respectively: {5} the golden bisection proportional to 𝜙; {7} an analogous trisection; {9} an analogous quadrasection; {11} an analogous pentasection.}} The chord length &lt;math&gt;l&lt;/math&gt; is related to the number of sides of the regular polygon formed by the chord (its numerator &lt;math&gt;k&lt;/math&gt;), and to the winding number or density of its regular skew Petrie polygon (its denominator &lt;math&gt;d&lt;/math&gt;).

The regular 4-polytopes do not contain an infinite variety of these chord flavors, but only 30 of them: only 15 unique pairs of 180° degree complementary chords, which combine to make every regular ''n''-dimensional thing found in nature.{{Efn|The 15 ''major chords'' suffice to construct all the regular ''n''-polytopes. Their 180° complements include 15 ''minor chords'' which occur only in the 120-cell, not in any smaller regular polytope.{{Efn|name=Major and minor chords}} Including the complement of the 180° {{radic|4}} digon {2} chord (the {0} chord {{radic|0}}), and the skew variant of the 90° {{radic|2}} square {4} chord (the {30/7} chord), there are 32 distinct natural numbers which are all the chords of all the regular polytopes in all dimensions.|name=32 chords}} All the regular polytopes (including the largest one on the cover of the box) can be built not only from a box containing a number of blocks of 3 shapes glued together by gravity, they can be also be built from a box containing a number of sticks of 15 lengths glued together by rubber joints.

[[File:15 major chords.png|500px|thumb|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the others occur only in the interior of 120-cell, not as edges of any of the regular 4-polytopes. There are 30 distinct 4-space chordal distances between vertices within the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to identify any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=Major and minor chords}} chords are properly labelled #1 — #15 in the order they fan out by size.{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''; Figure 3|ps=; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with a &quot;fan of chords&quot; diagram like the one here.|p=23}} They join vertex pairs which are 1 — 15 edges apart on a &lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon. A #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; chord forms &lt;small&gt;&lt;math&gt;f+1&lt;/math&gt;&lt;/small&gt; regular geodesic polygons which include a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; disjoint regular {{nowrap|&lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt;-gons}} (e.g. &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; a compound of 5 disjoint hexagons) plus a single [[W:Triacontagon#Triacontagram|regular skew {{nowrap|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram}}]] that wraps around the sphere &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times before closing its circle. Notice the maverick #11 chord, with its non-conforming origin point for its {{nowrap|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;}} polygram.]]

These 15 chords (natural numbers) include the integers 2, 3, 4, 5, 6, 10, 11, 15 and 30. The others are the fractions 5/2, 10/3, 13/7, 15/2, 15/4, and 15/7. Represented among the chord lengths (the square roots) measured in ''unit radii'' (not in {{radic|2}} radii, as metrics are often given elsewhere in this article) are the {{background color|paleturquoise|roots of the first 4 positive integers {{radic|1}}, {{radic|2}}, {{radic|3}} and {{radic|4}}}}, the {{background color|#FFCCCC|root of the very distinguished rational fraction {{radic|5/2}}}}, and the roots of various distinguished irrational fractions as Steinbach discovered: for example, 4 of them are {{background color|yellow|roots of golden numbers (a small integer ± 𝜙)}}. The 15 chords each form their regular polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, which is just &lt;small&gt;&lt;math&gt;\{k\}&lt;/math&gt;&lt;/small&gt; in 8 cases. Arranged in order #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; from shortest edge-length square root to longest, the #1 — #15 chords bridge vertex pairs which are &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; vertices apart on a [[W:Triacontagon#Petrie polygons|&lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon]]. Each chord makes a [[W:Triacontagon#Triacontagram|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; polygram]] of its edge length, a compound of the smaller &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. The 15 distinct polygons form a sequence you would not anticipate exactly unless you have Steinbach's formula:

{| style=&quot;background-color:transparent; white-space:nowrap;&quot;
|-
| {30/1} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|.073}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{30} 
| 120-cell edges
|-
| {30/2}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.191}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15}
|- style=&quot;background: yellow;&quot;|
| {30/3}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.382}} 
| &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi}&lt;/math&gt;&lt;/small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{10}''' 
| 600-cell edges
|-
| {30/4} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.573}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/2}
|- style=&quot;background: paleturquoise;&quot;|
| {30/5}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|1}}'''
| &lt;math&gt;1&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{6}''' 
| 24-cell, 8-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/6}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|1.382}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{5}'''
|- style=&quot;background: paleturquoise;&quot;|
| {30/7}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|2}}'''
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{4}''' 
| 16-cell edges
|- style=&quot;background: #FFCCCC;&quot;|
| {30/8}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.5}} 
| &lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;&lt;small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/4} 
| 5-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/9}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.618}} 
| &lt;math&gt;\phi&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{10/3}
|- style=&quot;background: paleturquoise;&quot;|
| {30/10}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|3}}''' 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{3}'''
|-
| {30/11}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.426}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{11/11}
|- style=&quot;background: yellow;&quot;|
| {30/12}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.618}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{5/2}
|-
| {30/13}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.810}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{13/7}
|-
| {30/14}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.928}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/7}
|- style=&quot;background: paleturquoise;&quot;|
| {30/15}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|4}}''' 
| &lt;math&gt;2&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{2}'''
|}

{| class=wikitable style=&quot;white-space:nowrap; float:right;&quot;
!colspan=6|Natural number &lt;small&gt;&lt;math&gt;h=\{k/d\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Fibers&lt;br&gt;&lt;small&gt;&lt;math&gt;f\{h\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Cells&lt;br&gt;&lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|4-tope&lt;br&gt;&lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Verts&lt;br&gt;&lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|[[W:Triacontagon#Triacontagram|30-gram]]&lt;br&gt;&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;
|-
!Chord
!Arc
!colspan=2|&lt;small&gt;&lt;math&gt;l/\sqrt{1}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;l/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;h&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#1
|15.5~°
|{{radic|0.073}}
|0.270
|{{radic|0.146}}
|&lt;small&gt;&lt;math&gt;30&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1\{30\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;5\{3,3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/1\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#2
|25.2~°
|{{radic|0.191}}
|0.437
|{{radic|0.382}}
|&lt;small&gt;&lt;math&gt;15&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;..\{3,3\}..&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/2\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#3
|36°
|{{radic|0.382}}
|0.618
|{{radic|0.764}}
|&lt;small&gt;&lt;math&gt;10&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;30\{3,3\}20&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/3\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#4
|44.5~°
|{{radic|0.573}}
|0.757
|{{radic|1.146}}
|&lt;small&gt;&lt;math&gt;15/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/4\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5
|60°
|{{radic|1}}
|1
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;6&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/5\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#6
|72°
|{{radic|1.382}}
|1.175
|{{radic|2.624}}
|&lt;small&gt;&lt;math&gt;5&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}} 
|&lt;small&gt;&lt;math&gt;10\{5,3\}12&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/6\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#7
|90°
|{{radic|2}}
|1.414
|{{radic|4}}
|&lt;small&gt;&lt;math&gt;4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7\{4\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;4\{4,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;9\{3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/7\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: #FFCCCC;&quot;|
|#8
|104.5~°
|{{radic|2.5}}
|1.581
|{{radic|5}}
|&lt;small&gt;&lt;math&gt;15/4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/4\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}} 
|&lt;small&gt;&lt;math&gt;15\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/8\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#9
|108°
|{{radic|2.618}}
|1.618
|{{radic|5.236}}
|&lt;small&gt;&lt;math&gt;10/3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10/3\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/9\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#10
|120°
|{{radic|3}}
|1.732
|{{radic|6}}
|&lt;small&gt;&lt;math&gt;3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;10\{3\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/10\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#11
|135.5~°
|{{radic|3.426}}
|1.851~
|{{radic|6.852}}
|&lt;small&gt;&lt;math&gt;11/11&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;11\{11/11\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;11\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#12
|144°
|{{radic|3.618}}
|1.902
|{{radic|7.235}}
|&lt;small&gt;&lt;math&gt;5/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2},3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/12\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#13
|154.8~°
|{{radic|3.810}}
|1.952
|{{radic|7.621}}
|&lt;small&gt;&lt;math&gt;13/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;13\{13/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;7\{5,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/13\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#14
|164.5~°
|{{radic|3.928}}
|1.982
|{{radic|7.857}}
|&lt;small&gt;&lt;math&gt;15/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;137\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/14\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{radic|8}}
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15\{2\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐𝜙&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;8\{3,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/15\}&lt;/math&gt;&lt;/small&gt;
|}

The polygons '''{10}''', '''{6}''', '''{5}''', '''{4}''', '''{3}''', and '''{2}''' are flat polygons that each lie on a great circle in a central plane of the 3-sphere. The polygons {30}, {15}, {15/2}, {15/4}, {10/3}, {11/11}, {5/2}, {13/7}, {15/7} and all the {30/''n''} polygons are skew polygrams spiraling through curved 3-space, each successive edge in a different central plane of the 3-sphere. Projected to the 1-sphere (a circle in the plane), they are [[W:Star polygon#Regular star polygon|regular star polygrams]], the simplest being the pentagram {5/2}. In 4-space they wind around the 3-sphere more than once ''without self-intersecting'', before they close their circuit. They lie on helical geodesic circles or ''isoclines'' of circumference greater than 2𝝅''r''.

Even the 6 integer chords which form flat polygons also form skew polygrams. Each of the 15 chords occurs as a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, a fiber bundle of Clifford parallel polygons that totals 30 vertices per bundle. Each chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; also forms a single skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram that winds through another, shorter chord's bundle of parallel polygons, intersecting them in succession multiple times each, and weaving that whole bundle of parallel fibers together as the cross-thread does on a loom. The polygram cross-thread has to wind multiple times through the whole bundle (&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times around the [[W:Triacontagon#Triacontagram|30-gram]] in which each edge connects every &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt;th vertex) to complete an enumeration of all 30 vertices in the bundle. For example, the #5 chord forms hexagon &lt;small&gt;&lt;math&gt;\{6\}&lt;/math&gt;&lt;/small&gt; flat planes in parallel bundles of 5, written &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt;, and also a skew &lt;small&gt;&lt;math&gt;\{15/2\}&lt;/math&gt;&lt;/small&gt;-gram of #5 edges which winds 2 times through the #3 chord's bundle of 3 decagon &lt;small&gt;&lt;math&gt;\{10\}&lt;/math&gt;&lt;/small&gt; flat planes. Every chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; forms both its &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram and its fiber bundle of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, and the edges of one chord's parallel polygons are also the edges of its skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram weaving through some shorter chord's bundle of parallel polygons. Each numbered chord is a natural number, a ratio of two integers and their polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, either flat or skew. Each chord number defines a &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; skew polygon that is a compound bundle of the &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. All these flat or skew polygons are regular polygons made of their numerator instances of a single chord #, whose length is that square root, Steinbach's function of the natural number.

The relationship between each bundle of flat Clifford parallel great circle polygons of a distinct length, and its corresponding bundle of skew polygrams of a longer length, is precisely that of a discrete simple rotation and its corresponding discrete isoclinic rotation. For example, the flat great circle hexagons of edge length {{radic|1}} lie in the invariant rotation planes of the 24-cell's characteristic 60° isoclinic rotation, in which the vertices move on skew hexagram rotation paths of edge length {{radic|3}}. The 24-cell contains 4 bundles of 4 Clifford parallel (disjoint) hexagon central planes each. The 24 vertices of the 24-cell in each bundle are linked by rotation in the 4 hexagon invariant planes in two different ways: simply and isoclinically. In a simple rotation the vertices of a single invariant hexagon exchange places on that great circle hexagon of edge {{radic|1}}. In an isoclinic rotation in which any one of these 4 disjoint hexagon central planes is invariant, all 4 of them are necessarily invariant planes in the rotation, because they rotate in parallel. In that isoclinic rotation the vertices of all 4 hexagon invariant planes (all 24 vertices of the 24-cell) exchange places concurrently, moving along 4 disjoint helical hexagrams of edge {{radic|3}} in parallel. It is a characteristic only of this distinct rotation in hexagon invariant planes that the helical vertex path or ''isocline'' is a hexagram of period 6, edge length {{radic|3}}, and circumference 720°. Rotations in other kinds of polygon invariant planes have different characteristic isoclines. The period of their isoclinic rotation must be ''the same or greater'' than the period of their simple rotation; in most cases (unlike this one) it is greater.

The 15 chords form not only distinct fiber bundles of polygons, but also distinct bundles of polyhedra: cell rings &lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt; and compound vertex figures &lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;. These combine to form their characteristic regular 4-polytope &lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Coxeter|1973|p=269|loc=Compounds|ps=; &quot;It is remarkable that the vertices of {5, 3, 3} include the vertices of all the other fifteen regular polytopes in four dimensions.&quot;}} All these elements of a regular 4-polytope constructed from one chord are themselves regular polytopes, and each of the 16 regular 4-polytopes is a function of one characteristic chord. If the regular 4-polytope is convex, its characteristic chord is its unit-radius edge. For a non-convex regular 4-polytope, the characteristic chord is its unit-radius ''mid-edge diameter'', the distance from the midpoint of each edge to the midpoint of the completely orthogonal edge. Each of the characteristic chords (each distinct natural number) has its expression as one of the 16 regular 4-polytopes and all the products of the distinct regular 1-polytopes, 2-polytopes and 3-polytopes that construct it, and the whole distinct construction from the natural number to the 4-polytope is an [[W:SO(4)|SO(4) symmetry group]] expression of that 4-polytope's characteristic discrete [[W:SO(4)#Isoclinic rotations|isoclinic rotation]], in the invariant central planes of its characteristic chord, by the characteristic chord arc.

...demonstrate this theory by proper enumeration of the chord lengths in the table...

...color-code the △ ☐ 𝜙 ✩ symbols ... chord colors in diagram should match them ... some of the diagram colors are wrong and must be changed, but otherwise diagram is pretty good ... i think in a colored diagram all chords of the same length should have the same size line dash, the polygons are not obvious enough and the 15 fanning-out chords are readily identified by their labels...

... add back the counts of chords in the box (very small font number under each table entry), and present these two images as the front and back a box of building sticks and rubber joints...What's in the other box...

...say the rational-root major chords {{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|5/2}} are the ''n''-edges: their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge...

...abbreviated text of the [[120-cell#Relationships among interior polytopes|''§Relationships among interior polytopes'']] section from the 120-cell article...to explain the color-coding

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius=1}}

See also the elements, properties and metrics of the sequence of regular 4-polytopes summarized in tabular form in the section ''[[#Build with the blocks|§Build with the blocks]]'', below.

== Alicia Boole Stott's original formulation of dimensional analogy ==
&lt;blockquote&gt;&quot;...a method by which bodies having a certain kind of semi-regularity may be derived from regular bodies in an Euclidean space of any number of dimensions...&quot;{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''; presents cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues|pp=12-45}}&lt;/blockquote&gt;

Alica Boole Stott's 1910 paper reads like an exceptionally well-written patent application. It is a comprehensive stepwise description of her original invention, a general method of dimensional analogy between regular and semiregular polytopes of the same edge length.

[[File:The Roots of Three.png|thumb|An illustration of Boole Stott's expansion of unit-edge triangle to hexagon captures the geometric basis in 3 of the &lt;math&gt;A_n&lt;/math&gt;root system.]]

The engineering-like articulation of her new principles is entirely discrete and group-theoretic. Boole Stott repeatedly refers to sets of polytope elements as ''groups''. All her transformations are discrete instances of one or another special case of her &quot;two inverse operations which may be called ''expansion'' and ''contraction''.&quot; Hers is an exploration of ''n''-dimensional space at its most basic: space defined by the discrete objects which fill it, themselves defined directly as groups of elements with a few operations on them, which transform them into each other. This is geometry which seems to precede Euclid's postulates, and does not depend on them. Instead it corresponds directly to the deeper group theory mathematics which underlies geometry, even though Boole Stott, like the ancients, did not have that mathematics.

Boole Stott enumerated her expansion and contraction operations as transformations between regular and semi-regular polytopes of a single edge length. In her 1910 paper she did not consider any polytopes of multiple edge lengths, or expansions of the subject polytope by a distance other than that exactly required to produce new edges of the original edge length, so of course she did not enumerate Moxness's non-uniform rhombicosidodecahedron (Hull #8). This may be why section 8&lt;sub&gt;3&lt;/sub&gt; is missing from her earlier drawings of the sections of {5,3,3}; she may not have had a construction for it.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The Boole Stott expansion (or contraction) operation is as precisely defined a transformation as Coxeter's reflection, rotation and translation operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions). Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;
where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;
where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;
For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} An instance of the expansion (or contraction) operation is an application of the general operation to a particular subject polytope, operating on all of its elements symmetrically at once. The general operation is restricted, such that one may only expand (or contract) the subject polytope by pushing all its edges, faces or cells directly away from (or toward) its center, without changing the size of the moved elements. This splits each vertex into several (or coalesces several vertices into one), and has corresponding effects on the other elements, producing new edges ''of the same size'' between pairs of separated vertices (or reducing some edges to single vertices). The distance all the elements are moved away from (or toward) the center must be such that the vertices separate producing new edges of only the original edge length (or come together reducing edges to single vertices). It is permitted to combine these three operations, e.g. by pushing the edges ''and'' the faces, or by pushing all three element sets, provided the combination results in a semi-regular polytope of only one edge length. The three operations are all commutative with one another, so combined operations can be considered either as a sequence of expansions (or contractions) linking several related polytopes, or as their concurrent movements, a single transformation of the subject polytope. Since there are 8 combinations of one to three operations there are 8 polytope variants (including the original regular polytope), and all are adjacent to one another by a single transformation. The variants of the original regular polytope are named for which element set(s) are moved. By enumerating all 8 variants of expansion (and contraction) for each of the six regular convex 4-polytopes, Boole Stott produced a table of regular and semi-regular 4-polytopes divided into 12 sets of 8: six expansion sets and six contraction sets.{{Sfn|Boole Stott|1910|loc=''Table of Polytopes in S&lt;sub&gt;4&lt;/sub&gt;''; note that Boole Stott's usage of ''S&lt;sub&gt;4&lt;/sub&gt;'' denotes a Euclidean 4-space ''E&lt;sub&gt;4&lt;/sub&gt;'', or a 3-sphere space embedded in it, ''not'' a 4-sphere space embedded in a Euclidean 5-space|p=30}} Each set includes just one of the six regular convex 4-polytopes, with its distinct unit-radius edge length, and the 7 possible semi-regular expansions (or contractions) of it, which all preserve that characteristic edge length. The polytopes in each set of 8 all have the same total number of elements as their original regular polytope, but each polytope in the set has a distinct combination of cell, face, and vertex figures. The 12 sets are not completely disjoint, and even within one set the 8 operations do not necessarily produce 8 unique polytopes. For example, the regular 5-cell (the 4-simplex polytope) has 5 cells, 10 faces, 10 edges, and 5 vertices (30 elements). Expanding the 5-cell produces 7 related semi-regular 4-polytopes of 30 elements each (two of which happen to be the same polytope). In each expansion, the 5 cells have expanded into semi-regular or regular polyhedra, the 10 faces and 10 edges ''may'' have expanded into prisms, and the 5 original vertices have expanded into regular or semi-regular cells.

== The regular 11-cell 4-polytope ==

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosadodecahedron). Each non-convex Jessen's icosahedron is a set of 6 reflex edges that are {{radic|5}} 5-cell edges, each set symmetrically arranged in 3 parallel pairs, not as far apart ({{radic|3}}) as they are long, in 3 orthogonal central planes. The 12-point (Jessen's sets of 6 reflex 5-cell edges) are inscribed in the 60-point with their endpoints as a vertex of a pentagon face. The 12 pentagon faces are joined to 5 other pentagon faces by the 5-cell edges; not to their nearest neighbor pentagons but to their next-to-nearest neighbor pentagons; and since there are 5 vertices to each pentagon face, there are 10 12-point (Jessen's sets of 6 reflex 5-cell edges). Each pentagon face is joined to another pentagon face by 5 reflex edges of five distinct Jessen's. The 5 reflex edges are Clifford parallel, the 5 edges of a pentagonal prism column that has a twist to it. The contraction of those 5 edges into 1, and those 10 vertices (two pentagons) into 2, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing all their edges as a single Jessen's. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (octahedron) by conflating its parallel pairs of edges, leaving the 11-cell's abstract 6-point (hexad cell). The hexad's situation is that it occurs in bundles of 5, disjoint but close together in the sense of adjacent, like the great circle rings in a Hopf fiber bundle. 

To summarize, the 12-point (2-hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell). We shall now be able to see how Fuller's perfect hexads combine with Todd's perfect pentads to form the perfect heptads of the 11-cell and 137-cell.  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=alias the hemi-cuboctahedron}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces. (The ''heptad'' arrives at the party, anonymously; they are our distinguished anti-celebrity, who most of the other guests have never even heard of.) The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. This 12-point (truncated tetrahedron) is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 100 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp: Truncated tetrahedron|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp: Truncated tetrahedron|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}} Of course Moxness has been there, seen that, already with his graphics software;{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} see his parameterized illustration of it exactly as it occurs in the 11-cell, below in his bestiary of A3, B3, and H3 Archimedian and Catalan polyhedra in the 120-cell.]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:Archimedean_and_Catalan_solid_hulls_with_their_Weyl_orbit_definitions.svg|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of the 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. In this catalog of Archimedean and Catalan polyhedra sighted in the 120-cell by Moxness's software,{{Sfn|Moxness: Archimedean and Catalan hulls|2023|loc=Hull #1 Archimedean Name A3 110 Truncated Tetrahedron A (upper left)}} it is the first entry (upper left), Moxness's Hull # = 1 with 12 vertices of 3D norm {{radic|11}}/2. On a {{radic|2}}-radius 3-sphere, 5 of these with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell. 20 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges are cells in the [[#The rings of the 11-cells|cell rings of the quasi-regular 11-cell]], which also has 12 regular 5-cells as cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

...

Both kinds of cells in the 11-cell have expressions in two different dimensions (on the 2-sphere and 3-sphere), because they are all centered on the center of the 3-sphere, unlike the cells of most 4-polytopes, which are centered on various off-center points within the 4-polytope's interior. 

One kind of 11-cell cell appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (semi-regular truncated cuboctahedron), a.k.a. the truncated tetrahedron. The other kind of cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now, except in the 120-cell. But this is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron, and a regular tetrahedron behind each face. The 11-cell is thus a honeycomb of mixed regular tetrahedra and truncated cuboctahedra (a.k.a. truncated tetrahedra) that meet at their shared 5-cell faces. 

Both the 5-cells (decahedra) and the truncated cuboctahedra (truncated tetrahedra) are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks in their 4-polytopes (off-center).{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} Therefore all the 11-cell's cells are concentric, which indeed makes a different kind of honeycomb, an inside-out cellular honeycomb with all the cells piled on top of each other overlapping in 4-space. They do not overlap in 3-sphere space however; their 3-volumes (but not their 4-contents) are disjoint. They make a proper 4D honeycomb like the cells of any 4-polytope, with volumetrically disjoint 3D cells that meet each other in the proper way at their shared 2D faces.

An 11-cell contains 6 completely disjoint 5-point (1-pentad 5-cell decahedron cells), and 5 volumetrically disjoint 12-point (2-hexad truncated cuboctahedron cells), 11 volumetrically disjoint cells altogether. Of course it is supposed to have 11 identical 11-point (elevenad hemi-icosahedral cells), and that is also true in its abstract sense, which is not the same sense as disjoint cellular containment: each pentad or hexad cell is inscribed in one of 11 larger 60-point (Moxness's rhombicosadodecahedron cells). The abstract pentads and hexads are only hemi-polyhedra, so the 11-cell's pentad cells are 12 (out of 120) disjoint instances of a pentad 5-point, and its hexad cells are 10 (out of 100) disjoint instances of a hexad 6-point, 1/10th of the 120-cell's cellular volume in both cases. The 11-cell's hemi-icosahedral cells are 11 (out of 120) non-disjoint instances of an elevenad 60-point. Each 11-cell shares each of its hexad cells with 9 other 11-cells, and each of its 11 vertices (out of 600) occurs in 9 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.|name=Tetrahedral vertex figures}}

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

...insert image thumb of 11-cell...

...Construct the 11-cell from the heptad....

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...Four concentric 4-polytopes may be mutually orthogonal. In any set of 4 mutually orthogonal regular 5-cells, one pair of faces in each orthogonal pair are completely orthogonal to each other. Thus three edges in one 5-cell in an orthogonal pair are opposite (completely orthogonal) to a face of the orthogonal 5-cell. .... A 5-cell has 5 vertices in its abstract hemi-icosahedral cells, but only 4 vertices in one concrete icosahedral cell; the 5th lies in an icosahedral cell of another 11-cell (not in another icosahedral cell of the same 11-cell).

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

....

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 11-cell ===

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon.... (wrong, 11-cell's Hopf map is the truncated icosahedron)

....

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk: Truncated icosahedron|2018}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

...

Since the rhombicosadodecahedron has 120 5-cell edges, the 120-cell has 120 completely disjoint 5-cells, and the rhombicosadodecahedron shares each vertex with 11 other rhombicosadodecahedra, how a rhombicosadodecahedron is bound to its neighbor 5-cells is constrained. A further constraint is that the abstract hemi-icosahedron has only 15 5-cell edges, so the 120 actual 5-cell edges of the rhombicosadodecahedron are conflated 8-to-1 in the hemi-icosahedron. To be conflated by contraction into a single edge, those 8 edges must lie parallel to each other in the 120-cell; either actually parallel in the same plane, or Clifford parallel in separate planes, as two edges in completely orthogonal planes may be parallel and orthogonal at the same time. Each 5-cell inscribed in the 120-cell contains 5 such tetrahedra occur in parallel pairs in 4 orthogonal planes (none of them completely orthogonal planes). 

.... 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
Each conflated vertex is the 10 vertices of two orthogonal regular 5-cells...
The 60-point contains 6 pairs of these orthogonal regular 5-cell cells...
...its 20 hexad cells are ..., a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere.

== The regular 137-cell 4-polytope ==

(description originally written for the 11-cell, which applies only to a fibration of 11 11-cells)

It is quasi-regular with two kinds of cells, the 12-point (regular icosahedron) and the 5-point (regular 5-cell), which have just one kind of regular face between them. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of regular facets (it is semi-regular with regular facets), it also means that its facets have expressions of two different dimensionalities because they are concentric in the 3-sphere. The 11-cell has two kinds of regular cells. One appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (regular icosahedron), but since it is embedded at the center of the 3-sphere (unlike the cells of most 4-polytopes) it also acts as an inscribed 4-polytope.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} The other cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now except in the 120-cell. This is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron. The 11-cell is a honeycomb of mixed icosahedron and half-icosahedron cells. Like the icosahedra, the decahedra are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks, off-center in their 4-polytopes. All the 11-cell's cells are concentric, which indeed makes an unusual honeycomb of cells, an inside-out honeycomb. It is a proper 4D honeycomb however, with volumetrically disjoint 3D cells that meet each other in the proper way at shared 2D faces. The embedded 12-point (regular icosahedron) cell is an icosahedral pyramid (as a 4-polytope), which is filled by 20 4-point (irregular tetrahedra) that meet at its center; the center point of the 11-cell acts as a 12th vertex, just as the center point of the 24-cell acts as a 25th vertex in some honeycombs. The 5-point (regular 5-cell) cell contains five 4-point (regular tetrahedra) which meet the 12-point (regular icosahedra)'s irregular tetrahedral cells face-to-face, so the 11-cell is quasi-regular in the usual sense of having two varieties of regular 3-polytope cells, but additionally both varieties of of its cells are bi-dimensional: another clue that the 11-cell has something to do with dimensional analogy

....

The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

....

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur to make an 11-cell. 11 of them occur in a 137-cell, all 120 of them occur in the 120-cell, and they occur nowhere else.

How can this regular 4-polytope possibly exist? By Schläfli's criterion {3, 3, ..} exists, but not as a finite polytope.{{Sfn|Coxeter|1973|loc=§7.7 Schläfli's criterion}} Coxeter found {3, 5, 3} as a finite hyperbolic honeycomb, but not in Euclidean 4-space.{{Sfn|Coxeter|1968|title=The Beauty of Geometry: Twelve Geometric Essays|loc=''Regular Honeycombs in Hyperbolic Space''|ps=; &quot;The honeycombs considered by the above authors [Schlegel, Schläfli, Klein, Sommerville] have finite cells and finite vertex figures. ... A further extension allows a cell or vertex figure to be a ''star''-polytope, so that the honeycomb covers the space several times. .... We shall find that there are four regular star-honeycombs in hyperbolic 4-space, as well as two infinite families of them in the hyperbolic plane.&quot;}} We find that {3, 3, ..} is a central expression of the &lt;math&gt;H_4&lt;/math&gt; root system, being the regular convex hull in 4 dimensions of 137 points, 96 pentads and 600 hexads. The minimal expression of &lt;math&gt;H_4&lt;/math&gt; is the regular convex hull {3, 3, 5} of 120 points, 600 tetrads and 75 hexads, and its maximal expression is the regular convex hull {5, 3, 3} of 600 points, 120 pentads and 675 hexads. {3, 3, ..} is a third regular &lt;math&gt;H_4&lt;/math&gt; polytope which fits between them. The properties and metrics of this regular sequence are presented in tabular form below in ''[[#Build with the blocks|§Build with the blocks]]''.

{|class=&quot;wikitable&quot;
!colspan=3|Regular &lt;math&gt;H_4&lt;/math&gt; polytopes
|-
|[[File:600-cell.gif|200px]]
|[[File:5-cell.gif|200px]]
|[[File:120-cell.gif|200px]]
|-
!120-point 600-cell
!137-point 137-cell
!600-point 120-cell
|}

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 137-cell ===
....

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=Apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|2022|loc=''[[Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]''}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie: On Fuller's use of language|2024|loc=''[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry'']]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center, 1948-1949|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with their mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the gender neutral shad can open their six beaks all the way, until they become flat squares and they becomes a cuboctahedron, or they can shut them all tight like a turtle retracting into their octahedron shell. The six mouths always move in unison. This is [[Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do their ''really'' odd trick -- where they flip their 6 jaws 90 degrees in their 6 faces and shut their 6 beaks on the opposite axis of their squares than the one they opened them on -- or whether they will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than their normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make their edge length exactly the same size as their radius, when they open their mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing their mouths in spherical synchrony, their 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron (the hexad), then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=''[[W:SO(4)|SO(4)]]''}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=Isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its orthogonal hexad's planes (wz, wx, wy).{{Efn|name=Six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of (expanded octahederon 16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh &amp; Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== Legendre's 12-point hexad binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a 12-point double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's perfectly realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, this 2-hexad expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre hexad) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction.{{Efn|name=Concentric polyhedra}} In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== ''n''-simplexes ===
....

The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== ''n''-orthoplexes ===
....
[[File:Great 5-cell √5 digons rectangle.png|thumb|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} ''regular'' great hexagons. These are the same 200 dodecagon central planes illustrated above, which also contain 6 120-cell edges and two opposing ''irregular'' great hexagons (truncated {{radic|6}} triangles). The 1200 {{radic|5}} 5-cell face triangles occupy 1200 other, non-central planes.]]

...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's Hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[16--cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[16--cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=Concentric polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[16--cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== ''n''-taliesins ===
The 4-simplex pentad building block combines with the 4-orthoplex hexad building block to make the 4-taliesin heptad building block. The heptad is the essential building block of the ''taliesin'' family of polytopes. The 11-cell and 57-cell hemi-polytopes and the 137-cell and 120-cell regular 4-polytopes are nuclear members of this trans-dimensional family. The 24-cell and 600-cell are extended members of it, because they also contain instances of the heptad building block.

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its spherical symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* [[W:Taliesin Myrddin Namkai-Meche|Taliesin Myrddin Namkai-Meche]], a young Reed College scholar who was stabbed to death by a crazy man in 2017 on a Portland, Oregon light rail train, when he rose to the nonviolent defense of two black teenagers who were being violently assaulted in racist and anti-Muslim language by a right-wing extremist.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in orthogonal [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].
* '''Taliesan polytope''', an (''n+1)''-polytope whose ''n''-polytope dimensional analogue inscribed concentrically in the ''n''-sphere forms a convex honeycomb of itself which is the (''n+1)''-polytope.{{Efn|name=Embedding point}}
* The taliesan polytope's honeycomb, a Hopf fibration of its great circles.
* The ''n''-dimensional family of dimensionally analogous polytopes related to a taliesan polytope by expansion and contraction operations.
* The characteristic [[w:Orthoscheme|orthoscheme]] of a teliesan polytope, an irregular simplex that is the complete gene for the entire expression of trans-dimensional symmetry relationships among the polytopes of the ''n''-dimensional family, as well as the complete gene for the teliesan polytope's rotational symmetry.{{Efn|name=Characteristic orthoscheme}}

=== ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== ''n''-equilaterals ===
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed at four different [[#n-leviathon|chord]] lengths from each other. In the {{radic|2}}-radius polytopes we have using as examples in this paper they are {{radic|2}}, {{radic|4}}, {{radic|6}} and {{radic|8}}, but in a ''unit-radius'' polytope they are: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}.

Each vertex is joined to 8 others by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}. The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== ''n''-pentagonals ===

[[File:600-cell.gif|thumb|The 120-point (600-cell), &lt;math&gt;H^4&lt;/math&gt;.]]
...

[[File:600-cell vertex geometry.png|thumb|Vertex geometry of the 600-cell, showing the 5 regular great circle polygons and the 8 vertex-to-vertex chord lengths with angles of arc. The golden ratio{{Efn|name=golden chords|group=}} governs the fractional roots of every other chord,{{Efn|name=fractional root chords}} and the radial golden triangles which meet at the center.|alt=|400x400px]]
The 120 vertices of the 600-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏&lt;sup&gt;−1&lt;/sup&gt;) beginning with a vertex; see column ''a''}} at eight different [[#n-leviathon|chord]] lengths from each other. These edges and chords of the 600-cell are simply the edges and chords of its five great circle polygons. In ascending order of length in a ''unit-radius'' polytope (not in a {{radic|2}} radius polytopes like our usual examples), they are {{radic|0.𝚫}}, {{radic|1}}, {{radic|1.𝚫}}, {{radic|2}}, {{radic|2.𝚽}}, {{radic|3}}, {{radic|3.𝚽}}, and {{radic|4}}.{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=Decimal fractions of square roots|group=}}

Notice that the four [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#''n-''equilaterals|hypercubic chords]] of the 24-cell ({{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|4}}) alternate with the four new chords of the 600-cell's additional great circles, the decagons and pentagons. The new ''golden chord'' lengths are necessarily square roots of fractions, but very special fractions related to the golden ratio{{Efn|1=The fractional-root ''golden chords'' are irrational fractions that are functions of {{radic|5}}. They exemplify that the [[W:golden ratio|golden ratio]] &lt;big&gt;𝜙&lt;/big&gt; {{=}} {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618 is a circle ratio related to &lt;big&gt;𝜋&lt;/big&gt;:&lt;br&gt;
: {{sfrac|𝜋|5}} = arccos (𝜙/2)
is one decagon edge, the 𝚽 = {{radic|0.𝚫}} = {{radic|0.382~}} ≈ 0.618 chord.
Reciprocally, in this function discovered by Robert Everest expressing &lt;big&gt;𝜙&lt;/big&gt; as a function of &lt;big&gt;𝜋&lt;/big&gt; and the numbers 1, 2, 3 and 5 of the Fibonacci series:&lt;br&gt;
: &lt;big&gt;𝜙&lt;/big&gt; = 1 – 2 cos ({{sfrac|3𝜋|5}})
{{sfrac|3𝜋|5}} is the arc length of the &lt;big&gt;𝜙&lt;/big&gt; = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618 chord.|name=golden chords|group=}} including the two [[W:golden section|golden section]]s of {{radic|5}}, as shown in the diagram.{{Efn|The 600-cell edges are decagon edges of length {{radic|0.𝚫}}, which is 𝚽, the ''smaller'' golden section of {{radic|5}}; the edges are in the inverse [[W:golden ratio|golden ratio]] {{sfrac|1|φ}} to the {{radic|1}} hexagon chords (the 24-cell edges).
The other fractional-root chords exhibit golden relationships as well. The chord of length {{radic|1.𝚫}} is a pentagon edge.
The next fractional-root chord is a decagon diagonal of length {{radic|2.𝚽}} which is &lt;big&gt;𝜙&lt;/big&gt;, the ''larger'' golden section of {{radic|5}}; it is in the golden ratio{{Efn|name=golden chords|group=}} to the {{radic|1}} chord (and the radius).{{Efn|Notice in the diagram how the &lt;big&gt;𝜙&lt;/big&gt; chord (the ''larger'' golden section) sums with the adjacent 𝚽 edge (the ''smaller'' golden section) to {{radic|5}}, as if together they were a {{radic|5}} chord bent to fit inside the {{radic|4}} diameter.}}
The last fractional-root chord is the pentagon diagonal of length {{radic|3.𝚽}}.
The [[W:Pentagon#Side length is given|diagonal of a regular pentagon]] is always in the golden ratio to its edge, and indeed &lt;big&gt;𝜙&lt;/big&gt;{{radic|1.𝚫}} is {{radic|3.𝚽}}.|name=fractional root chords|group=}}

....


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== ''n''-leviathon ===
...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of its &lt;math&gt;10^2&lt;/math&gt; disjoint instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction, before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]].{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}} .... It is only because she did not have the 11-cell to start with that she did not discover the regular 137-cell {3, 3, ..} by her method.

Between Alicia Boole Stott and her identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and her identification of symmetry groups with conservation laws (in her 1st theorem), we owe the deepest mathematics of geometry and physics originally to two pioneering women, colleagues of their contemporaries the greatest men of the academy, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just an [[W:abstract polytope|abstract 4-polytope]], not just a singleton convex 4-polytope, and not just a honeycomb.{{Sfn|Coxeter|1970|loc=''Twisted Honeycombs''}} The 11-point (11-cell) has a concrete ...regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}&lt;nowiki/&gt;} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, 3, 3} as the 137-point (137-cell).

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 11-cell, 137-cell and 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=''Clifford Spinors and Root System Induction: H4 and the Grand Antiprism''}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=Apology}}

== Build with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Duveneck|1978|loc=Carl Jung, quoted in ''Life on Two Levels''|p=ii|ps=.{{Sfn|Jung|1961|ps=: &quot;The best of truths is of no use unless it has become one's most personal inner experience. It is the duty of everyone who takes a solitary path to share with society what he finds on his journey of discovery.&quot;}}}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1954|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius={{radic|2}}}}

== Acknowledgements ==
...

== Notes ==
{{Regular convex 4-polytopes Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Citation|last=Boole Stott|first=Alicia|author-link=W:Alicia Boole Stott|title=Geometrical deduction of semiregular from regular polytopes and space fillings|date=1910|pages=12-45|publisher=Johannes Muller|place=Amsterdam|url=https://dwc.knaw.nl/DL/publications/PU00011492.pdf}}
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      <text bytes="208258" sha1="6fhk0fdqc5zsrb1cq0se21wr4x5k4a6" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|June 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a regular 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the &lt;s&gt;120-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 5, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 11 [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] cells. Eleven 11-cells (plural) are the 137-cell, the &lt;s&gt;600-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 3, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 137 [[W:Rhombic triacontahedron|triacontahedron]] cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=''Regularity of Graphs, Complexes and Designs''}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=''A Symmetrical Arrangement of Eleven Hemi-Icosahedra''}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]]
The 4-dimensional regular polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:Simplex|simplex]] (called the [[5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:Orthoplex|orthoplex]] (called the [[16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular convex 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the [[w:Tesseract|16-point (8-cell) tesseract]], the 4-cubic, which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found within any of the other regular convex 4-polytopes, except in the largest and most complex one, the [[120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). However, we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the 16-cell is their very starting point, and the most frequently used tool in the box.

== 5-cells and hemi-icosahedra in the 11-cell ==
[[File:Symmetrical_5-set_Venn_diagram.svg|thumb|The 5-point (10-face) regular 5-cell (the regular 4-simplex). Grünbaum's rotationally symmetrical 5-set Venn diagram{{Sfn|Grünbaum|1975|loc=''Rotationally symmetrical 5-set Venn diagram'', Fig 1 (e)|ps=; partitions the individual elements of the 5-cell.}} is an illustration of the 5-cell labeling each of its &lt;math&gt;2^5&lt;/math&gt; elements.{{Sfn|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled; image includes the Python code to render it, optimising for maximum area of the smallest regions.}}]]
[[File:Hemi-icosahedron.png|thumb|The 6-point (10 face) [[W:hemi-icosahedron|hemi-icosahedron]], an abstraction of the regular icosahedron, has half as many faces, edges and vertices. Each element of the abstract polyhedron represents two real elements found in different places in the 11-cell's realization.{{Sfn|Ruen: hemi-icosahedron|2007}}]]
The most apparent relationship between the pentad 5-cell and the hexad hemi-icosahedron is that they both have 10 triangular faces. When we see a facet congruence between a 4-polytope and a 3-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 6-point 5-simplex. It is also the same size as the 6-point 3-orthoplex; thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no chord congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals, pentad and hexad, who are famously stand-offish with each other. Interesting!

The cell of the 11-cell is an abstract hexad hemi-icosahedron containing pentads in its interior, most handsomely illustrated by Séquin.{{Sfn|Séquin &amp; Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing rings of polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The 11-cell's 11 hemi-icosahedral cells have 10 triangle faces each. If we assume 11-cell and 5-cell faces are congruent, a 5-cell tetrahedron is behind each hemi-icosahedron face. At least 3, and no more than 4, of the 5-cell's 5 vertices belong to the same hemi-icosahedron. After the 3 face vertices, the other 2 of the 5-cell's vertices must define an 11-cell edge lying opposite the face. Coxeter determined that the 11-cell does indeed have an edge opposite each face, and that it does not belong to the same hemi-icosahedral cell as its opposing face.{{Sfn|Coxeter|1984|p=110|loc=§6. The Petrie polygon [of the 11-cell]|ps=; &quot;We may reasonably call this edge and face ''opposites''. It is easy to find the face opposite to a given edge by looking at the faces to which a given edge belongs. ... Conversely, given a face, we can find the opposite edge by seeing which vertices belong to neither of the hemi-icosahedra which share that face. The ten edges opposite to the ten faces of one hemi-icosahedron are the edges of the complementary &lt;math&gt;a_4&lt;/math&gt; [4-simplex], that is, the joins of all pairs of the five vertices [of the 11-cell] not belonging to the given hemi-icosahedron.&quot;}}

There are 11 disjoint 5-cells in each 11-cell, which is comprised of 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. The real 11-cell is harder to see than the abstract 11-cell that represents it, because the real hemi-icosahedron that the abstract hemi-icosahedron represents is hard to see, but seeing these objects will be easier once we have identified the real hemi-icosahedron and exactly where the 11-cell's real elements reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each hemi-icosahedron's 10 faces come from 10 disjoint 5-cells, and there is just one 5-cell with which it does not share a face.{{Efn|As Coxeter observes (in the previous citation), that unrepresented 5-point (5-cell) is the other 5 vertices of the 11-point (11-cell) that are not vertices of this 6-point (hemi-icosahedron), the hemi-icosahedron's disjoint complement.}} Each 5-cell has 10 faces from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share a face.

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in two different planes which are completely orthogonal to each other. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must be sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness: 8 concentric hulls|2022|loc=Hull #8 (lower right)|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell of the 11-cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Compounds in the 120-cell|§Compounds in the 120-cell]]'' that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software which uses Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s to render the polyhedra which are found in the interior of ''n''-dimensional polytopes.{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the tetrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is a real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (Hull #8) is a concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this {12} irregular great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing {6} irregular great hexagons (truncated triangles) of alternating 120-cell edges and 5-cell edges. The great dodecagon is a great circle of the hemi-icosahedral cell, and also a great circle of the 120-cell, because unlike the cells of most 4-polytopes, the 11-cell's cells are embedded in 4-space at the exact center of the 4-polytope, making them inscribed 4-polytopes themselves.{{Efn|name=Embedding point}} There are 25 great dodecagons and 120 5-cell edges in Moxness's 60-point (Hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (Hull #8) polyhedron altogether.

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point (Hull #8) is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] similar to the one from the catalog shown here,{{Sfn|Piesk: Rhombicosidodecahedron|2018}} but a slightly shallower truncation with smaller red pentagons (120-cell faces) and narrower rhombs. Rhombicosidodecahedra are made by truncating the [[W:Rhombic triacontahedron|rhombic triacontahedron]], which is the 30-sided polyhedron with only one kind of face, the dual of the 30-point icosidodecahedron. The 120-cell contains 120 of these 11-cell cells.]]
Moxness's 60-point (Hull #8) is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (Hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|The 5-point (regular 5-cell) can be another abstraction of Moxness's 60-point (Hull #8), 12-vertices-into-1, but it is not strictly the result of a contraction operation as described by Boole Stott, since she did not apply her expansion and contraction operations to uniform polytopes with more than one edge length.}} In fact Moxness was not the first to find a rhombicosidodecahedron in the 120-cell; Alicia Boole Stott was. It is the 6th section of {5,3,3} beginning with a cell, which Boole Stott identified as the ''e&lt;sub&gt;2&lt;/sub&gt; expansion'' of the icosahedron (or equivalently of its dual polyhedron the dodecahedron).{{Sfn|Boole Stott|1910|loc=§Examples of the e&lt;sub&gt;2&lt;/sub&gt; expansion|p=7}} But that rhombicosidodecahedron identified by Boole Stott is not Moxness's Hull #8: rather, it is the semi-regular Archimedean solid, with a single edge length and square faces. Moxness's Hull #8 with its two distinct edge lengths and rectangular faces is Coxeter's 8th section of {5,3,3} beginning with a cell, which is missing from the sections illustrated by Boole Stott.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than tetrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. The 120-cell contains 120 of Moxness's 60-point (11-cell cells). They are non-disjoint, with 12 60-points sharing each vertex.

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (Hull #8) is the real hemi-icosahedron cell, with 120 5-cell edges in it as chords lurking under its surface. The rhombicosadodecahedron has enough of those 5-cell edges in its interior to make 20 disjoint tetrahedral cells which belong to 20 distinct 5-cells.

The 600-point (120-cell) may be constructed as a semi-regular honeycomb of two kinds of cells, the 120 non-disjoint 60-point (Moxness's rhombicosadodecahedra) and their neighbors the 120 disjoint 5-point (5-cells), cell-bonded together by shared tetrahedral cells of edge-length {{radic|5}}. Each rhombicosadodecahedron is bonded to 20 islanded 5-cells, sharing with each a 5-cell tetrahedron (4 of its interior 5-cell faces, and 6 of its interior 5-cell edges). The rhombicosadodecahedron is also bonded directly to 10 other rhombicosadodecahedra (since each 11-cell has 11 rhombicosadodecahedron cells), again not by sharing one of its exterior faces, but this time by sharing an interior face, a {{radic|6}} triangle which lies in a central plane.{{Efn|The {{radic|6}} chord is another hidden chord lurking beneath the 60-point rhombicosadodecahedron's surface, like the {{radic|5}} 5-cell edge. Both these hidden chords lie on the same great circle with the 120-cell edges (the pentagon face edges). Three {{radic|6}} chords make a triangle which lies in a central plane. This {{radic|6}} great triangle must not be confused with the {{radic|5}} face triangle of the 5-cell, which does not lie in a central plane.}} The 120 rhombicosadodecahedra are also bonded directly together at ''all'' their exterior faces: at their pentagon faces (which are 120-cell faces of dodecahedral cells); at their small triangular faces (which are 600-cell faces of small tetrahedral cells, 10 of which are inscribed in each dodecahedral cell); and at their rectangular faces (which are faces of 10-point [[w:Pentagonal_prism|pentagonal prism]] cells, where the pentagonal prism is a column inscribed in a dodecahedral cell, joining its opposite pentagon faces). The 120 rhombicosadodecahedra are bonded directly to each other at their own exterior faces; no dodecahedra, small tetrahedra, or pentagonal prisms lie between them, and those objects are not cells of this honeycomb; we mention them here only to help locate the rhombicosadodecahedra within the 120-cell. The only objects which fill holes between the rhombicosadodecahedra are the 120 islanded 5-cells.{{Efn|This honeycomb of the 120-cell resembles the [[600-cell#Icosahedra|honeycomb of the 600-cell consisting of 24 face-bonded regular icosahedra]] with 24 clusters of 5 tetrahedra filling the holes between them. In the 120-cell there are 5 times as many of everything as in the 600-cell, so there are 120 (rhombicosadodecahedron) icosahedral cells with 600 tetrahedra filling the holes between them.}}

It is possible for rhombicosadodecahedra to meet each other at more than one kind of exterior face because they are not meeting there as cells of the same 11-cell; they belong to different 11-cells. The 11 rhombicosadodecahedra in the same 11-cell do not meet at their own exterior faces, they meet at their interior {{radic|6}} triangle faces. As a cellular 4-polytope, the 11-cell is inside-out. Its cells meet each other at interior planes, sharing interior {{radic|6}} triangle faces, instead of in the usual way that 4-polytopes' cells share an exterior face. It is a deeper kind of intertwining to be sure, but the 11 rhombicosadodecahedra in the same 11-cell are still volumetrically disjoint cells, and the 11-cell is a perfectly real 4-polytope with real cells. It is a semi-regular 4-polytope, because it has two different kinds of cells: 11 rhombicosadodecahedra and 11 5-cells. The 11 5-cells are islanded (not bound to each other), and the 11 rhombicosadodecahedra are bound both to 5-cells (at {{radic|5}} 5-cell faces) and to each other (at {{radic|6}} central triangle faces). In the section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#What's in the box|§What's in the box]]'', below, we will look closely at the relation between the {{radic|5}} chord and the {{radic|6}} chord, which is the relation between the pentad and the hexad, and at the general phenomenon of hemi-polytopes which meet each other at a central plane. Then in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Cell rings of the 11-cells|§Cell rings of the 11-cells]]'' we will look at the two-cell cellular structure of the 11-cell in more detail.

Considered as a single complex of overlapping cells rather than as distinct honeycombs of volumetrically disjoint cells, the 120-cell contains 60-point (rhombicosadodecahedral cells), islanded 5-point (5-cells), 20-point (dodecahedral cells), and 10-point ([[w:Pentagonal_prism|pentagonal prism]] cells). The 60-points meet each other in pairs at their pentagon faces, meet regular dodecahedra at those pentagon faces also, meet each other in quads at their 600-cell faces where they share an interior 4-point (600-cell tetrahedral cell) and its four faces (just as they share a larger interior 5-cell tetrahedron and its faces), and meet each other and the 10-point (pentagonal prisms) at their rectangle faces. We might try to imagine such a complex with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with parts truncated from 30 neighboring truncated dodecahedra pentagonal prisms). There are 120 of each of these 3 kinds of dodecahedron. 

Moxness's 60-point (rhombicosadodecahedron cell) is a central object of the 120-cell, and contains many wonders. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra by expansion and contraction operations, as Alicia Boole Stott discovered before 1910,{{Sfn|Polo-Blanco: ''Theory and history of geometric models of Alicia Boole Stott''|2007|loc=§5.3.2 1910 paper on semi-regular polytopes|pp=152-158|ps=; summarizes Boole Stott's method and results from {{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}}, including her tables of expansion-contraction dimensional analogies and a few of her illustrations.}} and those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, rediscovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter &amp; Petrie|1938|p=4|loc=''[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]''}}|name=Concentric polyhedra}} Before we move on from the 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice one more thing about the hidden {{radic|5}} chords lurking below its surface. The 12 little pentagon faces are spanned by sets of 6 {{radic|5}} chords (5-cell edges). The 6 chords of each set are disjoint (they don't touch or form 5-cell faces), but they are symmetrically arranged in 3 parallel pairs, {{radic|3}} apart, which lie in 3 orthogonal central planes. It turns out they are the 6 reflex edges of a 12-point non-convex polyhedron called the [[w:Jessen's_icosahedron|Jessen's icosahedron]], inscribed in the 60-point cell. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again and work our way outwards, before eventually in ''[[#The perfection of Fuller's cyclic design|§The perfection of Fuller's cyclic design]]'' we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cell)s ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks, showing its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Pentahemidemicube.png|120px]]
|[[File:Tetrahemihexahedron_rotation.gif|120px]]
|[[File:Pentahemicosahedron.png|120px]]
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts needed to build the 120-cell (the picture on the cover of the box), and everything else that's invisible in that picture which fits inside the 120-cell: all the astonishing regular 4-polytopes. The pentad building block is half a pair of orthogonal 5-point (regular 5-cells), the hexad building block is half an 8-point (16-cell), and the heptad building block is a quarter of an 11-point (11-cell). Everything can be built from these blocks by embedding them at the center of the 3-sphere, inscribing them as 4-polytopes to make compounds as with the 16-cell building block, rather than by assembling them face-to-face as 3-polytopes to make honeycombs in which they are cells or vertex figures.{{Efn|name=Embedding point}}

The pentad building block is the 5-point '''pentahemidemicube''', an abstract polyhedron with 10 equilateral triangle faces, 10 {{radic|5}} edges, and 5 vertices.{{Sfn|Christie: Pentahemidemicube|2024|loc=File:Pentahemidemicube.png|ps=; &quot;The 5-point ''pentahemidemicube'' is an abstract polyhedron with 10 equilateral triangle faces, 10 edges, and 5 vertices. It has a real presentation as the regular 5-cell.&quot;}} It has a real presentation as the 5-point (regular 5-cell 4-polytope), a regular tetrahedron expanded by addition of a fifth vertex equidistant from all the others in a fourth dimension. The pentad block is a tetrahedral pyramid (a 5-cell) inscribed in a cube: a tetrahedron inscribed in a cube (a demicube) expanded by addition of a fifth vertex, one of the cube's other 4 vertices. Each pentad (5-cell) building block lies in the 120-cell orthogonal to three other pentad (5-cell) building blocks. They are four completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges. Each regular 5-cell in the 120-cell is a pentad (pentahemidemicube) building block, in one of 30 orthogonal sets of 4. There are 120 pentad building blocks in the box.

Every vertex of the 16-cell, and therefore every vertex of the 8-cell, 24-cell, 600-cell and 120-cell, is surrounded by a 6-point (octahedral vertex figure). The hexad building block is that 6-point octahedron embedded at the center of the 4-polytope instead of at a vertex, that is, inscribed in the 3-sphere as a 4-polytope, the 6-point '''[[w:Tetrahemihexahedron|tetrahemihexahedron]]''' or '''hemi-cuboctahedron'''.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; &quot;The tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U&lt;sub&gt;4&lt;/sub&gt;. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices.&quot;}} The hexads have 4 red triangle 5-cell faces, bounded by 12 {{radic|5}} edges. Their 3 yellow square faces lie in central planes, because the building block is an abstract 4-polytope. The hexad is an irregular abstract polyhedron that also has 6 {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

Two orthogonal 5-point (pentad) building blocks intersect in a 4-point (inscribed tetrahedron) to form a 6-point (hexad) building block. Two orthogonal 6-point (hexad) building blocks intersect in a 4-point (central square) to form an 8-point (16-cell) building block, the most usable building block, from which everything larger can be built by compounding.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 120 pentads to pair with a completely orthogonal hexad, and although there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed) they only result in 120 pairs two different ways, so there are 120 11-cells. The 11-cell is 6 pentads' and 5 hexads' smallest joint compound, and the 120-cell is their largest, 120 pentads and 100 hexads.

The heptad{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''|pages=22–31}} building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|4}} edges and 9 {{radic|5}} edges), and 7 vertices.{{Sfn|Christie: Pentahemicosahedron|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;}} It is a compound of a 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]) and a 6-point (hexad [[W:hemi-icosahedron|hemi-icosahedron]]) intersecting in a 4-point (tetrahedron), that is, of three orthogonal 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]&lt;nowiki/&gt;s) intersecting in their common 4-point (tetrahedron). It is also the abstract cube contracted by removal of one vertex, which reduces one of the cube's two inscribed tetrahedra (demicubes) to a single remaining triangular face. As with all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Four &lt;small&gt;&lt;math&gt;(11-7)&lt;/math&gt;&lt;/small&gt; orthogonal 7-point (pentahemicosahedron) blocks can be joined at a common Petrie polygon (a skew hexagon which contains their orthogonal 6-edge paths) to construct a real 11-point ([[W:11-cell|11-cell]] semi-regular 4-polytope), which magically contains six &lt;small&gt;&lt;math&gt;(11-5)&lt;/math&gt;&lt;/small&gt; 5-point (1-pentad regular decahedra) as cells and five &lt;small&gt;&lt;math&gt;(11-6)&lt;/math&gt;&lt;/small&gt; 12-point (2-hexad semi-regular cuboctahedra) as cells. There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like [[W:Euclid's postulates|Euclid's postulates]], and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as honeycombs of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles. It possesses the complete symmetry of the polytope without any redundancy, because it contains one of each of the polytope's characteristic root elements.{{Efn|A [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be partitioned radially by its planes of symmetry (Coxeter's &quot;reflecting circles&quot;) into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated rotationally to generate its regular 4-polytope because it is the complete [[W:gene|gene]] for it, containing all of its elements and capturing all of its symmetry without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring that triggers the reflections.|name=Characteristic orthoscheme}} It is the gene for the polytope, which can be replicated to construct the polytope.

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its &lt;math&gt;A_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the regular compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;small&gt;&lt;math&gt;120\times 120 = 14400&lt;/math&gt;&lt;/small&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[16--cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its &lt;math&gt;B_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 384. The 120-cell is the convex hull of the regular compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 24-point (24-cell) is subdivided into 1152 instances of its [[24-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;F_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 1152. The 120-cell is the convex hull of the regular compound of 25 disjoint 24-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;25\times 1152 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has three equivalent origins in distinct root systems, though most are obviously constructed from one root, and not so obviously from the others. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything larger than the pentad can be constructed by compounding hexads, and we shall see that everything as large as a 24-cell has a construction from heptads which are a product of a pentad and a hexad. Surely every uniform polytope can be constructed by some function of expansions and contractions from any of three root systems, however indirect the recipe. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes, related by an equals sign between them, then give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics by [[W:Noether's theorem|Noether's theorem]].

=== Pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 octahedral cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell contains the 600 octahedral cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It also contains the 600 cubical cells of its 75 8-cells. It is a 600-cell in three ways, although the octahedra and cubes are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell and the 8-cell, respectively.|name=600 cells three ways}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (1-pentad 5-cell), out of the 8-point (2-hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (1-pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope that contains 5-point (1-pentad 5-cell) building blocks beside the largest one, it just isn't a ''regular'' 4-polytope. It is semi-regular with two kinds of cells, the 4-point (regular tetrahedron) and the 12-point (truncated cuboctahedron), which have just one kind of regular face between them, a 5-cell face. That semi-regular 4-polytope is the 11-cell.

We said earlier that the 11-cell had 11 disjoint 5-cells and 11 hemi-icosahedral cells in it, and that its construction from them might be hard to see, but now we are going to account numerically, at least, for how it can have 6 pentad cells plus 5 hexad cells magically adding to 11 of the same kind of elevenad cell. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, 11. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (5-cell regular decahedron) cells, and the 12 6-point (hexad) building blocks into 100 12-point (regular icosahedron) cells, and then magically adding one whole 5-point (5-cell regular decahedron) cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 12-point (regular icosahedron) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such (cell bundles of 6 + 5 = 11 cells), with each 11-cell bundle containing 11 of both kinds of cell. Very, very strange!{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden'': §II To Add is to Multiply|p=38|ps=; the sequence of regular polygons with harmonious proportions beginning {5}, {7}, {9}, {11} as a family of natural numbers; &quot;The numbers in this optimal family have remarkable properties [that allow] similar figures to be arranged easily or allow a figure to be dissected into a set of similar figures. This is the origin of the Greek idea of geometric progression — that ''multiplication can be accomplished instead by addition''.... For instance, it is well known that the square of the golden ratio is one more than itself (written above as 𝜙&lt;sup&gt;2&lt;/sup&gt; = 𝜙 + 1) and one less than 𝜙 is its reciprocal (1/𝜙 = 𝜙 - 1). The first relation equates multiplication with addition, while the second accomplishes division through subtraction. [These are pentagon {5} bisectional golden ratios, but] the heptagonal [{7} tri-sectional] ratios also behave in surprising ways — adding to multiply, subtracting to divide.... There are analogous constructions for [nonagon {9}] quadrisectional and [hendecagon {11}] pentasectional systems.&quot;}}

The 120-cell has three interchangeable cyclic expansion-contraction cycles, the pentad, hexad and heptad cyclic designs, rooted in the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems, respectively. Each of them constructs the whole sequence of regular 4-polytopes from a single building block. The key to all three cycles is that 5 + 6 = 11, and there is an 11-point quasi-regular 4-polytope in the sequence of convex 4-polytopes. It lies between the 8-point (16-cell) and the 16-point (8-cell), and is itself a &lt;math&gt;B_4&lt;/math&gt; polytope. Its role is to be a unified single building block common to all three cyclic expansion-contraction cycles. It is involved with the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems and expresses in one building block their relations to each other.

We have already seen how the hexad building cycle works. It is plain compounding of 4-orthoplexes at the beginning, increasing to densely intricate non-disjoint symmetry toward the end, making it difficult to see why it works once past the 24-cell. This is how the build works, in terms of sixes. The 16-cell is a 2-hexad (3 pairs of completely orthogonal great square planes). The 24-cell is 3 16-cells and a 4-hexad, 4 different ways: 16 non-disjoint hexads (great hexagon planes). The 120-point (600-cell) is 25 24-cells and a 20-hexad, 10 different ways: 200 non-disjoint hexads (great hexagon planes). The 600-point (120-cell) is 225 (9 times 25) 24-cells. ....

The pentad building cycle doesn't do plain compounding to regular 4-polytopes until the last one, the 120-cell. Fives don't even appear as pentagons until after the 24-cell. We have seen how there are 11 5-cells in the quasi-regular 11-cell, which occurs before the 8-cell, and also how everything can be compounded from irregular 5-cells (4-orthoschemes specialized for each 4-polytope). It is still not evident how a pentad cycle can work to build all the regular 4-polytopes from the regular 4-simplex. This is how is how the build works, in terms of fives. ....

This is how the build works, in terms of elevens. The 11-cell is 4 orthogonal heptads (intersecting in their skew hexagon Petrie polygon). The 24-cell is also 4 heptads (intersecting in 4 orthogonal digon axes, so each heptad contributes 6 unique points to a single fibration of 24 points). Heptads are quarter elevenads, so the 11-cell and 24-cell are both 1 elevenad. ....

See the tabular sequence of convex 4-polytopes in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Build with the blocks|§Build with the blocks]]'', below: table rows ''Inscribed, Pentad, Hexad, Heptad''.

=== 11 of them ===
There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of eleven 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum eleven of them. There is only a single Hopf fibration of the 11 great circles of an 11-cell. 

The 5-cell and the 11-cell are limit cases. The 5-cells are completely disjoint, but the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 10 other instances of itself.

== Cell rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer &amp; Segerman|2013|loc=''Puzzling the 120-cell''}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his section of the Wikipedia article on the 120-cell entitled ''[[120-cell#Visualization|§Visualization]]''.{{Sfn|Ruen &amp; Goucher et al. eds. 120-cell|2024|loc=Goucher's ''[[120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious place to look for the 11-cell's Hopf map is in Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other place it could be hiding. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The quasi-regular convex 11-cell|§The quasi-regular convex 11-cell]]'', and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen: triacontagon|2011|loc=''Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them except his illustrations. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Four orthogonal projections of the 120-cell to a central plane, selected from the table of 3 regular and 11 compound 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|100 icosahedra&lt;br&gt;120 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|5 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 icosahedra&lt;br&gt;6 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|120
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|4
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|50
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|7.2°
|align=center|36°
|align=center|6°
|align=center|7.2°
|}

....

Here we summarize our findings on the 11-cells so far, including in advance some results to be obtained in the next section two sections.

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section....

== The 15 major chords ring everything ==
The natural numbers are each a distinct flavor.

Each integer has its own distinct taste. The first six integers {0..5} are the six strongest, most familiar tastes in nature, instantly distinguished and utterly unique.

The seventh flavor {6} is the first ratio of two of the others. Thereafter follows in strange sequence all the ratios of all the flavors, each marriage of every two flavors, most of which are fractions &lt;math&gt;\{k/d\}&lt;/math&gt;, but some of which are integers &lt;math&gt;\{k\}&lt;/math&gt;.

Some of these ratios are successful flavors in their own right, as distinct and inimitable as the marriage of chocolate and oranges, but the marriage of many flavor pairs is mud. It is a rule of thumb among flavor specialists that nearly any taste can be successfully alloyed with either onions or chocolate, but the direct marriage of these two flavors is notably unsuccessful. Some flavors make famous pairings or infamous ones, and subtle new pairings are always to be discovered that work surprisingly well for some purpose, but the majority of flavor marriages are undistinguished compounds.

In polytope geometry, each chord of a polytope is both is a distinct 1-dimensional object, a chord of the sphere of a distinct length &lt;math&gt;l&lt;/math&gt;, and a distinct natural number &lt;math&gt;h&lt;/math&gt;, a unique flavor. If the polytope is regular, it is a noteworthy and successful flavor. The chord's length &lt;math&gt;l&lt;/math&gt; is a square root, related to the natural number &lt;math&gt;h = k/d&lt;/math&gt; and to a set of polytopes they both represent, by a formula discovered by Steinbach.{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden''; Figure 5. Optimal sections and proportions|p=37|ps=; the regular polygons {5}, {7}, {9} and {11} with their diagonals define respectively: {5} the golden bisection proportional to 𝜙; {7} an analogous trisection; {9} an analogous quadrasection; {11} an analogous pentasection.}} The chord length &lt;math&gt;l&lt;/math&gt; is related to the number of sides of the regular polygon formed by the chord (its numerator &lt;math&gt;k&lt;/math&gt;), and to the winding number or density of its regular skew Petrie polygon (its denominator &lt;math&gt;d&lt;/math&gt;).

The regular 4-polytopes do not contain an infinite variety of these chord flavors, but only 30 of them: only 15 unique pairs of 180° degree complementary chords, which combine to make every regular ''n''-dimensional thing found in nature.{{Efn|The 15 ''major chords'' suffice to construct all the regular ''n''-polytopes. Their 180° complements include 15 ''minor chords'' which occur only in the 120-cell, not in any smaller regular polytope.{{Efn|name=Major and minor chords}} Including the complement of the 180° {{radic|4}} digon {2} chord (the {0} chord {{radic|0}}), and the skew variant of the 90° {{radic|2}} square {4} chord (the {30/7} chord), there are 32 distinct natural numbers which are all the chords of all the regular polytopes in all dimensions.|name=32 chords}} All the regular polytopes (including the largest one on the cover of the box) can be built not only from a box containing a number of blocks of 3 shapes glued together by gravity, they can be also be built from a box containing a number of sticks of 15 lengths glued together by rubber joints.

[[File:15 major chords.png|500px|thumb|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the others occur only in the interior of 120-cell, not as edges of any of the regular 4-polytopes. There are 30 distinct 4-space chordal distances between vertices within the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to identify any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=Major and minor chords}} chords are properly labelled #1 — #15 in the order they fan out by size.{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''; Figure 3|ps=; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with a &quot;fan of chords&quot; diagram like the one here.|p=23}} They join vertex pairs which are 1 — 15 edges apart on a &lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon. A #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; chord forms &lt;small&gt;&lt;math&gt;f+1&lt;/math&gt;&lt;/small&gt; regular geodesic polygons which include a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; disjoint regular {{nowrap|&lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt;-gons}} (e.g. &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; a compound of 5 disjoint hexagons) plus a single [[W:Triacontagon#Triacontagram|regular skew {{nowrap|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram}}]] that wraps around the sphere &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times before closing its circle. Notice the maverick #11 chord, with its non-conforming origin point for its {{nowrap|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;}} polygram.]]

These 15 chords (natural numbers) include the integers 2, 3, 4, 5, 6, 10, 11, 15 and 30. The others are the fractions 5/2, 10/3, 13/7, 15/2, 15/4, and 15/7. Represented among the chord lengths (the square roots) measured in ''unit radii'' (not in {{radic|2}} radii, as metrics are often given elsewhere in this article) are the {{background color|paleturquoise|roots of the first 4 positive integers {{radic|1}}, {{radic|2}}, {{radic|3}} and {{radic|4}}}}, the {{background color|#FFCCCC|root of the very distinguished rational fraction {{radic|5/2}}}}, and the roots of various distinguished irrational fractions as Steinbach discovered: for example, 4 of them are {{background color|yellow|roots of golden numbers (a small integer ± 𝜙)}}. The 15 chords each form their regular polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, which is just &lt;small&gt;&lt;math&gt;\{k\}&lt;/math&gt;&lt;/small&gt; in 8 cases. Arranged in order #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; from shortest edge-length square root to longest, the #1 — #15 chords bridge vertex pairs which are &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; vertices apart on a [[W:Triacontagon#Petrie polygons|&lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon]]. Each chord makes a [[W:Triacontagon#Triacontagram|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; polygram]] of its edge length, a compound of the smaller &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. The 15 distinct polygons form a sequence you would not anticipate exactly unless you have Steinbach's formula:

{| style=&quot;background-color:transparent; white-space:nowrap;&quot;
|-
| {30/1} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|.073}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{30} 
| 120-cell edges
|-
| {30/2}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.191}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15}
|- style=&quot;background: yellow;&quot;|
| {30/3}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.382}} 
| &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi}&lt;/math&gt;&lt;/small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{10}''' 
| 600-cell edges
|-
| {30/4} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.573}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/2}
|- style=&quot;background: paleturquoise;&quot;|
| {30/5}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|1}}'''
| &lt;math&gt;1&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{6}''' 
| 24-cell, 8-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/6}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|1.382}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{5}'''
|- style=&quot;background: paleturquoise;&quot;|
| {30/7}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|2}}'''
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{4}''' 
| 16-cell edges
|- style=&quot;background: #FFCCCC;&quot;|
| {30/8}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.5}} 
| &lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;&lt;small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/4} 
| 5-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/9}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.618}} 
| &lt;math&gt;\phi&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{10/3}
|- style=&quot;background: paleturquoise;&quot;|
| {30/10}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|3}}''' 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{3}'''
|-
| {30/11}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.426}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{11/11}
|- style=&quot;background: yellow;&quot;|
| {30/12}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.618}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{5/2}
|-
| {30/13}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.810}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{13/7}
|-
| {30/14}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.928}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/7}
|- style=&quot;background: paleturquoise;&quot;|
| {30/15}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|4}}''' 
| &lt;math&gt;2&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{2}'''
|}

{| class=wikitable style=&quot;white-space:nowrap; float:right;&quot;
!colspan=6|Natural number &lt;small&gt;&lt;math&gt;h=\{k/d\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Fibers&lt;br&gt;&lt;small&gt;&lt;math&gt;f\{h\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Cells&lt;br&gt;&lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|4-tope&lt;br&gt;&lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Verts&lt;br&gt;&lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|[[W:Triacontagon#Triacontagram|30-gram]]&lt;br&gt;&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;
|-
!Chord
!Arc
!colspan=2|&lt;small&gt;&lt;math&gt;l/\sqrt{1}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;l/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;h&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#1
|15.5~°
|{{radic|0.073}}
|0.270
|{{radic|0.146}}
|&lt;small&gt;&lt;math&gt;30&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1\{30\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;5\{3,3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/1\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#2
|25.2~°
|{{radic|0.191}}
|0.437
|{{radic|0.382}}
|&lt;small&gt;&lt;math&gt;15&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;..\{3,3\}..&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/2\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#3
|36°
|{{radic|0.382}}
|0.618
|{{radic|0.764}}
|&lt;small&gt;&lt;math&gt;10&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;30\{3,3\}20&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/3\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#4
|44.5~°
|{{radic|0.573}}
|0.757
|{{radic|1.146}}
|&lt;small&gt;&lt;math&gt;15/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/4\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5
|60°
|{{radic|1}}
|1
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;6&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/5\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#6
|72°
|{{radic|1.382}}
|1.175
|{{radic|2.624}}
|&lt;small&gt;&lt;math&gt;5&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}} 
|&lt;small&gt;&lt;math&gt;10\{5,3\}12&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/6\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#7
|90°
|{{radic|2}}
|1.414
|{{radic|4}}
|&lt;small&gt;&lt;math&gt;4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7\{4\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;4\{4,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;9\{3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/7\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: #FFCCCC;&quot;|
|#8
|104.5~°
|{{radic|2.5}}
|1.581
|{{radic|5}}
|&lt;small&gt;&lt;math&gt;15/4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/4\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}} 
|&lt;small&gt;&lt;math&gt;15\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/8\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#9
|108°
|{{radic|2.618}}
|1.618
|{{radic|5.236}}
|&lt;small&gt;&lt;math&gt;10/3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10/3\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/9\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#10
|120°
|{{radic|3}}
|1.732
|{{radic|6}}
|&lt;small&gt;&lt;math&gt;3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;10\{3\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/10\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#11
|135.5~°
|{{radic|3.426}}
|1.851~
|{{radic|6.852}}
|&lt;small&gt;&lt;math&gt;11/11&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;11\{11/11\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;11\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#12
|144°
|{{radic|3.618}}
|1.902
|{{radic|7.235}}
|&lt;small&gt;&lt;math&gt;5/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2},3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/12\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#13
|154.8~°
|{{radic|3.810}}
|1.952
|{{radic|7.621}}
|&lt;small&gt;&lt;math&gt;13/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;13\{13/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;7\{5,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/13\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#14
|164.5~°
|{{radic|3.928}}
|1.982
|{{radic|7.857}}
|&lt;small&gt;&lt;math&gt;15/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;137\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/14\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{radic|8}}
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15\{2\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐𝜙&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;8\{3,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/15\}&lt;/math&gt;&lt;/small&gt;
|}

The polygons '''{10}''', '''{6}''', '''{5}''', '''{4}''', '''{3}''', and '''{2}''' are flat polygons that each lie on a great circle in a central plane of the 3-sphere. The polygons {30}, {15}, {15/2}, {15/4}, {10/3}, {11/11}, {5/2}, {13/7}, {15/7} and all the {30/''n''} polygons are skew polygrams spiraling through curved 3-space, each successive edge in a different central plane of the 3-sphere. Projected to the 1-sphere (a circle in the plane), they are [[W:Star polygon#Regular star polygon|regular star polygrams]], the simplest being the pentagram {5/2}. In 4-space they wind around the 3-sphere more than once ''without self-intersecting'', before they close their circuit. They lie on helical geodesic circles or ''isoclines'' of circumference greater than 2𝝅''r''.

Even the 6 integer chords which form flat polygons also form skew polygrams. Each of the 15 chords occurs as a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, a fiber bundle of Clifford parallel polygons that totals 30 vertices per bundle. Each chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; also forms a single skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram that winds through another, shorter chord's bundle of parallel polygons, intersecting them in succession multiple times each, and weaving that whole bundle of parallel fibers together as the cross-thread does on a loom. The polygram cross-thread has to wind multiple times through the whole bundle (&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times around the [[W:Triacontagon#Triacontagram|30-gram]] in which each edge connects every &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt;th vertex) to complete an enumeration of all 30 vertices in the bundle. For example, the #5 chord forms hexagon &lt;small&gt;&lt;math&gt;\{6\}&lt;/math&gt;&lt;/small&gt; flat planes in parallel bundles of 5, written &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt;, and also a skew &lt;small&gt;&lt;math&gt;\{15/2\}&lt;/math&gt;&lt;/small&gt;-gram of #5 edges which winds 2 times through the #3 chord's bundle of 3 decagon &lt;small&gt;&lt;math&gt;\{10\}&lt;/math&gt;&lt;/small&gt; flat planes. Every chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; forms both its &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram and its fiber bundle of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, and the edges of one chord's parallel polygons are also the edges of its skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram weaving through some shorter chord's bundle of parallel polygons. Each numbered chord is a natural number, a ratio of two integers and their polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, either flat or skew. Each chord number defines a &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; skew polygon that is a compound bundle of the &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. All these flat or skew polygons are regular polygons made of their numerator instances of a single chord #, whose length is that square root, Steinbach's function of the natural number.

The relationship between each bundle of flat Clifford parallel great circle polygons of a distinct length, and its corresponding bundle of skew polygrams of a longer length, is precisely that of a discrete simple rotation and its corresponding discrete isoclinic rotation. For example, the flat great circle hexagons of edge length {{radic|1}} lie in the invariant rotation planes of the 24-cell's characteristic 60° isoclinic rotation, in which the vertices move on skew hexagram rotation paths of edge length {{radic|3}}. The 24-cell contains 4 bundles of 4 Clifford parallel (disjoint) hexagon central planes each. The 24 vertices of the 24-cell in each bundle are linked by rotation in the 4 hexagon invariant planes in two different ways: simply and isoclinically. In a simple rotation the vertices of a single invariant hexagon exchange places on that great circle hexagon of edge {{radic|1}}. In an isoclinic rotation in which any one of these 4 disjoint hexagon central planes is invariant, all 4 of them are necessarily invariant planes in the rotation, because they rotate in parallel. In that isoclinic rotation the vertices of all 4 hexagon invariant planes (all 24 vertices of the 24-cell) exchange places concurrently, moving along 4 disjoint helical hexagrams of edge {{radic|3}} in parallel. It is a characteristic only of this distinct rotation in hexagon invariant planes that the helical vertex path or ''isocline'' is a hexagram of period 6, edge length {{radic|3}}, and circumference 720°. Rotations in other kinds of polygon invariant planes have different characteristic isoclines. The period of their isoclinic rotation must be ''the same or greater'' than the period of their simple rotation; in most cases (unlike this one) it is greater.

The 15 chords form not only distinct fiber bundles of polygons, but also distinct bundles of polyhedra: cell rings &lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt; and compound vertex figures &lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;. These combine to form their characteristic regular 4-polytope &lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Coxeter|1973|p=269|loc=Compounds|ps=; &quot;It is remarkable that the vertices of {5, 3, 3} include the vertices of all the other fifteen regular polytopes in four dimensions.&quot;}} All these elements of a regular 4-polytope constructed from one chord are themselves regular polytopes, and each of the 16 regular 4-polytopes is a function of one characteristic chord. If the regular 4-polytope is convex, its characteristic chord is its unit-radius edge. For a non-convex regular 4-polytope, the characteristic chord is its unit-radius ''mid-edge diameter'', the distance from the midpoint of each edge to the midpoint of the completely orthogonal edge. Each of the characteristic chords (each distinct natural number) has its expression as one of the 16 regular 4-polytopes and all the products of the distinct regular 1-polytopes, 2-polytopes and 3-polytopes that construct it, and the whole distinct construction from the natural number to the 4-polytope is an [[W:SO(4)|SO(4) symmetry group]] expression of that 4-polytope's characteristic discrete [[W:SO(4)#Isoclinic rotations|isoclinic rotation]], in the invariant central planes of its characteristic chord, by the characteristic chord arc.

...demonstrate this theory by proper enumeration of the chord lengths in the table...

...color-code the △ ☐ 𝜙 ✩ symbols ... chord colors in diagram should match them ... some of the diagram colors are wrong and must be changed, but otherwise diagram is pretty good ... i think in a colored diagram all chords of the same length should have the same size line dash, the polygons are not obvious enough and the 15 fanning-out chords are readily identified by their labels...

... add back the counts of chords in the box (very small font number under each table entry), and present these two images as the front and back a box of building sticks and rubber joints...What's in the other box...

...say the rational-root major chords {{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|5/2}} are the ''n''-edges: their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge...

...abbreviated text of the [[120-cell#Relationships among interior polytopes|''§Relationships among interior polytopes'']] section from the 120-cell article...to explain the color-coding

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius=1}}

See also the elements, properties and metrics of the sequence of regular 4-polytopes summarized in tabular form in the section ''[[#Build with the blocks|§Build with the blocks]]'', below.

== Alicia Boole Stott's original formulation of dimensional analogy ==
&lt;blockquote&gt;&quot;...a method by which bodies having a certain kind of semi-regularity may be derived from regular bodies in an Euclidean space of any number of dimensions...&quot;{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''; presents cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues|pp=12-45}}&lt;/blockquote&gt;

Alica Boole Stott's 1910 paper reads like an exceptionally well-written patent application. It is a comprehensive stepwise description of her original invention, a general method of dimensional analogy between regular and semiregular polytopes of the same edge length.

[[File:The Roots of Three.png|thumb|An illustration of Boole Stott's expansion of unit-edge triangle to hexagon captures the geometric basis in 3 of the &lt;math&gt;A_n&lt;/math&gt;root system.]]

The engineering-like articulation of her new principles is entirely discrete and group-theoretic. Boole Stott repeatedly refers to sets of polytope elements as ''groups''. All her transformations are discrete instances of one or another special case of her &quot;two inverse operations which may be called ''expansion'' and ''contraction''.&quot; Hers is an exploration of ''n''-dimensional space at its most basic: space defined by the discrete objects which fill it, themselves defined directly as groups of elements with a few operations on them, which transform them into each other. This is geometry which seems to precede Euclid's postulates, and does not depend on them. Instead it corresponds directly to the deeper group theory mathematics which underlies geometry, even though Boole Stott, like the ancients, did not have that mathematics.

Boole Stott enumerated her expansion and contraction operations as transformations between regular and semi-regular polytopes of a single edge length. In her 1910 paper she did not consider any polytopes of multiple edge lengths, or expansions of the subject polytope by a distance other than that exactly required to produce new edges of the original edge length, so of course she did not enumerate Moxness's non-uniform rhombicosidodecahedron (Hull #8). This may be why section 8&lt;sub&gt;3&lt;/sub&gt; is missing from her earlier drawings of the sections of {5,3,3}; she may not have had a construction for it.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The Boole Stott expansion (or contraction) operation is as precisely defined a transformation as Coxeter's reflection, rotation and translation operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions). Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;
where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;
where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;
For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} An instance of the expansion (or contraction) operation is an application of the general operation to a particular subject polytope, operating on all of its elements symmetrically at once. The general operation is restricted, such that one may only expand (or contract) the subject polytope by pushing all its edges, faces or cells directly away from (or toward) its center, without changing the size of the moved elements. This splits each vertex into several (or coalesces several vertices into one), and has corresponding effects on the other elements, producing new edges ''of the same size'' between pairs of separated vertices (or reducing some edges to single vertices). The distance all the elements are moved away from (or toward) the center must be such that the vertices separate producing new edges of only the original edge length (or come together reducing edges to single vertices). It is permitted to combine these three operations, e.g. by pushing the edges ''and'' the faces, or by pushing all three element sets, provided the combination results in a semi-regular polytope of only one edge length. The three operations are all commutative with one another, so combined operations can be considered either as a sequence of expansions (or contractions) linking several related polytopes, or as their concurrent movements, a single transformation of the subject polytope. Since there are 8 combinations of one to three operations there are 8 polytope variants (including the original regular polytope), and all are adjacent to one another by a single transformation. The variants of the original regular polytope are named for which element set(s) are moved. By enumerating all 8 variants of expansion (and contraction) for each of the six regular convex 4-polytopes, Boole Stott produced a table of regular and semi-regular 4-polytopes divided into 12 sets of 8: six expansion sets and six contraction sets.{{Sfn|Boole Stott|1910|loc=''Table of Polytopes in S&lt;sub&gt;4&lt;/sub&gt;''; note that Boole Stott's usage of ''S&lt;sub&gt;4&lt;/sub&gt;'' denotes a Euclidean 4-space ''E&lt;sub&gt;4&lt;/sub&gt;'', or a 3-sphere space embedded in it, ''not'' a 4-sphere space embedded in a Euclidean 5-space|p=30}} Each set includes just one of the six regular convex 4-polytopes, with its distinct unit-radius edge length, and the 7 possible semi-regular expansions (or contractions) of it, which all preserve that characteristic edge length. The polytopes in each set of 8 all have the same total number of elements as their original regular polytope, but each polytope in the set has a distinct combination of cell, face, and vertex figures. The 12 sets are not completely disjoint, and even within one set the 8 operations do not necessarily produce 8 unique polytopes. For example, the regular 5-cell (the 4-simplex polytope) has 5 cells, 10 faces, 10 edges, and 5 vertices (30 elements). Expanding the 5-cell produces 7 related semi-regular 4-polytopes of 30 elements each (two of which happen to be the same polytope). In each expansion, the 5 cells have expanded into semi-regular or regular polyhedra, the 10 faces and 10 edges ''may'' have expanded into prisms, and the 5 original vertices have expanded into regular or semi-regular cells.

== The regular 11-cell 4-polytope ==

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosadodecahedron). Each non-convex Jessen's icosahedron is a set of 6 reflex edges that are {{radic|5}} 5-cell edges, each set symmetrically arranged in 3 parallel pairs, not as far apart ({{radic|3}}) as they are long, in 3 orthogonal central planes. The 12-point (Jessen's sets of 6 reflex 5-cell edges) are inscribed in the 60-point with their endpoints as a vertex of a pentagon face. The 12 pentagon faces are joined to 5 other pentagon faces by the 5-cell edges; not to their nearest neighbor pentagons but to their next-to-nearest neighbor pentagons; and since there are 5 vertices to each pentagon face, there are 10 12-point (Jessen's sets of 6 reflex 5-cell edges). Each pentagon face is joined to another pentagon face by 5 reflex edges of five distinct Jessen's. The 5 reflex edges are Clifford parallel, the 5 edges of a pentagonal prism column that has a twist to it. The contraction of those 5 edges into 1, and those 10 vertices (two pentagons) into 2, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing all their edges as a single Jessen's. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (octahedron) by conflating its parallel pairs of edges, leaving the 11-cell's abstract 6-point (hexad cell). The hexad's situation is that it occurs in bundles of 5, disjoint but close together in the sense of adjacent, like the great circle rings in a Hopf fiber bundle. 

To summarize, the 12-point (2-hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell). We shall now be able to see how Fuller's perfect hexads combine with Todd's perfect pentads to form the perfect heptads of the 11-cell and 137-cell.  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=alias the hemi-cuboctahedron}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces. (The ''heptad'' arrives at the party, anonymously; they are our distinguished anti-celebrity, who most of the other guests have never even heard of.) The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. This 12-point (truncated tetrahedron) is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 100 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp: Truncated tetrahedron|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp: Truncated tetrahedron|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}} Of course Moxness has been there, seen that, already with his graphics software;{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} see his parameterized illustration of it exactly as it occurs in the 11-cell, below in his bestiary of A3, B3, and H3 Archimedian and Catalan polyhedra in the 120-cell.]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:Archimedean_and_Catalan_solid_hulls_with_their_Weyl_orbit_definitions.svg|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of the 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. In this catalog of Archimedean and Catalan polyhedra sighted in the 120-cell by Moxness's software,{{Sfn|Moxness: Archimedean and Catalan hulls|2023|loc=Hull #1 Archimedean Name A3 110 Truncated Tetrahedron A (upper left)}} it is the first entry (upper left), Moxness's Hull # = 1 with 12 vertices of 3D norm {{radic|11}}/2. On a {{radic|2}}-radius 3-sphere, 5 of these with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell. 20 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges are cells in the [[#The rings of the 11-cells|cell rings of the quasi-regular 11-cell]], which also has 12 regular 5-cells as cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

...

Both kinds of cells in the 11-cell have expressions in two different dimensions (on the 2-sphere and 3-sphere), because they are all centered on the center of the 3-sphere, unlike the cells of most 4-polytopes, which are centered on various off-center points within the 4-polytope's interior. 

One kind of 11-cell cell appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (semi-regular truncated cuboctahedron), a.k.a. the truncated tetrahedron. The other kind of cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now, except in the 120-cell. But this is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron, and a regular tetrahedron behind each face. The 11-cell is thus a honeycomb of mixed regular tetrahedra and truncated cuboctahedra (a.k.a. truncated tetrahedra) that meet at their shared 5-cell faces. 

Both the 5-cells (decahedra) and the truncated cuboctahedra (truncated tetrahedra) are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks in their 4-polytopes (off-center).{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} Therefore all the 11-cell's cells are concentric, which indeed makes a different kind of honeycomb, an inside-out cellular honeycomb with all the cells piled on top of each other overlapping in 4-space. They do not overlap in 3-sphere space however; their 3-volumes (but not their 4-contents) are disjoint. They make a proper 4D honeycomb like the cells of any 4-polytope, with volumetrically disjoint 3D cells that meet each other in the proper way at their shared 2D faces.

An 11-cell contains 6 completely disjoint 5-point (1-pentad 5-cell decahedron cells), and 5 volumetrically disjoint 12-point (2-hexad truncated cuboctahedron cells), 11 volumetrically disjoint cells altogether. Of course it is supposed to have 11 identical 11-point (elevenad hemi-icosahedral cells), and that is also true in its abstract sense, which is not the same sense as disjoint cellular containment: each pentad or hexad cell is inscribed in one of 11 larger 60-point (Moxness's rhombicosadodecahedron cells). The abstract pentads and hexads are only hemi-polyhedra, so the 11-cell's pentad cells are 12 (out of 120) disjoint instances of a pentad 5-point, and its hexad cells are 10 (out of 100) disjoint instances of a hexad 6-point, 1/10th of the 120-cell's cellular volume in both cases. The 11-cell's hemi-icosahedral cells are 11 (out of 120) non-disjoint instances of an elevenad 60-point. Each 11-cell shares each of its hexad cells with 9 other 11-cells, and each of its 11 vertices (out of 600) occurs in 9 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.|name=Tetrahedral vertex figures}}

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

...insert image thumb of 11-cell...

...Construct the 11-cell from the heptad....

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...Four concentric 4-polytopes may be mutually orthogonal. In any set of 4 mutually orthogonal regular 5-cells, one pair of faces in each orthogonal pair are completely orthogonal to each other. Thus three edges in one 5-cell in an orthogonal pair are opposite (completely orthogonal) to a face of the orthogonal 5-cell. .... A 5-cell has 5 vertices in its abstract hemi-icosahedral cells, but only 4 vertices in one concrete icosahedral cell; the 5th lies in an icosahedral cell of another 11-cell (not in another icosahedral cell of the same 11-cell).

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

....

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 11-cell ===

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon.... (wrong, 11-cell's Hopf map is the truncated icosahedron)

....

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk: Truncated icosahedron|2018}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

...

Since the rhombicosadodecahedron has 120 5-cell edges, the 120-cell has 120 completely disjoint 5-cells, and the rhombicosadodecahedron shares each vertex with 11 other rhombicosadodecahedra, how a rhombicosadodecahedron is bound to its neighbor 5-cells is constrained. A further constraint is that the abstract hemi-icosahedron has only 15 5-cell edges, so the 120 actual 5-cell edges of the rhombicosadodecahedron are conflated 8-to-1 in the hemi-icosahedron. To be conflated by contraction into a single edge, those 8 edges must lie parallel to each other in the 120-cell; either actually parallel in the same plane, or Clifford parallel in separate planes, as two edges in completely orthogonal planes may be parallel and orthogonal at the same time. Each 5-cell inscribed in the 120-cell contains 5 such tetrahedra occur in parallel pairs in 4 orthogonal planes (none of them completely orthogonal planes). 

.... 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
Each conflated vertex is the 10 vertices of two orthogonal regular 5-cells...
The 60-point contains 6 pairs of these orthogonal regular 5-cell cells...
...its 20 hexad cells are ..., a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere.

== The regular 137-cell 4-polytope ==

(description originally written for the 11-cell, which applies only to a fibration of 11 11-cells)

It is quasi-regular with two kinds of cells, the 12-point (regular icosahedron) and the 5-point (regular 5-cell), which have just one kind of regular face between them. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of regular facets (it is semi-regular with regular facets), it also means that its facets have expressions of two different dimensionalities because they are concentric in the 3-sphere. The 11-cell has two kinds of regular cells. One appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (regular icosahedron), but since it is embedded at the center of the 3-sphere (unlike the cells of most 4-polytopes) it also acts as an inscribed 4-polytope.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} The other cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now except in the 120-cell. This is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron. The 11-cell is a honeycomb of mixed icosahedron and half-icosahedron cells. Like the icosahedra, the decahedra are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks, off-center in their 4-polytopes. All the 11-cell's cells are concentric, which indeed makes an unusual honeycomb of cells, an inside-out honeycomb. It is a proper 4D honeycomb however, with volumetrically disjoint 3D cells that meet each other in the proper way at shared 2D faces. The embedded 12-point (regular icosahedron) cell is an icosahedral pyramid (as a 4-polytope), which is filled by 20 4-point (irregular tetrahedra) that meet at its center; the center point of the 11-cell acts as a 12th vertex, just as the center point of the 24-cell acts as a 25th vertex in some honeycombs. The 5-point (regular 5-cell) cell contains five 4-point (regular tetrahedra) which meet the 12-point (regular icosahedra)'s irregular tetrahedral cells face-to-face, so the 11-cell is quasi-regular in the usual sense of having two varieties of regular 3-polytope cells, but additionally both varieties of of its cells are bi-dimensional: another clue that the 11-cell has something to do with dimensional analogy

....

The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

....

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur to make an 11-cell. 11 of them occur in a 137-cell, all 120 of them occur in the 120-cell, and they occur nowhere else.

How can this regular 4-polytope possibly exist? By Schläfli's criterion {3, 3, ..} exists, but not as a finite polytope.{{Sfn|Coxeter|1973|loc=§7.7 Schläfli's criterion}} Coxeter found {3, 5, 3} as a finite hyperbolic honeycomb, but not in Euclidean 4-space.{{Sfn|Coxeter|1968|title=The Beauty of Geometry: Twelve Geometric Essays|loc=''Regular Honeycombs in Hyperbolic Space''|ps=; &quot;The honeycombs considered by the above authors [Schlegel, Schläfli, Klein, Sommerville] have finite cells and finite vertex figures. ... A further extension allows a cell or vertex figure to be a ''star''-polytope, so that the honeycomb covers the space several times. .... We shall find that there are four regular star-honeycombs in hyperbolic 4-space, as well as two infinite families of them in the hyperbolic plane.&quot;}} We find that {3, 3, ..} is a central expression of the &lt;math&gt;H_4&lt;/math&gt; root system, being the regular convex hull in 4 dimensions of 137 points, 96 pentads and 600 hexads. The minimal expression of &lt;math&gt;H_4&lt;/math&gt; is the regular convex hull {3, 3, 5} of 120 points, 600 tetrads and 75 hexads, and its maximal expression is the regular convex hull {5, 3, 3} of 600 points, 120 pentads and 675 hexads. {3, 3, ..} is a third regular &lt;math&gt;H_4&lt;/math&gt; polytope which fits between them. The properties and metrics of this regular sequence are presented in tabular form below in ''[[#Build with the blocks|§Build with the blocks]]''.

{|class=&quot;wikitable&quot;
!colspan=3|Regular &lt;math&gt;H_4&lt;/math&gt; polytopes
|-
|[[File:600-cell.gif|200px]]
|[[File:5-cell.gif|200px]]
|[[File:120-cell.gif|200px]]
|-
!120-point 600-cell
!137-point 137-cell
!600-point 120-cell
|}

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 137-cell ===
....

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=Apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|2022|loc=''[[Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]''}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie: On Fuller's use of language|2024|loc=''[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry'']]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center, 1948-1949|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with their mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the gender neutral shad can open their six beaks all the way, until they become flat squares and they becomes a cuboctahedron, or they can shut them all tight like a turtle retracting into their octahedron shell. The six mouths always move in unison. This is [[Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do their ''really'' odd trick -- where they flip their 6 jaws 90 degrees in their 6 faces and shut their 6 beaks on the opposite axis of their squares than the one they opened them on -- or whether they will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than their normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make their edge length exactly the same size as their radius, when they open their mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing their mouths in spherical synchrony, their 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron (the hexad), then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=''[[W:SO(4)|SO(4)]]''}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=Isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its orthogonal hexad's planes (wz, wx, wy).{{Efn|name=Six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of (expanded octahederon 16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh &amp; Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== Legendre's 12-point hexad binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a 12-point double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's perfectly realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, this 2-hexad expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre hexad) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction.{{Efn|name=Concentric polyhedra}} In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== ''n''-simplexes ===
....

The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== ''n''-orthoplexes ===
....
[[File:Great 5-cell √5 digons rectangle.png|thumb|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} ''regular'' great hexagons. These are the same 200 dodecagon central planes illustrated above, which also contain 6 120-cell edges and two opposing ''irregular'' great hexagons (truncated {{radic|6}} triangles). The 1200 {{radic|5}} 5-cell face triangles occupy 1200 other, non-central planes.]]

...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's Hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[16--cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[16--cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=Concentric polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[16--cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== ''n''-taliesins ===
The 4-simplex pentad building block combines with the 4-orthoplex hexad building block to make the 4-taliesin heptad building block. The heptad is the essential building block of the ''taliesin'' family of polytopes. The 11-cell and 57-cell hemi-polytopes and the 137-cell and 120-cell regular 4-polytopes are nuclear members of this trans-dimensional family. The 24-cell and 600-cell are extended members of it, because they also contain instances of the heptad building block.

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its spherical symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* [[W:Taliesin Myrddin Namkai-Meche|Taliesin Myrddin Namkai-Meche]], a young Reed College scholar who was stabbed to death by a crazy man in 2017 on a Portland, Oregon light rail train, when he rose to the nonviolent defense of two black teenagers who were being violently assaulted in racist and anti-Muslim language by a right-wing extremist.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in orthogonal [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].
* '''Taliesan polytope''', an (''n+1)''-polytope whose ''n''-polytope dimensional analogue inscribed concentrically in the ''n''-sphere forms a convex honeycomb of itself which is the (''n+1)''-polytope.{{Efn|name=Embedding point}}
* The taliesan polytope's honeycomb, a Hopf fibration of its great circles.
* The ''n''-dimensional family of dimensionally analogous polytopes related to a taliesan polytope by expansion and contraction operations.
* The characteristic [[w:Orthoscheme|orthoscheme]] of a teliesan polytope, an irregular simplex that is the complete gene for the entire expression of trans-dimensional symmetry relationships among the polytopes of the ''n''-dimensional family, as well as the complete gene for the teliesan polytope's rotational symmetry.{{Efn|name=Characteristic orthoscheme}}

=== ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== ''n''-equilaterals ===
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed at four different [[#n-leviathon|chord]] lengths from each other. In the {{radic|2}}-radius polytopes we have using as examples in this paper they are {{radic|2}}, {{radic|4}}, {{radic|6}} and {{radic|8}}, but in a ''unit-radius'' polytope they are: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}.

Each vertex is joined to 8 others by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}. The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== ''n''-pentagonals ===

[[File:600-cell.gif|thumb|The 120-point (600-cell), &lt;math&gt;H^4&lt;/math&gt;.]]
...

[[File:600-cell vertex geometry.png|thumb|Vertex geometry of the 600-cell, showing the 5 regular great circle polygons and the 8 vertex-to-vertex chord lengths with angles of arc. The golden ratio{{Efn|name=golden chords|group=}} governs the fractional roots of every other chord,{{Efn|name=fractional root chords}} and the radial golden triangles which meet at the center.|alt=|400x400px]]
The 120 vertices of the 600-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏&lt;sup&gt;−1&lt;/sup&gt;) beginning with a vertex; see column ''a''}} at eight different [[#n-leviathon|chord]] lengths from each other. These edges and chords of the 600-cell are simply the edges and chords of its five great circle polygons. In ascending order of length in a ''unit-radius'' polytope (not in a {{radic|2}} radius polytopes like our usual examples), they are {{radic|0.𝚫}}, {{radic|1}}, {{radic|1.𝚫}}, {{radic|2}}, {{radic|2.𝚽}}, {{radic|3}}, {{radic|3.𝚽}}, and {{radic|4}}.{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=Decimal fractions of square roots|group=}}

Notice that the four [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#''n-''equilaterals|hypercubic chords]] of the 24-cell ({{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|4}}) alternate with the four new chords of the 600-cell's additional great circles, the decagons and pentagons. The new ''golden chord'' lengths are necessarily square roots of fractions, but very special fractions related to the golden ratio{{Efn|1=The fractional-root ''golden chords'' are irrational fractions that are functions of {{radic|5}}. They exemplify that the [[W:golden ratio|golden ratio]] &lt;big&gt;𝜙&lt;/big&gt; {{=}} {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618 is a circle ratio related to &lt;big&gt;𝜋&lt;/big&gt;:&lt;br&gt;
: {{sfrac|𝜋|5}} = arccos (𝜙/2)
is one decagon edge, the 𝚽 = {{radic|0.𝚫}} = {{radic|0.382~}} ≈ 0.618 chord.
Reciprocally, in this function discovered by Robert Everest expressing &lt;big&gt;𝜙&lt;/big&gt; as a function of &lt;big&gt;𝜋&lt;/big&gt; and the numbers 1, 2, 3 and 5 of the Fibonacci series:&lt;br&gt;
: &lt;big&gt;𝜙&lt;/big&gt; = 1 – 2 cos ({{sfrac|3𝜋|5}})
{{sfrac|3𝜋|5}} is the arc length of the &lt;big&gt;𝜙&lt;/big&gt; = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618 chord.|name=golden chords|group=}} including the two [[W:golden section|golden section]]s of {{radic|5}}, as shown in the diagram.{{Efn|The 600-cell edges are decagon edges of length {{radic|0.𝚫}}, which is 𝚽, the ''smaller'' golden section of {{radic|5}}; the edges are in the inverse [[W:golden ratio|golden ratio]] {{sfrac|1|φ}} to the {{radic|1}} hexagon chords (the 24-cell edges).
The other fractional-root chords exhibit golden relationships as well. The chord of length {{radic|1.𝚫}} is a pentagon edge.
The next fractional-root chord is a decagon diagonal of length {{radic|2.𝚽}} which is &lt;big&gt;𝜙&lt;/big&gt;, the ''larger'' golden section of {{radic|5}}; it is in the golden ratio{{Efn|name=golden chords|group=}} to the {{radic|1}} chord (and the radius).{{Efn|Notice in the diagram how the &lt;big&gt;𝜙&lt;/big&gt; chord (the ''larger'' golden section) sums with the adjacent 𝚽 edge (the ''smaller'' golden section) to {{radic|5}}, as if together they were a {{radic|5}} chord bent to fit inside the {{radic|4}} diameter.}}
The last fractional-root chord is the pentagon diagonal of length {{radic|3.𝚽}}.
The [[W:Pentagon#Side length is given|diagonal of a regular pentagon]] is always in the golden ratio to its edge, and indeed &lt;big&gt;𝜙&lt;/big&gt;{{radic|1.𝚫}} is {{radic|3.𝚽}}.|name=fractional root chords|group=}}

....


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== ''n''-leviathon ===
...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of its &lt;math&gt;10^2&lt;/math&gt; disjoint instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction, before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]].{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}} .... It is only because she did not have the 11-cell to start with that she did not discover the regular 137-cell {3, 3, ..} by her method.

Between Alicia Boole Stott and her identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and her identification of symmetry groups with conservation laws (in her 1st theorem), we owe the deepest mathematics of geometry and physics originally to two pioneering women, colleagues of their contemporaries the greatest men of the academy, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just an [[W:abstract polytope|abstract 4-polytope]], not just a singleton convex 4-polytope, and not just a honeycomb.{{Sfn|Coxeter|1970|loc=''Twisted Honeycombs''}} The 11-point (11-cell) has a concrete ...regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}&lt;nowiki/&gt;} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, 3, 3} as the 137-point (137-cell).

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 11-cell, 137-cell and 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=''Clifford Spinors and Root System Induction: H4 and the Grand Antiprism''}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=Apology}}

== Build with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Duveneck|1978|loc=Carl Jung, quoted in ''Life on Two Levels''|p=ii|ps=.{{Sfn|Jung|1961|ps=: &quot;The best of truths is of no use unless it has become one's most personal inner experience. It is the duty of everyone who takes a solitary path to share with society what he finds on his journey of discovery.&quot;}}}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1954|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius={{radic|2}}}}

== Acknowledgements ==
...

== Notes ==
{{Regular convex 4-polytopes Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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{{Refend}}</text>
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      <text bytes="208543" sha1="np20u1x3vxiq8asirykj3hssd3xvg3s" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|June 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a regular 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the &lt;s&gt;120-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 5, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 11 [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] cells. Eleven 11-cells (plural) are the 137-cell, the &lt;s&gt;600-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 3, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 137 [[W:Rhombic triacontahedron|triacontahedron]] cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=''Regularity of Graphs, Complexes and Designs''}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=''A Symmetrical Arrangement of Eleven Hemi-Icosahedra''}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]]
The 4-dimensional regular polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:Simplex|simplex]] (called the [[5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:Orthoplex|orthoplex]] (called the [[16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular convex 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the [[w:Tesseract|16-point (8-cell) tesseract]], the 4-cubic, which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found within any of the other regular convex 4-polytopes, except in the largest and most complex one, the [[120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). However, we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the 16-cell is their very starting point, and the most frequently used tool in the box.

== 5-cells and hemi-icosahedra in the 11-cell ==
[[File:Symmetrical_5-set_Venn_diagram.svg|thumb|The 5-point (10-face) regular 5-cell (the regular 4-simplex). Grünbaum's rotationally symmetrical 5-set Venn diagram{{Sfn|Grünbaum|1975|loc=''Rotationally symmetrical 5-set Venn diagram'', Fig 1 (e)|ps=; partitions the individual elements of the 5-cell.}} is an illustration of the 5-cell labeling each of its &lt;math&gt;2^5&lt;/math&gt; elements.{{Sfn|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled; image includes the Python code to render it, optimising for maximum area of the smallest regions.}}]]
[[File:Hemi-icosahedron.png|thumb|The 6-point (10 face) [[W:hemi-icosahedron|hemi-icosahedron]], an abstraction of the regular icosahedron, has half as many faces, edges and vertices. Each element of the abstract polyhedron represents two real elements found in different places in the 11-cell's realization.{{Sfn|Ruen: hemi-icosahedron|2007}}]]
The most apparent relationship between the pentad 5-cell and the hexad hemi-icosahedron is that they both have 10 triangular faces. When we see a facet congruence between a 4-polytope and a 3-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 6-point 5-simplex. It is also the same size as the 6-point 3-orthoplex; thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no chord congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals, pentad and hexad, who are famously stand-offish with each other. Interesting!

The cell of the 11-cell is an abstract hexad hemi-icosahedron with an apparently pentad interior, most handsomely illustrated by Séquin.{{Sfn|Séquin &amp; Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing rings of polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The 11-cell's 11 hemi-icosahedral cells have 10 triangle faces each. The only way 11-cell and 5-cell faces can be congruent is if each hemi-icosahedron's 10 faces belong to 10 distinct 5-cells. At least 3 of each 5-cell's 5 vertices belong to the hemi-icosahedron, and its other 2 vertices define an 11-cell edge lying opposite the face. Coxeter determined that the 11-cell does indeed have an edge opposite each face, and that it does not belong to the same hemi-icosahedral cell as its opposing face. He found that each hemi-icosahedron's 10 face-opposite edges are the 10 edges of the single 5-cell which does not share any vertices, edges or faces with the hemi-icosahedron. There is only one such 5-cell completely disjoint from the hemi-icosahedron: six hemi-icosahedron vertices plus five 5-cell vertices make the eleven vertex 11-cell.{{Sfn|Coxeter|1984|p=110|loc=§6. The Petrie polygon [of the 11-cell]|ps=; &quot;We may reasonably call this edge and face ''opposites''. It is easy to find the face opposite to a given edge by looking at the faces to which a given edge belongs. ... Conversely, given a face, we can find the opposite edge by seeing which vertices belong to neither of the hemi-icosahedra which share that face. The ten edges opposite to the ten faces of one hemi-icosahedron are the edges of the complementary &lt;math&gt;a_4&lt;/math&gt; [4-simplex], that is, the joins of all pairs of the five vertices [of the 11-cell] not belonging to the given hemi-icosahedron.&quot;}}

There are 11 disjoint 5-cells in each 11-cell, which is comprised of 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. The real 11-cell is harder to see than the abstract 11-cell that represents it, because the real hemi-icosahedron that the abstract hemi-icosahedron represents is hard to see, but seeing these objects will be easier once we have identified the real hemi-icosahedron and exactly where the 11-cell's real elements reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each hemi-icosahedron's 10 faces come from 10 disjoint 5-cells, and there is just one 5-cell with which it does not share a face.{{Efn|As Coxeter observes (in the previous citation), that unrepresented 5-point (5-cell) is the other 5 vertices of the 11-point (11-cell) that are not vertices of this 6-point (hemi-icosahedron), the hemi-icosahedron's disjoint complement.}} Each 5-cell has 10 faces from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share a face.

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in two different planes which are completely orthogonal to each other. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must be sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness: 8 concentric hulls|2022|loc=Hull #8 (lower right)|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell of the 11-cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Compounds in the 120-cell|§Compounds in the 120-cell]]'' that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software which uses Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s to render the polyhedra which are found in the interior of ''n''-dimensional polytopes.{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the tetrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is a real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (Hull #8) is a concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this {12} irregular great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing {6} irregular great hexagons (truncated triangles) of alternating 120-cell edges and 5-cell edges. The great dodecagon is a great circle of the hemi-icosahedral cell, and also a great circle of the 120-cell, because unlike the cells of most 4-polytopes, the 11-cell's cells are embedded in 4-space at the exact center of the 4-polytope, making them inscribed 4-polytopes themselves.{{Efn|name=Embedding point}} There are 25 great dodecagons and 120 5-cell edges in Moxness's 60-point (Hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (Hull #8) polyhedron altogether.

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point (Hull #8) is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] similar to the one from the catalog shown here,{{Sfn|Piesk: Rhombicosidodecahedron|2018}} but a slightly shallower truncation with smaller red pentagons (120-cell faces) and narrower rhombs. Rhombicosidodecahedra are made by truncating the [[W:Rhombic triacontahedron|rhombic triacontahedron]], which is the 30-sided polyhedron with only one kind of face, the dual of the 30-point icosidodecahedron. The 120-cell contains 120 of these 11-cell cells.]]
Moxness's 60-point (Hull #8) is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (Hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|The 5-point (regular 5-cell) can be another abstraction of Moxness's 60-point (Hull #8), 12-vertices-into-1, but it is not strictly the result of a contraction operation as described by Boole Stott, since she did not apply her expansion and contraction operations to uniform polytopes with more than one edge length.}} In fact Moxness was not the first to find a rhombicosidodecahedron in the 120-cell; Alicia Boole Stott was. It is the 6th section of {5,3,3} beginning with a cell, which Boole Stott identified as the ''e&lt;sub&gt;2&lt;/sub&gt; expansion'' of the icosahedron (or equivalently of its dual polyhedron the dodecahedron).{{Sfn|Boole Stott|1910|loc=§Examples of the e&lt;sub&gt;2&lt;/sub&gt; expansion|p=7}} But that rhombicosidodecahedron identified by Boole Stott is not Moxness's Hull #8: rather, it is the semi-regular Archimedean solid, with a single edge length and square faces. Moxness's Hull #8 with its two distinct edge lengths and rectangular faces is Coxeter's 8th section of {5,3,3} beginning with a cell, which is missing from the sections illustrated by Boole Stott.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than tetrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. The 120-cell contains 120 of Moxness's 60-point (11-cell cells). They are non-disjoint, with 12 60-points sharing each vertex.

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (Hull #8) is the real hemi-icosahedron cell, with 120 5-cell edges in it as chords lurking under its surface. The rhombicosadodecahedron has enough of those 5-cell edges in its interior to make 20 disjoint tetrahedral cells which belong to 20 distinct 5-cells.

The 600-point (120-cell) may be constructed as a semi-regular honeycomb of two kinds of cells, the 120 non-disjoint 60-point (Moxness's rhombicosadodecahedra) and their neighbors the 120 disjoint 5-point (5-cells), cell-bonded together by shared tetrahedral cells of edge-length {{radic|5}}. Each rhombicosadodecahedron is bonded to 20 islanded 5-cells, sharing with each a 5-cell tetrahedron (4 of its interior 5-cell faces, and 6 of its interior 5-cell edges). The rhombicosadodecahedron is also bonded directly to 10 other rhombicosadodecahedra (since each 11-cell has 11 rhombicosadodecahedron cells), again not by sharing one of its exterior faces, but this time by sharing an interior face, a {{radic|6}} triangle which lies in a central plane.{{Efn|The {{radic|6}} chord is another hidden chord lurking beneath the 60-point rhombicosadodecahedron's surface, like the {{radic|5}} 5-cell edge. Both these hidden chords lie on the same great circle with the 120-cell edges (the pentagon face edges). Three {{radic|6}} chords make a triangle which lies in a central plane. This {{radic|6}} great triangle must not be confused with the {{radic|5}} face triangle of the 5-cell, which does not lie in a central plane.}} The 120 rhombicosadodecahedra are also bonded directly together at ''all'' their exterior faces: at their pentagon faces (which are 120-cell faces of dodecahedral cells); at their small triangular faces (which are 600-cell faces of small tetrahedral cells, 10 of which are inscribed in each dodecahedral cell); and at their rectangular faces (which are faces of 10-point [[w:Pentagonal_prism|pentagonal prism]] cells, where the pentagonal prism is a column inscribed in a dodecahedral cell, joining its opposite pentagon faces). The 120 rhombicosadodecahedra are bonded directly to each other at their own exterior faces; no dodecahedra, small tetrahedra, or pentagonal prisms lie between them, and those objects are not cells of this honeycomb; we mention them here only to help locate the rhombicosadodecahedra within the 120-cell. The only objects which fill holes between the rhombicosadodecahedra are the 120 islanded 5-cells.{{Efn|This honeycomb of the 120-cell resembles the [[600-cell#Icosahedra|honeycomb of the 600-cell consisting of 24 face-bonded regular icosahedra]] with 24 clusters of 5 tetrahedra filling the holes between them. In the 120-cell there are 5 times as many of everything as in the 600-cell, so there are 120 (rhombicosadodecahedron) icosahedral cells with 600 tetrahedra filling the holes between them.}}

It is possible for rhombicosadodecahedra to meet each other at more than one kind of exterior face because they are not meeting there as cells of the same 11-cell; they belong to different 11-cells. The 11 rhombicosadodecahedra in the same 11-cell do not meet at their own exterior faces, they meet at their interior {{radic|6}} triangle faces. As a cellular 4-polytope, the 11-cell is inside-out. Its cells meet each other at interior planes, sharing interior {{radic|6}} triangle faces, instead of in the usual way that 4-polytopes' cells share an exterior face. It is a deeper kind of intertwining to be sure, but the 11 rhombicosadodecahedra in the same 11-cell are still volumetrically disjoint cells, and the 11-cell is a perfectly real 4-polytope with real cells. It is a semi-regular 4-polytope, because it has two different kinds of cells: 11 rhombicosadodecahedra and 11 5-cells. The 11 5-cells are islanded (not bound to each other), and the 11 rhombicosadodecahedra are bound both to 5-cells (at {{radic|5}} 5-cell faces) and to each other (at {{radic|6}} central triangle faces). In the section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#What's in the box|§What's in the box]]'', below, we will look closely at the relation between the {{radic|5}} chord and the {{radic|6}} chord, which is the relation between the pentad and the hexad, and at the general phenomenon of hemi-polytopes which meet each other at a central plane. Then in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Cell rings of the 11-cells|§Cell rings of the 11-cells]]'' we will look at the two-cell cellular structure of the 11-cell in more detail.

Considered as a single complex of overlapping cells rather than as distinct honeycombs of volumetrically disjoint cells, the 120-cell contains 60-point (rhombicosadodecahedral cells), islanded 5-point (5-cells), 20-point (dodecahedral cells), and 10-point ([[w:Pentagonal_prism|pentagonal prism]] cells). The 60-points meet each other in pairs at their pentagon faces, meet regular dodecahedra at those pentagon faces also, meet each other in quads at their 600-cell faces where they share an interior 4-point (600-cell tetrahedral cell) and its four faces (just as they share a larger interior 5-cell tetrahedron and its faces), and meet each other and the 10-point (pentagonal prisms) at their rectangle faces. We might try to imagine such a complex with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with parts truncated from 30 neighboring truncated dodecahedra pentagonal prisms). There are 120 of each of these 3 kinds of dodecahedron. 

Moxness's 60-point (rhombicosadodecahedron cell) is a central object of the 120-cell, and contains many wonders. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra by expansion and contraction operations, as Alicia Boole Stott discovered before 1910,{{Sfn|Polo-Blanco: ''Theory and history of geometric models of Alicia Boole Stott''|2007|loc=§5.3.2 1910 paper on semi-regular polytopes|pp=152-158|ps=; summarizes Boole Stott's method and results from {{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}}, including her tables of expansion-contraction dimensional analogies and a few of her illustrations.}} and those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, rediscovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter &amp; Petrie|1938|p=4|loc=''[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]''}}|name=Concentric polyhedra}} Before we move on from the 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice one more thing about the hidden {{radic|5}} chords lurking below its surface. The 12 little pentagon faces are spanned by sets of 6 {{radic|5}} chords (5-cell edges). The 6 chords of each set are disjoint (they don't touch or form 5-cell faces), but they are symmetrically arranged in 3 parallel pairs, {{radic|3}} apart, which lie in 3 orthogonal central planes. It turns out they are the 6 reflex edges of a 12-point non-convex polyhedron called the [[w:Jessen's_icosahedron|Jessen's icosahedron]], inscribed in the 60-point cell. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again and work our way outwards, before eventually in ''[[#The perfection of Fuller's cyclic design|§The perfection of Fuller's cyclic design]]'' we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cell)s ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks, showing its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Pentahemidemicube.png|120px]]
|[[File:Tetrahemihexahedron_rotation.gif|120px]]
|[[File:Pentahemicosahedron.png|120px]]
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts needed to build the 120-cell (the picture on the cover of the box), and everything else that's invisible in that picture which fits inside the 120-cell: all the astonishing regular 4-polytopes. The pentad building block is half a pair of orthogonal 5-point (regular 5-cells), the hexad building block is half an 8-point (16-cell), and the heptad building block is a quarter of an 11-point (11-cell). Everything can be built from these blocks by embedding them at the center of the 3-sphere, inscribing them as 4-polytopes to make compounds as with the 16-cell building block, rather than by assembling them face-to-face as 3-polytopes to make honeycombs in which they are cells or vertex figures.{{Efn|name=Embedding point}}

The pentad building block is the 5-point '''pentahemidemicube''', an abstract polyhedron with 10 equilateral triangle faces, 10 {{radic|5}} edges, and 5 vertices.{{Sfn|Christie: Pentahemidemicube|2024|loc=File:Pentahemidemicube.png|ps=; &quot;The 5-point ''pentahemidemicube'' is an abstract polyhedron with 10 equilateral triangle faces, 10 edges, and 5 vertices. It has a real presentation as the regular 5-cell.&quot;}} It has a real presentation as the 5-point (regular 5-cell 4-polytope), a regular tetrahedron expanded by addition of a fifth vertex equidistant from all the others in a fourth dimension. The pentad block is a tetrahedral pyramid (a 5-cell) inscribed in a cube: a tetrahedron inscribed in a cube (a demicube) expanded by addition of a fifth vertex, one of the cube's other 4 vertices. Each pentad (5-cell) building block lies in the 120-cell orthogonal to three other pentad (5-cell) building blocks. They are four completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges. Each regular 5-cell in the 120-cell is a pentad (pentahemidemicube) building block, in one of 30 orthogonal sets of 4. There are 120 pentad building blocks in the box.

Every vertex of the 16-cell, and therefore every vertex of the 8-cell, 24-cell, 600-cell and 120-cell, is surrounded by a 6-point (octahedral vertex figure). The hexad building block is that 6-point octahedron embedded at the center of the 4-polytope instead of at a vertex, that is, inscribed in the 3-sphere as a 4-polytope, the 6-point '''[[w:Tetrahemihexahedron|tetrahemihexahedron]]''' or '''hemi-cuboctahedron'''.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; &quot;The tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U&lt;sub&gt;4&lt;/sub&gt;. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices.&quot;}} The hexads have 4 red triangle 5-cell faces, bounded by 12 {{radic|5}} edges. Their 3 yellow square faces lie in central planes, because the building block is an abstract 4-polytope. The hexad is an irregular abstract polyhedron that also has 6 {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

Two orthogonal 5-point (pentad) building blocks intersect in a 4-point (inscribed tetrahedron) to form a 6-point (hexad) building block. Two orthogonal 6-point (hexad) building blocks intersect in a 4-point (central square) to form an 8-point (16-cell) building block, the most usable building block, from which everything larger can be built by compounding.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 120 pentads to pair with a completely orthogonal hexad, and although there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed) they only result in 120 pairs two different ways, so there are 120 11-cells. The 11-cell is 6 pentads' and 5 hexads' smallest joint compound, and the 120-cell is their largest, 120 pentads and 100 hexads.

The heptad{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''|pages=22–31}} building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|4}} edges and 9 {{radic|5}} edges), and 7 vertices.{{Sfn|Christie: Pentahemicosahedron|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;}} It is a compound of a 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]) and a 6-point (hexad [[W:hemi-icosahedron|hemi-icosahedron]]) intersecting in a 4-point (tetrahedron), that is, of three orthogonal 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]&lt;nowiki/&gt;s) intersecting in their common 4-point (tetrahedron). It is also the abstract cube contracted by removal of one vertex, which reduces one of the cube's two inscribed tetrahedra (demicubes) to a single remaining triangular face. As with all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Four &lt;small&gt;&lt;math&gt;(11-7)&lt;/math&gt;&lt;/small&gt; orthogonal 7-point (pentahemicosahedron) blocks can be joined at a common Petrie polygon (a skew hexagon which contains their orthogonal 6-edge paths) to construct a real 11-point ([[W:11-cell|11-cell]] semi-regular 4-polytope), which magically contains six &lt;small&gt;&lt;math&gt;(11-5)&lt;/math&gt;&lt;/small&gt; 5-point (1-pentad regular decahedra) as cells and five &lt;small&gt;&lt;math&gt;(11-6)&lt;/math&gt;&lt;/small&gt; 12-point (2-hexad semi-regular cuboctahedra) as cells. There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like [[W:Euclid's postulates|Euclid's postulates]], and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as honeycombs of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles. It possesses the complete symmetry of the polytope without any redundancy, because it contains one of each of the polytope's characteristic root elements.{{Efn|A [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be partitioned radially by its planes of symmetry (Coxeter's &quot;reflecting circles&quot;) into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated rotationally to generate its regular 4-polytope because it is the complete [[W:gene|gene]] for it, containing all of its elements and capturing all of its symmetry without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring that triggers the reflections.|name=Characteristic orthoscheme}} It is the gene for the polytope, which can be replicated to construct the polytope.

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its &lt;math&gt;A_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the regular compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;small&gt;&lt;math&gt;120\times 120 = 14400&lt;/math&gt;&lt;/small&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[16--cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its &lt;math&gt;B_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 384. The 120-cell is the convex hull of the regular compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 24-point (24-cell) is subdivided into 1152 instances of its [[24-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;F_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 1152. The 120-cell is the convex hull of the regular compound of 25 disjoint 24-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;25\times 1152 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has three equivalent origins in distinct root systems, though most are obviously constructed from one root, and not so obviously from the others. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything larger than the pentad can be constructed by compounding hexads, and we shall see that everything as large as a 24-cell has a construction from heptads which are a product of a pentad and a hexad. Surely every uniform polytope can be constructed by some function of expansions and contractions from any of three root systems, however indirect the recipe. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes, related by an equals sign between them, then give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics by [[W:Noether's theorem|Noether's theorem]].

=== Pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 octahedral cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell contains the 600 octahedral cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It also contains the 600 cubical cells of its 75 8-cells. It is a 600-cell in three ways, although the octahedra and cubes are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell and the 8-cell, respectively.|name=600 cells three ways}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (1-pentad 5-cell), out of the 8-point (2-hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (1-pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope that contains 5-point (1-pentad 5-cell) building blocks beside the largest one, it just isn't a ''regular'' 4-polytope. It is semi-regular with two kinds of cells, the 4-point (regular tetrahedron) and the 12-point (truncated cuboctahedron), which have just one kind of regular face between them, a 5-cell face. That semi-regular 4-polytope is the 11-cell.

We said earlier that the 11-cell had 11 disjoint 5-cells and 11 hemi-icosahedral cells in it, and that its construction from them might be hard to see, but now we are going to account numerically, at least, for how it can have 6 pentad cells plus 5 hexad cells magically adding to 11 of the same kind of elevenad cell. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, 11. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (5-cell regular decahedron) cells, and the 12 6-point (hexad) building blocks into 100 12-point (regular icosahedron) cells, and then magically adding one whole 5-point (5-cell regular decahedron) cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 12-point (regular icosahedron) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such (cell bundles of 6 + 5 = 11 cells), with each 11-cell bundle containing 11 of both kinds of cell. Very, very strange!{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden'': §II To Add is to Multiply|p=38|ps=; the sequence of regular polygons with harmonious proportions beginning {5}, {7}, {9}, {11} as a family of natural numbers; &quot;The numbers in this optimal family have remarkable properties [that allow] similar figures to be arranged easily or allow a figure to be dissected into a set of similar figures. This is the origin of the Greek idea of geometric progression — that ''multiplication can be accomplished instead by addition''.... For instance, it is well known that the square of the golden ratio is one more than itself (written above as 𝜙&lt;sup&gt;2&lt;/sup&gt; = 𝜙 + 1) and one less than 𝜙 is its reciprocal (1/𝜙 = 𝜙 - 1). The first relation equates multiplication with addition, while the second accomplishes division through subtraction. [These are pentagon {5} bisectional golden ratios, but] the heptagonal [{7} tri-sectional] ratios also behave in surprising ways — adding to multiply, subtracting to divide.... There are analogous constructions for [nonagon {9}] quadrisectional and [hendecagon {11}] pentasectional systems.&quot;}}

The 120-cell has three interchangeable cyclic expansion-contraction cycles, the pentad, hexad and heptad cyclic designs, rooted in the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems, respectively. Each of them constructs the whole sequence of regular 4-polytopes from a single building block. The key to all three cycles is that 5 + 6 = 11, and there is an 11-point quasi-regular 4-polytope in the sequence of convex 4-polytopes. It lies between the 8-point (16-cell) and the 16-point (8-cell), and is itself a &lt;math&gt;B_4&lt;/math&gt; polytope. Its role is to be a unified single building block common to all three cyclic expansion-contraction cycles. It is involved with the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems and expresses in one building block their relations to each other.

We have already seen how the hexad building cycle works. It is plain compounding of 4-orthoplexes at the beginning, increasing to densely intricate non-disjoint symmetry toward the end, making it difficult to see why it works once past the 24-cell. This is how the build works, in terms of sixes. The 16-cell is a 2-hexad (3 pairs of completely orthogonal great square planes). The 24-cell is 3 16-cells and a 4-hexad, 4 different ways: 16 non-disjoint hexads (great hexagon planes). The 120-point (600-cell) is 25 24-cells and a 20-hexad, 10 different ways: 200 non-disjoint hexads (great hexagon planes). The 600-point (120-cell) is 225 (9 times 25) 24-cells. ....

The pentad building cycle doesn't do plain compounding to regular 4-polytopes until the last one, the 120-cell. Fives don't even appear as pentagons until after the 24-cell. We have seen how there are 11 5-cells in the quasi-regular 11-cell, which occurs before the 8-cell, and also how everything can be compounded from irregular 5-cells (4-orthoschemes specialized for each 4-polytope). It is still not evident how a pentad cycle can work to build all the regular 4-polytopes from the regular 4-simplex. This is how is how the build works, in terms of fives. ....

This is how the build works, in terms of elevens. The 11-cell is 4 orthogonal heptads (intersecting in their skew hexagon Petrie polygon). The 24-cell is also 4 heptads (intersecting in 4 orthogonal digon axes, so each heptad contributes 6 unique points to a single fibration of 24 points). Heptads are quarter elevenads, so the 11-cell and 24-cell are both 1 elevenad. ....

See the tabular sequence of convex 4-polytopes in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Build with the blocks|§Build with the blocks]]'', below: table rows ''Inscribed, Pentad, Hexad, Heptad''.

=== 11 of them ===
There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of eleven 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum eleven of them. There is only a single Hopf fibration of the 11 great circles of an 11-cell. 

The 5-cell and the 11-cell are limit cases. The 5-cells are completely disjoint, but the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 10 other instances of itself.

== Cell rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer &amp; Segerman|2013|loc=''Puzzling the 120-cell''}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his section of the Wikipedia article on the 120-cell entitled ''[[120-cell#Visualization|§Visualization]]''.{{Sfn|Ruen &amp; Goucher et al. eds. 120-cell|2024|loc=Goucher's ''[[120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious place to look for the 11-cell's Hopf map is in Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other place it could be hiding. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The quasi-regular convex 11-cell|§The quasi-regular convex 11-cell]]'', and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen: triacontagon|2011|loc=''Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them except his illustrations. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Four orthogonal projections of the 120-cell to a central plane, selected from the table of 3 regular and 11 compound 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|100 icosahedra&lt;br&gt;120 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|5 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 icosahedra&lt;br&gt;6 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|120
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|4
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|50
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|7.2°
|align=center|36°
|align=center|6°
|align=center|7.2°
|}

....

Here we summarize our findings on the 11-cells so far, including in advance some results to be obtained in the next section two sections.

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section....

== The 15 major chords ring everything ==
The natural numbers are each a distinct flavor.

Each integer has its own distinct taste. The first six integers {0..5} are the six strongest, most familiar tastes in nature, instantly distinguished and utterly unique.

The seventh flavor {6} is the first ratio of two of the others. Thereafter follows in strange sequence all the ratios of all the flavors, each marriage of every two flavors, most of which are fractions &lt;math&gt;\{k/d\}&lt;/math&gt;, but some of which are integers &lt;math&gt;\{k\}&lt;/math&gt;.

Some of these ratios are successful flavors in their own right, as distinct and inimitable as the marriage of chocolate and oranges, but the marriage of many flavor pairs is mud. It is a rule of thumb among flavor specialists that nearly any taste can be successfully alloyed with either onions or chocolate, but the direct marriage of these two flavors is notably unsuccessful. Some flavors make famous pairings or infamous ones, and subtle new pairings are always to be discovered that work surprisingly well for some purpose, but the majority of flavor marriages are undistinguished compounds.

In polytope geometry, each chord of a polytope is both is a distinct 1-dimensional object, a chord of the sphere of a distinct length &lt;math&gt;l&lt;/math&gt;, and a distinct natural number &lt;math&gt;h&lt;/math&gt;, a unique flavor. If the polytope is regular, it is a noteworthy and successful flavor. The chord's length &lt;math&gt;l&lt;/math&gt; is a square root, related to the natural number &lt;math&gt;h = k/d&lt;/math&gt; and to a set of polytopes they both represent, by a formula discovered by Steinbach.{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden''; Figure 5. Optimal sections and proportions|p=37|ps=; the regular polygons {5}, {7}, {9} and {11} with their diagonals define respectively: {5} the golden bisection proportional to 𝜙; {7} an analogous trisection; {9} an analogous quadrasection; {11} an analogous pentasection.}} The chord length &lt;math&gt;l&lt;/math&gt; is related to the number of sides of the regular polygon formed by the chord (its numerator &lt;math&gt;k&lt;/math&gt;), and to the winding number or density of its regular skew Petrie polygon (its denominator &lt;math&gt;d&lt;/math&gt;).

The regular 4-polytopes do not contain an infinite variety of these chord flavors, but only 30 of them: only 15 unique pairs of 180° degree complementary chords, which combine to make every regular ''n''-dimensional thing found in nature.{{Efn|The 15 ''major chords'' suffice to construct all the regular ''n''-polytopes. Their 180° complements include 15 ''minor chords'' which occur only in the 120-cell, not in any smaller regular polytope.{{Efn|name=Major and minor chords}} Including the complement of the 180° {{radic|4}} digon {2} chord (the {0} chord {{radic|0}}), and the skew variant of the 90° {{radic|2}} square {4} chord (the {30/7} chord), there are 32 distinct natural numbers which are all the chords of all the regular polytopes in all dimensions.|name=32 chords}} All the regular polytopes (including the largest one on the cover of the box) can be built not only from a box containing a number of blocks of 3 shapes glued together by gravity, they can be also be built from a box containing a number of sticks of 15 lengths glued together by rubber joints.

[[File:15 major chords.png|500px|thumb|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the others occur only in the interior of 120-cell, not as edges of any of the regular 4-polytopes. There are 30 distinct 4-space chordal distances between vertices within the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to identify any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=Major and minor chords}} chords are properly labelled #1 — #15 in the order they fan out by size.{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''; Figure 3|ps=; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with a &quot;fan of chords&quot; diagram like the one here.|p=23}} They join vertex pairs which are 1 — 15 edges apart on a &lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon. A #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; chord forms &lt;small&gt;&lt;math&gt;f+1&lt;/math&gt;&lt;/small&gt; regular geodesic polygons which include a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; disjoint regular {{nowrap|&lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt;-gons}} (e.g. &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; a compound of 5 disjoint hexagons) plus a single [[W:Triacontagon#Triacontagram|regular skew {{nowrap|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram}}]] that wraps around the sphere &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times before closing its circle. Notice the maverick #11 chord, with its non-conforming origin point for its {{nowrap|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;}} polygram.]]

These 15 chords (natural numbers) include the integers 2, 3, 4, 5, 6, 10, 11, 15 and 30. The others are the fractions 5/2, 10/3, 13/7, 15/2, 15/4, and 15/7. Represented among the chord lengths (the square roots) measured in ''unit radii'' (not in {{radic|2}} radii, as metrics are often given elsewhere in this article) are the {{background color|paleturquoise|roots of the first 4 positive integers {{radic|1}}, {{radic|2}}, {{radic|3}} and {{radic|4}}}}, the {{background color|#FFCCCC|root of the very distinguished rational fraction {{radic|5/2}}}}, and the roots of various distinguished irrational fractions as Steinbach discovered: for example, 4 of them are {{background color|yellow|roots of golden numbers (a small integer ± 𝜙)}}. The 15 chords each form their regular polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, which is just &lt;small&gt;&lt;math&gt;\{k\}&lt;/math&gt;&lt;/small&gt; in 8 cases. Arranged in order #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; from shortest edge-length square root to longest, the #1 — #15 chords bridge vertex pairs which are &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; vertices apart on a [[W:Triacontagon#Petrie polygons|&lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon]]. Each chord makes a [[W:Triacontagon#Triacontagram|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; polygram]] of its edge length, a compound of the smaller &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. The 15 distinct polygons form a sequence you would not anticipate exactly unless you have Steinbach's formula:

{| style=&quot;background-color:transparent; white-space:nowrap;&quot;
|-
| {30/1} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|.073}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{30} 
| 120-cell edges
|-
| {30/2}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.191}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15}
|- style=&quot;background: yellow;&quot;|
| {30/3}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.382}} 
| &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi}&lt;/math&gt;&lt;/small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{10}''' 
| 600-cell edges
|-
| {30/4} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.573}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/2}
|- style=&quot;background: paleturquoise;&quot;|
| {30/5}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|1}}'''
| &lt;math&gt;1&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{6}''' 
| 24-cell, 8-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/6}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|1.382}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{5}'''
|- style=&quot;background: paleturquoise;&quot;|
| {30/7}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|2}}'''
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{4}''' 
| 16-cell edges
|- style=&quot;background: #FFCCCC;&quot;|
| {30/8}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.5}} 
| &lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;&lt;small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/4} 
| 5-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/9}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.618}} 
| &lt;math&gt;\phi&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{10/3}
|- style=&quot;background: paleturquoise;&quot;|
| {30/10}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|3}}''' 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{3}'''
|-
| {30/11}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.426}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{11/11}
|- style=&quot;background: yellow;&quot;|
| {30/12}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.618}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{5/2}
|-
| {30/13}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.810}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{13/7}
|-
| {30/14}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.928}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/7}
|- style=&quot;background: paleturquoise;&quot;|
| {30/15}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|4}}''' 
| &lt;math&gt;2&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{2}'''
|}

{| class=wikitable style=&quot;white-space:nowrap; float:right;&quot;
!colspan=6|Natural number &lt;small&gt;&lt;math&gt;h=\{k/d\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Fibers&lt;br&gt;&lt;small&gt;&lt;math&gt;f\{h\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Cells&lt;br&gt;&lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|4-tope&lt;br&gt;&lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Verts&lt;br&gt;&lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|[[W:Triacontagon#Triacontagram|30-gram]]&lt;br&gt;&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;
|-
!Chord
!Arc
!colspan=2|&lt;small&gt;&lt;math&gt;l/\sqrt{1}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;l/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;h&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#1
|15.5~°
|{{radic|0.073}}
|0.270
|{{radic|0.146}}
|&lt;small&gt;&lt;math&gt;30&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1\{30\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;5\{3,3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/1\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#2
|25.2~°
|{{radic|0.191}}
|0.437
|{{radic|0.382}}
|&lt;small&gt;&lt;math&gt;15&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;..\{3,3\}..&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/2\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#3
|36°
|{{radic|0.382}}
|0.618
|{{radic|0.764}}
|&lt;small&gt;&lt;math&gt;10&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;30\{3,3\}20&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/3\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#4
|44.5~°
|{{radic|0.573}}
|0.757
|{{radic|1.146}}
|&lt;small&gt;&lt;math&gt;15/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/4\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5
|60°
|{{radic|1}}
|1
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;6&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/5\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#6
|72°
|{{radic|1.382}}
|1.175
|{{radic|2.624}}
|&lt;small&gt;&lt;math&gt;5&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}} 
|&lt;small&gt;&lt;math&gt;10\{5,3\}12&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/6\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#7
|90°
|{{radic|2}}
|1.414
|{{radic|4}}
|&lt;small&gt;&lt;math&gt;4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7\{4\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;4\{4,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;9\{3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/7\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: #FFCCCC;&quot;|
|#8
|104.5~°
|{{radic|2.5}}
|1.581
|{{radic|5}}
|&lt;small&gt;&lt;math&gt;15/4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/4\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}} 
|&lt;small&gt;&lt;math&gt;15\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/8\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#9
|108°
|{{radic|2.618}}
|1.618
|{{radic|5.236}}
|&lt;small&gt;&lt;math&gt;10/3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10/3\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/9\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#10
|120°
|{{radic|3}}
|1.732
|{{radic|6}}
|&lt;small&gt;&lt;math&gt;3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;10\{3\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/10\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#11
|135.5~°
|{{radic|3.426}}
|1.851~
|{{radic|6.852}}
|&lt;small&gt;&lt;math&gt;11/11&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;11\{11/11\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;11\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#12
|144°
|{{radic|3.618}}
|1.902
|{{radic|7.235}}
|&lt;small&gt;&lt;math&gt;5/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2},3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/12\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#13
|154.8~°
|{{radic|3.810}}
|1.952
|{{radic|7.621}}
|&lt;small&gt;&lt;math&gt;13/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;13\{13/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;7\{5,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/13\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#14
|164.5~°
|{{radic|3.928}}
|1.982
|{{radic|7.857}}
|&lt;small&gt;&lt;math&gt;15/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;137\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/14\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{radic|8}}
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15\{2\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐𝜙&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;8\{3,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/15\}&lt;/math&gt;&lt;/small&gt;
|}

The polygons '''{10}''', '''{6}''', '''{5}''', '''{4}''', '''{3}''', and '''{2}''' are flat polygons that each lie on a great circle in a central plane of the 3-sphere. The polygons {30}, {15}, {15/2}, {15/4}, {10/3}, {11/11}, {5/2}, {13/7}, {15/7} and all the {30/''n''} polygons are skew polygrams spiraling through curved 3-space, each successive edge in a different central plane of the 3-sphere. Projected to the 1-sphere (a circle in the plane), they are [[W:Star polygon#Regular star polygon|regular star polygrams]], the simplest being the pentagram {5/2}. In 4-space they wind around the 3-sphere more than once ''without self-intersecting'', before they close their circuit. They lie on helical geodesic circles or ''isoclines'' of circumference greater than 2𝝅''r''.

Even the 6 integer chords which form flat polygons also form skew polygrams. Each of the 15 chords occurs as a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, a fiber bundle of Clifford parallel polygons that totals 30 vertices per bundle. Each chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; also forms a single skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram that winds through another, shorter chord's bundle of parallel polygons, intersecting them in succession multiple times each, and weaving that whole bundle of parallel fibers together as the cross-thread does on a loom. The polygram cross-thread has to wind multiple times through the whole bundle (&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times around the [[W:Triacontagon#Triacontagram|30-gram]] in which each edge connects every &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt;th vertex) to complete an enumeration of all 30 vertices in the bundle. For example, the #5 chord forms hexagon &lt;small&gt;&lt;math&gt;\{6\}&lt;/math&gt;&lt;/small&gt; flat planes in parallel bundles of 5, written &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt;, and also a skew &lt;small&gt;&lt;math&gt;\{15/2\}&lt;/math&gt;&lt;/small&gt;-gram of #5 edges which winds 2 times through the #3 chord's bundle of 3 decagon &lt;small&gt;&lt;math&gt;\{10\}&lt;/math&gt;&lt;/small&gt; flat planes. Every chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; forms both its &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram and its fiber bundle of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, and the edges of one chord's parallel polygons are also the edges of its skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram weaving through some shorter chord's bundle of parallel polygons. Each numbered chord is a natural number, a ratio of two integers and their polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, either flat or skew. Each chord number defines a &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; skew polygon that is a compound bundle of the &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. All these flat or skew polygons are regular polygons made of their numerator instances of a single chord #, whose length is that square root, Steinbach's function of the natural number.

The relationship between each bundle of flat Clifford parallel great circle polygons of a distinct length, and its corresponding bundle of skew polygrams of a longer length, is precisely that of a discrete simple rotation and its corresponding discrete isoclinic rotation. For example, the flat great circle hexagons of edge length {{radic|1}} lie in the invariant rotation planes of the 24-cell's characteristic 60° isoclinic rotation, in which the vertices move on skew hexagram rotation paths of edge length {{radic|3}}. The 24-cell contains 4 bundles of 4 Clifford parallel (disjoint) hexagon central planes each. The 24 vertices of the 24-cell in each bundle are linked by rotation in the 4 hexagon invariant planes in two different ways: simply and isoclinically. In a simple rotation the vertices of a single invariant hexagon exchange places on that great circle hexagon of edge {{radic|1}}. In an isoclinic rotation in which any one of these 4 disjoint hexagon central planes is invariant, all 4 of them are necessarily invariant planes in the rotation, because they rotate in parallel. In that isoclinic rotation the vertices of all 4 hexagon invariant planes (all 24 vertices of the 24-cell) exchange places concurrently, moving along 4 disjoint helical hexagrams of edge {{radic|3}} in parallel. It is a characteristic only of this distinct rotation in hexagon invariant planes that the helical vertex path or ''isocline'' is a hexagram of period 6, edge length {{radic|3}}, and circumference 720°. Rotations in other kinds of polygon invariant planes have different characteristic isoclines. The period of their isoclinic rotation must be ''the same or greater'' than the period of their simple rotation; in most cases (unlike this one) it is greater.

The 15 chords form not only distinct fiber bundles of polygons, but also distinct bundles of polyhedra: cell rings &lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt; and compound vertex figures &lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;. These combine to form their characteristic regular 4-polytope &lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Coxeter|1973|p=269|loc=Compounds|ps=; &quot;It is remarkable that the vertices of {5, 3, 3} include the vertices of all the other fifteen regular polytopes in four dimensions.&quot;}} All these elements of a regular 4-polytope constructed from one chord are themselves regular polytopes, and each of the 16 regular 4-polytopes is a function of one characteristic chord. If the regular 4-polytope is convex, its characteristic chord is its unit-radius edge. For a non-convex regular 4-polytope, the characteristic chord is its unit-radius ''mid-edge diameter'', the distance from the midpoint of each edge to the midpoint of the completely orthogonal edge. Each of the characteristic chords (each distinct natural number) has its expression as one of the 16 regular 4-polytopes and all the products of the distinct regular 1-polytopes, 2-polytopes and 3-polytopes that construct it, and the whole distinct construction from the natural number to the 4-polytope is an [[W:SO(4)|SO(4) symmetry group]] expression of that 4-polytope's characteristic discrete [[W:SO(4)#Isoclinic rotations|isoclinic rotation]], in the invariant central planes of its characteristic chord, by the characteristic chord arc.

...demonstrate this theory by proper enumeration of the chord lengths in the table...

...color-code the △ ☐ 𝜙 ✩ symbols ... chord colors in diagram should match them ... some of the diagram colors are wrong and must be changed, but otherwise diagram is pretty good ... i think in a colored diagram all chords of the same length should have the same size line dash, the polygons are not obvious enough and the 15 fanning-out chords are readily identified by their labels...

... add back the counts of chords in the box (very small font number under each table entry), and present these two images as the front and back a box of building sticks and rubber joints...What's in the other box...

...say the rational-root major chords {{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|5/2}} are the ''n''-edges: their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge...

...abbreviated text of the [[120-cell#Relationships among interior polytopes|''§Relationships among interior polytopes'']] section from the 120-cell article...to explain the color-coding

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius=1}}

See also the elements, properties and metrics of the sequence of regular 4-polytopes summarized in tabular form in the section ''[[#Build with the blocks|§Build with the blocks]]'', below.

== Alicia Boole Stott's original formulation of dimensional analogy ==
&lt;blockquote&gt;&quot;...a method by which bodies having a certain kind of semi-regularity may be derived from regular bodies in an Euclidean space of any number of dimensions...&quot;{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''; presents cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues|pp=12-45}}&lt;/blockquote&gt;

Alica Boole Stott's 1910 paper reads like an exceptionally well-written patent application. It is a comprehensive stepwise description of her original invention, a general method of dimensional analogy between regular and semiregular polytopes of the same edge length.

[[File:The Roots of Three.png|thumb|An illustration of Boole Stott's expansion of unit-edge triangle to hexagon captures the geometric basis in 3 of the &lt;math&gt;A_n&lt;/math&gt;root system.]]

The engineering-like articulation of her new principles is entirely discrete and group-theoretic. Boole Stott repeatedly refers to sets of polytope elements as ''groups''. All her transformations are discrete instances of one or another special case of her &quot;two inverse operations which may be called ''expansion'' and ''contraction''.&quot; Hers is an exploration of ''n''-dimensional space at its most basic: space defined by the discrete objects which fill it, themselves defined directly as groups of elements with a few operations on them, which transform them into each other. This is geometry which seems to precede Euclid's postulates, and does not depend on them. Instead it corresponds directly to the deeper group theory mathematics which underlies geometry, even though Boole Stott, like the ancients, did not have that mathematics.

Boole Stott enumerated her expansion and contraction operations as transformations between regular and semi-regular polytopes of a single edge length. In her 1910 paper she did not consider any polytopes of multiple edge lengths, or expansions of the subject polytope by a distance other than that exactly required to produce new edges of the original edge length, so of course she did not enumerate Moxness's non-uniform rhombicosidodecahedron (Hull #8). This may be why section 8&lt;sub&gt;3&lt;/sub&gt; is missing from her earlier drawings of the sections of {5,3,3}; she may not have had a construction for it.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The Boole Stott expansion (or contraction) operation is as precisely defined a transformation as Coxeter's reflection, rotation and translation operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions). Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;
where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;
where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;
For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} An instance of the expansion (or contraction) operation is an application of the general operation to a particular subject polytope, operating on all of its elements symmetrically at once. The general operation is restricted, such that one may only expand (or contract) the subject polytope by pushing all its edges, faces or cells directly away from (or toward) its center, without changing the size of the moved elements. This splits each vertex into several (or coalesces several vertices into one), and has corresponding effects on the other elements, producing new edges ''of the same size'' between pairs of separated vertices (or reducing some edges to single vertices). The distance all the elements are moved away from (or toward) the center must be such that the vertices separate producing new edges of only the original edge length (or come together reducing edges to single vertices). It is permitted to combine these three operations, e.g. by pushing the edges ''and'' the faces, or by pushing all three element sets, provided the combination results in a semi-regular polytope of only one edge length. The three operations are all commutative with one another, so combined operations can be considered either as a sequence of expansions (or contractions) linking several related polytopes, or as their concurrent movements, a single transformation of the subject polytope. Since there are 8 combinations of one to three operations there are 8 polytope variants (including the original regular polytope), and all are adjacent to one another by a single transformation. The variants of the original regular polytope are named for which element set(s) are moved. By enumerating all 8 variants of expansion (and contraction) for each of the six regular convex 4-polytopes, Boole Stott produced a table of regular and semi-regular 4-polytopes divided into 12 sets of 8: six expansion sets and six contraction sets.{{Sfn|Boole Stott|1910|loc=''Table of Polytopes in S&lt;sub&gt;4&lt;/sub&gt;''; note that Boole Stott's usage of ''S&lt;sub&gt;4&lt;/sub&gt;'' denotes a Euclidean 4-space ''E&lt;sub&gt;4&lt;/sub&gt;'', or a 3-sphere space embedded in it, ''not'' a 4-sphere space embedded in a Euclidean 5-space|p=30}} Each set includes just one of the six regular convex 4-polytopes, with its distinct unit-radius edge length, and the 7 possible semi-regular expansions (or contractions) of it, which all preserve that characteristic edge length. The polytopes in each set of 8 all have the same total number of elements as their original regular polytope, but each polytope in the set has a distinct combination of cell, face, and vertex figures. The 12 sets are not completely disjoint, and even within one set the 8 operations do not necessarily produce 8 unique polytopes. For example, the regular 5-cell (the 4-simplex polytope) has 5 cells, 10 faces, 10 edges, and 5 vertices (30 elements). Expanding the 5-cell produces 7 related semi-regular 4-polytopes of 30 elements each (two of which happen to be the same polytope). In each expansion, the 5 cells have expanded into semi-regular or regular polyhedra, the 10 faces and 10 edges ''may'' have expanded into prisms, and the 5 original vertices have expanded into regular or semi-regular cells.

== The regular 11-cell 4-polytope ==

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosadodecahedron). Each non-convex Jessen's icosahedron is a set of 6 reflex edges that are {{radic|5}} 5-cell edges, each set symmetrically arranged in 3 parallel pairs, not as far apart ({{radic|3}}) as they are long, in 3 orthogonal central planes. The 12-point (Jessen's sets of 6 reflex 5-cell edges) are inscribed in the 60-point with their endpoints as a vertex of a pentagon face. The 12 pentagon faces are joined to 5 other pentagon faces by the 5-cell edges; not to their nearest neighbor pentagons but to their next-to-nearest neighbor pentagons; and since there are 5 vertices to each pentagon face, there are 10 12-point (Jessen's sets of 6 reflex 5-cell edges). Each pentagon face is joined to another pentagon face by 5 reflex edges of five distinct Jessen's. The 5 reflex edges are Clifford parallel, the 5 edges of a pentagonal prism column that has a twist to it. The contraction of those 5 edges into 1, and those 10 vertices (two pentagons) into 2, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing all their edges as a single Jessen's. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (octahedron) by conflating its parallel pairs of edges, leaving the 11-cell's abstract 6-point (hexad cell). The hexad's situation is that it occurs in bundles of 5, disjoint but close together in the sense of adjacent, like the great circle rings in a Hopf fiber bundle. 

To summarize, the 12-point (2-hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell). We shall now be able to see how Fuller's perfect hexads combine with Todd's perfect pentads to form the perfect heptads of the 11-cell and 137-cell.  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=alias the hemi-cuboctahedron}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces. (The ''heptad'' arrives at the party, anonymously; they are our distinguished anti-celebrity, who most of the other guests have never even heard of.) The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. This 12-point (truncated tetrahedron) is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 100 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp: Truncated tetrahedron|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp: Truncated tetrahedron|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}} Of course Moxness has been there, seen that, already with his graphics software;{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} see his parameterized illustration of it exactly as it occurs in the 11-cell, below in his bestiary of A3, B3, and H3 Archimedian and Catalan polyhedra in the 120-cell.]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:Archimedean_and_Catalan_solid_hulls_with_their_Weyl_orbit_definitions.svg|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of the 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. In this catalog of Archimedean and Catalan polyhedra sighted in the 120-cell by Moxness's software,{{Sfn|Moxness: Archimedean and Catalan hulls|2023|loc=Hull #1 Archimedean Name A3 110 Truncated Tetrahedron A (upper left)}} it is the first entry (upper left), Moxness's Hull # = 1 with 12 vertices of 3D norm {{radic|11}}/2. On a {{radic|2}}-radius 3-sphere, 5 of these with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell. 20 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges are cells in the [[#The rings of the 11-cells|cell rings of the quasi-regular 11-cell]], which also has 12 regular 5-cells as cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

...

Both kinds of cells in the 11-cell have expressions in two different dimensions (on the 2-sphere and 3-sphere), because they are all centered on the center of the 3-sphere, unlike the cells of most 4-polytopes, which are centered on various off-center points within the 4-polytope's interior. 

One kind of 11-cell cell appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (semi-regular truncated cuboctahedron), a.k.a. the truncated tetrahedron. The other kind of cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now, except in the 120-cell. But this is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron, and a regular tetrahedron behind each face. The 11-cell is thus a honeycomb of mixed regular tetrahedra and truncated cuboctahedra (a.k.a. truncated tetrahedra) that meet at their shared 5-cell faces. 

Both the 5-cells (decahedra) and the truncated cuboctahedra (truncated tetrahedra) are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks in their 4-polytopes (off-center).{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} Therefore all the 11-cell's cells are concentric, which indeed makes a different kind of honeycomb, an inside-out cellular honeycomb with all the cells piled on top of each other overlapping in 4-space. They do not overlap in 3-sphere space however; their 3-volumes (but not their 4-contents) are disjoint. They make a proper 4D honeycomb like the cells of any 4-polytope, with volumetrically disjoint 3D cells that meet each other in the proper way at their shared 2D faces.

An 11-cell contains 6 completely disjoint 5-point (1-pentad 5-cell decahedron cells), and 5 volumetrically disjoint 12-point (2-hexad truncated cuboctahedron cells), 11 volumetrically disjoint cells altogether. Of course it is supposed to have 11 identical 11-point (elevenad hemi-icosahedral cells), and that is also true in its abstract sense, which is not the same sense as disjoint cellular containment: each pentad or hexad cell is inscribed in one of 11 larger 60-point (Moxness's rhombicosadodecahedron cells). The abstract pentads and hexads are only hemi-polyhedra, so the 11-cell's pentad cells are 12 (out of 120) disjoint instances of a pentad 5-point, and its hexad cells are 10 (out of 100) disjoint instances of a hexad 6-point, 1/10th of the 120-cell's cellular volume in both cases. The 11-cell's hemi-icosahedral cells are 11 (out of 120) non-disjoint instances of an elevenad 60-point. Each 11-cell shares each of its hexad cells with 9 other 11-cells, and each of its 11 vertices (out of 600) occurs in 9 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.|name=Tetrahedral vertex figures}}

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

...insert image thumb of 11-cell...

...Construct the 11-cell from the heptad....

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...Four concentric 4-polytopes may be mutually orthogonal. In any set of 4 mutually orthogonal regular 5-cells, one pair of faces in each orthogonal pair are completely orthogonal to each other. Thus three edges in one 5-cell in an orthogonal pair are opposite (completely orthogonal) to a face of the orthogonal 5-cell. .... A 5-cell has 5 vertices in its abstract hemi-icosahedral cells, but only 4 vertices in one concrete icosahedral cell; the 5th lies in an icosahedral cell of another 11-cell (not in another icosahedral cell of the same 11-cell).

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

....

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 11-cell ===

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon.... (wrong, 11-cell's Hopf map is the truncated icosahedron)

....

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk: Truncated icosahedron|2018}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

...

Since the rhombicosadodecahedron has 120 5-cell edges, the 120-cell has 120 completely disjoint 5-cells, and the rhombicosadodecahedron shares each vertex with 11 other rhombicosadodecahedra, how a rhombicosadodecahedron is bound to its neighbor 5-cells is constrained. A further constraint is that the abstract hemi-icosahedron has only 15 5-cell edges, so the 120 actual 5-cell edges of the rhombicosadodecahedron are conflated 8-to-1 in the hemi-icosahedron. To be conflated by contraction into a single edge, those 8 edges must lie parallel to each other in the 120-cell; either actually parallel in the same plane, or Clifford parallel in separate planes, as two edges in completely orthogonal planes may be parallel and orthogonal at the same time. Each 5-cell inscribed in the 120-cell contains 5 such tetrahedra occur in parallel pairs in 4 orthogonal planes (none of them completely orthogonal planes). 

.... 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
Each conflated vertex is the 10 vertices of two orthogonal regular 5-cells...
The 60-point contains 6 pairs of these orthogonal regular 5-cell cells...
...its 20 hexad cells are ..., a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere.

== The regular 137-cell 4-polytope ==

(description originally written for the 11-cell, which applies only to a fibration of 11 11-cells)

It is quasi-regular with two kinds of cells, the 12-point (regular icosahedron) and the 5-point (regular 5-cell), which have just one kind of regular face between them. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of regular facets (it is semi-regular with regular facets), it also means that its facets have expressions of two different dimensionalities because they are concentric in the 3-sphere. The 11-cell has two kinds of regular cells. One appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (regular icosahedron), but since it is embedded at the center of the 3-sphere (unlike the cells of most 4-polytopes) it also acts as an inscribed 4-polytope.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} The other cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now except in the 120-cell. This is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron. The 11-cell is a honeycomb of mixed icosahedron and half-icosahedron cells. Like the icosahedra, the decahedra are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks, off-center in their 4-polytopes. All the 11-cell's cells are concentric, which indeed makes an unusual honeycomb of cells, an inside-out honeycomb. It is a proper 4D honeycomb however, with volumetrically disjoint 3D cells that meet each other in the proper way at shared 2D faces. The embedded 12-point (regular icosahedron) cell is an icosahedral pyramid (as a 4-polytope), which is filled by 20 4-point (irregular tetrahedra) that meet at its center; the center point of the 11-cell acts as a 12th vertex, just as the center point of the 24-cell acts as a 25th vertex in some honeycombs. The 5-point (regular 5-cell) cell contains five 4-point (regular tetrahedra) which meet the 12-point (regular icosahedra)'s irregular tetrahedral cells face-to-face, so the 11-cell is quasi-regular in the usual sense of having two varieties of regular 3-polytope cells, but additionally both varieties of of its cells are bi-dimensional: another clue that the 11-cell has something to do with dimensional analogy

....

The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

....

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur to make an 11-cell. 11 of them occur in a 137-cell, all 120 of them occur in the 120-cell, and they occur nowhere else.

How can this regular 4-polytope possibly exist? By Schläfli's criterion {3, 3, ..} exists, but not as a finite polytope.{{Sfn|Coxeter|1973|loc=§7.7 Schläfli's criterion}} Coxeter found {3, 5, 3} as a finite hyperbolic honeycomb, but not in Euclidean 4-space.{{Sfn|Coxeter|1968|title=The Beauty of Geometry: Twelve Geometric Essays|loc=''Regular Honeycombs in Hyperbolic Space''|ps=; &quot;The honeycombs considered by the above authors [Schlegel, Schläfli, Klein, Sommerville] have finite cells and finite vertex figures. ... A further extension allows a cell or vertex figure to be a ''star''-polytope, so that the honeycomb covers the space several times. .... We shall find that there are four regular star-honeycombs in hyperbolic 4-space, as well as two infinite families of them in the hyperbolic plane.&quot;}} We find that {3, 3, ..} is a central expression of the &lt;math&gt;H_4&lt;/math&gt; root system, being the regular convex hull in 4 dimensions of 137 points, 96 pentads and 600 hexads. The minimal expression of &lt;math&gt;H_4&lt;/math&gt; is the regular convex hull {3, 3, 5} of 120 points, 600 tetrads and 75 hexads, and its maximal expression is the regular convex hull {5, 3, 3} of 600 points, 120 pentads and 675 hexads. {3, 3, ..} is a third regular &lt;math&gt;H_4&lt;/math&gt; polytope which fits between them. The properties and metrics of this regular sequence are presented in tabular form below in ''[[#Build with the blocks|§Build with the blocks]]''.

{|class=&quot;wikitable&quot;
!colspan=3|Regular &lt;math&gt;H_4&lt;/math&gt; polytopes
|-
|[[File:600-cell.gif|200px]]
|[[File:5-cell.gif|200px]]
|[[File:120-cell.gif|200px]]
|-
!120-point 600-cell
!137-point 137-cell
!600-point 120-cell
|}

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 137-cell ===
....

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=Apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|2022|loc=''[[Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]''}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie: On Fuller's use of language|2024|loc=''[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry'']]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center, 1948-1949|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with their mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the gender neutral shad can open their six beaks all the way, until they become flat squares and they becomes a cuboctahedron, or they can shut them all tight like a turtle retracting into their octahedron shell. The six mouths always move in unison. This is [[Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do their ''really'' odd trick -- where they flip their 6 jaws 90 degrees in their 6 faces and shut their 6 beaks on the opposite axis of their squares than the one they opened them on -- or whether they will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than their normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make their edge length exactly the same size as their radius, when they open their mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing their mouths in spherical synchrony, their 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron (the hexad), then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=''[[W:SO(4)|SO(4)]]''}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=Isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its orthogonal hexad's planes (wz, wx, wy).{{Efn|name=Six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of (expanded octahederon 16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh &amp; Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== Legendre's 12-point hexad binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a 12-point double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's perfectly realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, this 2-hexad expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre hexad) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction.{{Efn|name=Concentric polyhedra}} In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== ''n''-simplexes ===
....

The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== ''n''-orthoplexes ===
....
[[File:Great 5-cell √5 digons rectangle.png|thumb|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} ''regular'' great hexagons. These are the same 200 dodecagon central planes illustrated above, which also contain 6 120-cell edges and two opposing ''irregular'' great hexagons (truncated {{radic|6}} triangles). The 1200 {{radic|5}} 5-cell face triangles occupy 1200 other, non-central planes.]]

...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's Hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[16--cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[16--cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=Concentric polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[16--cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== ''n''-taliesins ===
The 4-simplex pentad building block combines with the 4-orthoplex hexad building block to make the 4-taliesin heptad building block. The heptad is the essential building block of the ''taliesin'' family of polytopes. The 11-cell and 57-cell hemi-polytopes and the 137-cell and 120-cell regular 4-polytopes are nuclear members of this trans-dimensional family. The 24-cell and 600-cell are extended members of it, because they also contain instances of the heptad building block.

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its spherical symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* [[W:Taliesin Myrddin Namkai-Meche|Taliesin Myrddin Namkai-Meche]], a young Reed College scholar who was stabbed to death by a crazy man in 2017 on a Portland, Oregon light rail train, when he rose to the nonviolent defense of two black teenagers who were being violently assaulted in racist and anti-Muslim language by a right-wing extremist.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in orthogonal [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].
* '''Taliesan polytope''', an (''n+1)''-polytope whose ''n''-polytope dimensional analogue inscribed concentrically in the ''n''-sphere forms a convex honeycomb of itself which is the (''n+1)''-polytope.{{Efn|name=Embedding point}}
* The taliesan polytope's honeycomb, a Hopf fibration of its great circles.
* The ''n''-dimensional family of dimensionally analogous polytopes related to a taliesan polytope by expansion and contraction operations.
* The characteristic [[w:Orthoscheme|orthoscheme]] of a teliesan polytope, an irregular simplex that is the complete gene for the entire expression of trans-dimensional symmetry relationships among the polytopes of the ''n''-dimensional family, as well as the complete gene for the teliesan polytope's rotational symmetry.{{Efn|name=Characteristic orthoscheme}}

=== ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== ''n''-equilaterals ===
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed at four different [[#n-leviathon|chord]] lengths from each other. In the {{radic|2}}-radius polytopes we have using as examples in this paper they are {{radic|2}}, {{radic|4}}, {{radic|6}} and {{radic|8}}, but in a ''unit-radius'' polytope they are: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}.

Each vertex is joined to 8 others by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}. The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== ''n''-pentagonals ===

[[File:600-cell.gif|thumb|The 120-point (600-cell), &lt;math&gt;H^4&lt;/math&gt;.]]
...

[[File:600-cell vertex geometry.png|thumb|Vertex geometry of the 600-cell, showing the 5 regular great circle polygons and the 8 vertex-to-vertex chord lengths with angles of arc. The golden ratio{{Efn|name=golden chords|group=}} governs the fractional roots of every other chord,{{Efn|name=fractional root chords}} and the radial golden triangles which meet at the center.|alt=|400x400px]]
The 120 vertices of the 600-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏&lt;sup&gt;−1&lt;/sup&gt;) beginning with a vertex; see column ''a''}} at eight different [[#n-leviathon|chord]] lengths from each other. These edges and chords of the 600-cell are simply the edges and chords of its five great circle polygons. In ascending order of length in a ''unit-radius'' polytope (not in a {{radic|2}} radius polytopes like our usual examples), they are {{radic|0.𝚫}}, {{radic|1}}, {{radic|1.𝚫}}, {{radic|2}}, {{radic|2.𝚽}}, {{radic|3}}, {{radic|3.𝚽}}, and {{radic|4}}.{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=Decimal fractions of square roots|group=}}

Notice that the four [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#''n-''equilaterals|hypercubic chords]] of the 24-cell ({{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|4}}) alternate with the four new chords of the 600-cell's additional great circles, the decagons and pentagons. The new ''golden chord'' lengths are necessarily square roots of fractions, but very special fractions related to the golden ratio{{Efn|1=The fractional-root ''golden chords'' are irrational fractions that are functions of {{radic|5}}. They exemplify that the [[W:golden ratio|golden ratio]] &lt;big&gt;𝜙&lt;/big&gt; {{=}} {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618 is a circle ratio related to &lt;big&gt;𝜋&lt;/big&gt;:&lt;br&gt;
: {{sfrac|𝜋|5}} = arccos (𝜙/2)
is one decagon edge, the 𝚽 = {{radic|0.𝚫}} = {{radic|0.382~}} ≈ 0.618 chord.
Reciprocally, in this function discovered by Robert Everest expressing &lt;big&gt;𝜙&lt;/big&gt; as a function of &lt;big&gt;𝜋&lt;/big&gt; and the numbers 1, 2, 3 and 5 of the Fibonacci series:&lt;br&gt;
: &lt;big&gt;𝜙&lt;/big&gt; = 1 – 2 cos ({{sfrac|3𝜋|5}})
{{sfrac|3𝜋|5}} is the arc length of the &lt;big&gt;𝜙&lt;/big&gt; = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618 chord.|name=golden chords|group=}} including the two [[W:golden section|golden section]]s of {{radic|5}}, as shown in the diagram.{{Efn|The 600-cell edges are decagon edges of length {{radic|0.𝚫}}, which is 𝚽, the ''smaller'' golden section of {{radic|5}}; the edges are in the inverse [[W:golden ratio|golden ratio]] {{sfrac|1|φ}} to the {{radic|1}} hexagon chords (the 24-cell edges).
The other fractional-root chords exhibit golden relationships as well. The chord of length {{radic|1.𝚫}} is a pentagon edge.
The next fractional-root chord is a decagon diagonal of length {{radic|2.𝚽}} which is &lt;big&gt;𝜙&lt;/big&gt;, the ''larger'' golden section of {{radic|5}}; it is in the golden ratio{{Efn|name=golden chords|group=}} to the {{radic|1}} chord (and the radius).{{Efn|Notice in the diagram how the &lt;big&gt;𝜙&lt;/big&gt; chord (the ''larger'' golden section) sums with the adjacent 𝚽 edge (the ''smaller'' golden section) to {{radic|5}}, as if together they were a {{radic|5}} chord bent to fit inside the {{radic|4}} diameter.}}
The last fractional-root chord is the pentagon diagonal of length {{radic|3.𝚽}}.
The [[W:Pentagon#Side length is given|diagonal of a regular pentagon]] is always in the golden ratio to its edge, and indeed &lt;big&gt;𝜙&lt;/big&gt;{{radic|1.𝚫}} is {{radic|3.𝚽}}.|name=fractional root chords|group=}}

....


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== ''n''-leviathon ===
...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of its &lt;math&gt;10^2&lt;/math&gt; disjoint instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction, before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]].{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}} .... It is only because she did not have the 11-cell to start with that she did not discover the regular 137-cell {3, 3, ..} by her method.

Between Alicia Boole Stott and her identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and her identification of symmetry groups with conservation laws (in her 1st theorem), we owe the deepest mathematics of geometry and physics originally to two pioneering women, colleagues of their contemporaries the greatest men of the academy, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just an [[W:abstract polytope|abstract 4-polytope]], not just a singleton convex 4-polytope, and not just a honeycomb.{{Sfn|Coxeter|1970|loc=''Twisted Honeycombs''}} The 11-point (11-cell) has a concrete ...regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}&lt;nowiki/&gt;} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, 3, 3} as the 137-point (137-cell).

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 11-cell, 137-cell and 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=''Clifford Spinors and Root System Induction: H4 and the Grand Antiprism''}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=Apology}}

== Build with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Duveneck|1978|loc=Carl Jung, quoted in ''Life on Two Levels''|p=ii|ps=.{{Sfn|Jung|1961|ps=: &quot;The best of truths is of no use unless it has become one's most personal inner experience. It is the duty of everyone who takes a solitary path to share with society what he finds on his journey of discovery.&quot;}}}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1954|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius={{radic|2}}}}

== Acknowledgements ==
...

== Notes ==
{{Regular convex 4-polytopes Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{Refend}}</text>
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    <revision>
      <id>2692236</id>
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      <comment>/* 5-cells and hemi-icosahedra in the 11-cell */</comment>
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      <text bytes="208609" sha1="tjm3dwugijnfyaokgmwotuxzd0ci3lv" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|June 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a regular 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the &lt;s&gt;120-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 5, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 11 [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] cells. Eleven 11-cells (plural) are the 137-cell, the &lt;s&gt;600-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 3, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 137 [[W:Rhombic triacontahedron|triacontahedron]] cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=''Regularity of Graphs, Complexes and Designs''}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=''A Symmetrical Arrangement of Eleven Hemi-Icosahedra''}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]]
The 4-dimensional regular polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:Simplex|simplex]] (called the [[5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:Orthoplex|orthoplex]] (called the [[16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular convex 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the [[w:Tesseract|16-point (8-cell) tesseract]], the 4-cubic, which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found within any of the other regular convex 4-polytopes, except in the largest and most complex one, the [[120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). However, we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the 16-cell is their very starting point, and the most frequently used tool in the box.

== 5-cells and hemi-icosahedra in the 11-cell ==
[[File:Symmetrical_5-set_Venn_diagram.svg|thumb|The 5-point (10-face) regular 5-cell (the regular 4-simplex). Grünbaum's rotationally symmetrical 5-set Venn diagram{{Sfn|Grünbaum|1975|loc=''Rotationally symmetrical 5-set Venn diagram'', Fig 1 (e)|ps=; partitions the individual elements of the 5-cell.}} is an illustration of the 5-cell labeling each of its &lt;math&gt;2^5&lt;/math&gt; elements.{{Sfn|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled; image includes the Python code to render it, optimising for maximum area of the smallest regions.}}]]
[[File:Hemi-icosahedron.png|thumb|The 6-point (10 face) [[W:hemi-icosahedron|hemi-icosahedron]], an abstraction of the regular icosahedron, has half as many faces, edges and vertices. Each element of the abstract polyhedron represents two real elements found in different places in the 11-cell's realization.{{Sfn|Ruen: hemi-icosahedron|2007}}]]
The most apparent relationship between the pentad 5-cell and the hexad hemi-icosahedron is that they both have 10 triangular faces. When we see a facet congruence between a 4-polytope and a 3-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 6-point 5-simplex. It is also the same size as the 6-point 3-orthoplex; thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no chord congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals, pentad and hexad, who are famously stand-offish with each other. Interesting!

The cell of the 11-cell is an abstract hexad hemi-icosahedron with an apparently pentad interior, most handsomely illustrated by Séquin.{{Sfn|Séquin &amp; Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing rings of polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The 11-cell's 11 hemi-icosahedral cells have 10 triangle faces each; the 5-cell's 5 tetrahedral cells have 10 faces and 10 edges all together. The only way an 11-cell and a 5-cell face can be the same face is if two hemi-icosahedral cells and two tetrahedral cells all meet at the face. Then 3 of each 5-cell's 5 vertices are a hemi-icosahedron face, and its other 2 vertices must be some 11-cell edge lying opposite the face. Coxeter determined that the 11-cell does indeed have an edge opposite each face, but he found that it does not belong to the same hemi-icosahedral cell as the opposing face. He found that the 10 edges opposite each hemi-icosahedron's faces are the 10 edges of the single 5-cell which does not share any vertices, edges or faces with the hemi-icosahedron. Of course there is just one such 5-point 5-cell completely disjoint from each 6-point hemi-icosahedron in the 11-point 11-cell.{{Sfn|Coxeter|1984|p=110|loc=§6. The Petrie polygon [of the 11-cell]|ps=; &quot;We may reasonably call this edge and face ''opposites''. It is easy to find the face opposite to a given edge by looking at the faces to which a given edge belongs. ... Conversely, given a face, we can find the opposite edge by seeing which vertices belong to neither of the hemi-icosahedra which share that face. The ten edges opposite to the ten faces of one hemi-icosahedron are the edges of the complementary &lt;math&gt;a_4&lt;/math&gt; [4-simplex], that is, the joins of all pairs of the five vertices [of the 11-cell] not belonging to the given hemi-icosahedron.&quot;}}

There are 11 disjoint 5-cells in each 11-cell, which is comprised of 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. The real 11-cell is harder to see than the abstract 11-cell that represents it, because the real hemi-icosahedron that the abstract hemi-icosahedron represents is hard to see, but seeing these objects will be easier once we have identified the real hemi-icosahedron and exactly where the 11-cell's real elements reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each hemi-icosahedron's 10 faces come from 10 disjoint 5-cells, and there is just one 5-cell with which it does not share a face.{{Efn|As Coxeter observes (in the previous citation), that unrepresented 5-point (5-cell) is the other 5 vertices of the 11-point (11-cell) that are not vertices of this 6-point (hemi-icosahedron), the hemi-icosahedron's disjoint complement.}} Each 5-cell has 10 faces from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share a face.

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in two different planes which are completely orthogonal to each other. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must be sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness: 8 concentric hulls|2022|loc=Hull #8 (lower right)|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell of the 11-cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Compounds in the 120-cell|§Compounds in the 120-cell]]'' that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software which uses Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s to render the polyhedra which are found in the interior of ''n''-dimensional polytopes.{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the tetrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is a real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (Hull #8) is a concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this {12} irregular great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing {6} irregular great hexagons (truncated triangles) of alternating 120-cell edges and 5-cell edges. The great dodecagon is a great circle of the hemi-icosahedral cell, and also a great circle of the 120-cell, because unlike the cells of most 4-polytopes, the 11-cell's cells are embedded in 4-space at the exact center of the 4-polytope, making them inscribed 4-polytopes themselves.{{Efn|name=Embedding point}} There are 25 great dodecagons and 120 5-cell edges in Moxness's 60-point (Hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (Hull #8) polyhedron altogether.

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point (Hull #8) is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] similar to the one from the catalog shown here,{{Sfn|Piesk: Rhombicosidodecahedron|2018}} but a slightly shallower truncation with smaller red pentagons (120-cell faces) and narrower rhombs. Rhombicosidodecahedra are made by truncating the [[W:Rhombic triacontahedron|rhombic triacontahedron]], which is the 30-sided polyhedron with only one kind of face, the dual of the 30-point icosidodecahedron. The 120-cell contains 120 of these 11-cell cells.]]
Moxness's 60-point (Hull #8) is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (Hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|The 5-point (regular 5-cell) can be another abstraction of Moxness's 60-point (Hull #8), 12-vertices-into-1, but it is not strictly the result of a contraction operation as described by Boole Stott, since she did not apply her expansion and contraction operations to uniform polytopes with more than one edge length.}} In fact Moxness was not the first to find a rhombicosidodecahedron in the 120-cell; Alicia Boole Stott was. It is the 6th section of {5,3,3} beginning with a cell, which Boole Stott identified as the ''e&lt;sub&gt;2&lt;/sub&gt; expansion'' of the icosahedron (or equivalently of its dual polyhedron the dodecahedron).{{Sfn|Boole Stott|1910|loc=§Examples of the e&lt;sub&gt;2&lt;/sub&gt; expansion|p=7}} But that rhombicosidodecahedron identified by Boole Stott is not Moxness's Hull #8: rather, it is the semi-regular Archimedean solid, with a single edge length and square faces. Moxness's Hull #8 with its two distinct edge lengths and rectangular faces is Coxeter's 8th section of {5,3,3} beginning with a cell, which is missing from the sections illustrated by Boole Stott.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than tetrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. The 120-cell contains 120 of Moxness's 60-point (11-cell cells). They are non-disjoint, with 12 60-points sharing each vertex.

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (Hull #8) is the real hemi-icosahedron cell, with 120 5-cell edges in it as chords lurking under its surface. The rhombicosadodecahedron has enough of those 5-cell edges in its interior to make 20 disjoint tetrahedral cells which belong to 20 distinct 5-cells.

The 600-point (120-cell) may be constructed as a semi-regular honeycomb of two kinds of cells, the 120 non-disjoint 60-point (Moxness's rhombicosadodecahedra) and their neighbors the 120 disjoint 5-point (5-cells), cell-bonded together by shared tetrahedral cells of edge-length {{radic|5}}. Each rhombicosadodecahedron is bonded to 20 islanded 5-cells, sharing with each a 5-cell tetrahedron (4 of its interior 5-cell faces, and 6 of its interior 5-cell edges). The rhombicosadodecahedron is also bonded directly to 10 other rhombicosadodecahedra (since each 11-cell has 11 rhombicosadodecahedron cells), again not by sharing one of its exterior faces, but this time by sharing an interior face, a {{radic|6}} triangle which lies in a central plane.{{Efn|The {{radic|6}} chord is another hidden chord lurking beneath the 60-point rhombicosadodecahedron's surface, like the {{radic|5}} 5-cell edge. Both these hidden chords lie on the same great circle with the 120-cell edges (the pentagon face edges). Three {{radic|6}} chords make a triangle which lies in a central plane. This {{radic|6}} great triangle must not be confused with the {{radic|5}} face triangle of the 5-cell, which does not lie in a central plane.}} The 120 rhombicosadodecahedra are also bonded directly together at ''all'' their exterior faces: at their pentagon faces (which are 120-cell faces of dodecahedral cells); at their small triangular faces (which are 600-cell faces of small tetrahedral cells, 10 of which are inscribed in each dodecahedral cell); and at their rectangular faces (which are faces of 10-point [[w:Pentagonal_prism|pentagonal prism]] cells, where the pentagonal prism is a column inscribed in a dodecahedral cell, joining its opposite pentagon faces). The 120 rhombicosadodecahedra are bonded directly to each other at their own exterior faces; no dodecahedra, small tetrahedra, or pentagonal prisms lie between them, and those objects are not cells of this honeycomb; we mention them here only to help locate the rhombicosadodecahedra within the 120-cell. The only objects which fill holes between the rhombicosadodecahedra are the 120 islanded 5-cells.{{Efn|This honeycomb of the 120-cell resembles the [[600-cell#Icosahedra|honeycomb of the 600-cell consisting of 24 face-bonded regular icosahedra]] with 24 clusters of 5 tetrahedra filling the holes between them. In the 120-cell there are 5 times as many of everything as in the 600-cell, so there are 120 (rhombicosadodecahedron) icosahedral cells with 600 tetrahedra filling the holes between them.}}

It is possible for rhombicosadodecahedra to meet each other at more than one kind of exterior face because they are not meeting there as cells of the same 11-cell; they belong to different 11-cells. The 11 rhombicosadodecahedra in the same 11-cell do not meet at their own exterior faces, they meet at their interior {{radic|6}} triangle faces. As a cellular 4-polytope, the 11-cell is inside-out. Its cells meet each other at interior planes, sharing interior {{radic|6}} triangle faces, instead of in the usual way that 4-polytopes' cells share an exterior face. It is a deeper kind of intertwining to be sure, but the 11 rhombicosadodecahedra in the same 11-cell are still volumetrically disjoint cells, and the 11-cell is a perfectly real 4-polytope with real cells. It is a semi-regular 4-polytope, because it has two different kinds of cells: 11 rhombicosadodecahedra and 11 5-cells. The 11 5-cells are islanded (not bound to each other), and the 11 rhombicosadodecahedra are bound both to 5-cells (at {{radic|5}} 5-cell faces) and to each other (at {{radic|6}} central triangle faces). In the section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#What's in the box|§What's in the box]]'', below, we will look closely at the relation between the {{radic|5}} chord and the {{radic|6}} chord, which is the relation between the pentad and the hexad, and at the general phenomenon of hemi-polytopes which meet each other at a central plane. Then in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Cell rings of the 11-cells|§Cell rings of the 11-cells]]'' we will look at the two-cell cellular structure of the 11-cell in more detail.

Considered as a single complex of overlapping cells rather than as distinct honeycombs of volumetrically disjoint cells, the 120-cell contains 60-point (rhombicosadodecahedral cells), islanded 5-point (5-cells), 20-point (dodecahedral cells), and 10-point ([[w:Pentagonal_prism|pentagonal prism]] cells). The 60-points meet each other in pairs at their pentagon faces, meet regular dodecahedra at those pentagon faces also, meet each other in quads at their 600-cell faces where they share an interior 4-point (600-cell tetrahedral cell) and its four faces (just as they share a larger interior 5-cell tetrahedron and its faces), and meet each other and the 10-point (pentagonal prisms) at their rectangle faces. We might try to imagine such a complex with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with parts truncated from 30 neighboring truncated dodecahedra pentagonal prisms). There are 120 of each of these 3 kinds of dodecahedron. 

Moxness's 60-point (rhombicosadodecahedron cell) is a central object of the 120-cell, and contains many wonders. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra by expansion and contraction operations, as Alicia Boole Stott discovered before 1910,{{Sfn|Polo-Blanco: ''Theory and history of geometric models of Alicia Boole Stott''|2007|loc=§5.3.2 1910 paper on semi-regular polytopes|pp=152-158|ps=; summarizes Boole Stott's method and results from {{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}}, including her tables of expansion-contraction dimensional analogies and a few of her illustrations.}} and those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, rediscovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter &amp; Petrie|1938|p=4|loc=''[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]''}}|name=Concentric polyhedra}} Before we move on from the 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice one more thing about the hidden {{radic|5}} chords lurking below its surface. The 12 little pentagon faces are spanned by sets of 6 {{radic|5}} chords (5-cell edges). The 6 chords of each set are disjoint (they don't touch or form 5-cell faces), but they are symmetrically arranged in 3 parallel pairs, {{radic|3}} apart, which lie in 3 orthogonal central planes. It turns out they are the 6 reflex edges of a 12-point non-convex polyhedron called the [[w:Jessen's_icosahedron|Jessen's icosahedron]], inscribed in the 60-point cell. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again and work our way outwards, before eventually in ''[[#The perfection of Fuller's cyclic design|§The perfection of Fuller's cyclic design]]'' we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cell)s ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks, showing its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Pentahemidemicube.png|120px]]
|[[File:Tetrahemihexahedron_rotation.gif|120px]]
|[[File:Pentahemicosahedron.png|120px]]
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts needed to build the 120-cell (the picture on the cover of the box), and everything else that's invisible in that picture which fits inside the 120-cell: all the astonishing regular 4-polytopes. The pentad building block is half a pair of orthogonal 5-point (regular 5-cells), the hexad building block is half an 8-point (16-cell), and the heptad building block is a quarter of an 11-point (11-cell). Everything can be built from these blocks by embedding them at the center of the 3-sphere, inscribing them as 4-polytopes to make compounds as with the 16-cell building block, rather than by assembling them face-to-face as 3-polytopes to make honeycombs in which they are cells or vertex figures.{{Efn|name=Embedding point}}

The pentad building block is the 5-point '''pentahemidemicube''', an abstract polyhedron with 10 equilateral triangle faces, 10 {{radic|5}} edges, and 5 vertices.{{Sfn|Christie: Pentahemidemicube|2024|loc=File:Pentahemidemicube.png|ps=; &quot;The 5-point ''pentahemidemicube'' is an abstract polyhedron with 10 equilateral triangle faces, 10 edges, and 5 vertices. It has a real presentation as the regular 5-cell.&quot;}} It has a real presentation as the 5-point (regular 5-cell 4-polytope), a regular tetrahedron expanded by addition of a fifth vertex equidistant from all the others in a fourth dimension. The pentad block is a tetrahedral pyramid (a 5-cell) inscribed in a cube: a tetrahedron inscribed in a cube (a demicube) expanded by addition of a fifth vertex, one of the cube's other 4 vertices. Each pentad (5-cell) building block lies in the 120-cell orthogonal to three other pentad (5-cell) building blocks. They are four completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges. Each regular 5-cell in the 120-cell is a pentad (pentahemidemicube) building block, in one of 30 orthogonal sets of 4. There are 120 pentad building blocks in the box.

Every vertex of the 16-cell, and therefore every vertex of the 8-cell, 24-cell, 600-cell and 120-cell, is surrounded by a 6-point (octahedral vertex figure). The hexad building block is that 6-point octahedron embedded at the center of the 4-polytope instead of at a vertex, that is, inscribed in the 3-sphere as a 4-polytope, the 6-point '''[[w:Tetrahemihexahedron|tetrahemihexahedron]]''' or '''hemi-cuboctahedron'''.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; &quot;The tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U&lt;sub&gt;4&lt;/sub&gt;. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices.&quot;}} The hexads have 4 red triangle 5-cell faces, bounded by 12 {{radic|5}} edges. Their 3 yellow square faces lie in central planes, because the building block is an abstract 4-polytope. The hexad is an irregular abstract polyhedron that also has 6 {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

Two orthogonal 5-point (pentad) building blocks intersect in a 4-point (inscribed tetrahedron) to form a 6-point (hexad) building block. Two orthogonal 6-point (hexad) building blocks intersect in a 4-point (central square) to form an 8-point (16-cell) building block, the most usable building block, from which everything larger can be built by compounding.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 120 pentads to pair with a completely orthogonal hexad, and although there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed) they only result in 120 pairs two different ways, so there are 120 11-cells. The 11-cell is 6 pentads' and 5 hexads' smallest joint compound, and the 120-cell is their largest, 120 pentads and 100 hexads.

The heptad{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''|pages=22–31}} building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|4}} edges and 9 {{radic|5}} edges), and 7 vertices.{{Sfn|Christie: Pentahemicosahedron|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;}} It is a compound of a 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]) and a 6-point (hexad [[W:hemi-icosahedron|hemi-icosahedron]]) intersecting in a 4-point (tetrahedron), that is, of three orthogonal 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]&lt;nowiki/&gt;s) intersecting in their common 4-point (tetrahedron). It is also the abstract cube contracted by removal of one vertex, which reduces one of the cube's two inscribed tetrahedra (demicubes) to a single remaining triangular face. As with all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Four &lt;small&gt;&lt;math&gt;(11-7)&lt;/math&gt;&lt;/small&gt; orthogonal 7-point (pentahemicosahedron) blocks can be joined at a common Petrie polygon (a skew hexagon which contains their orthogonal 6-edge paths) to construct a real 11-point ([[W:11-cell|11-cell]] semi-regular 4-polytope), which magically contains six &lt;small&gt;&lt;math&gt;(11-5)&lt;/math&gt;&lt;/small&gt; 5-point (1-pentad regular decahedra) as cells and five &lt;small&gt;&lt;math&gt;(11-6)&lt;/math&gt;&lt;/small&gt; 12-point (2-hexad semi-regular cuboctahedra) as cells. There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like [[W:Euclid's postulates|Euclid's postulates]], and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as honeycombs of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles. It possesses the complete symmetry of the polytope without any redundancy, because it contains one of each of the polytope's characteristic root elements.{{Efn|A [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be partitioned radially by its planes of symmetry (Coxeter's &quot;reflecting circles&quot;) into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated rotationally to generate its regular 4-polytope because it is the complete [[W:gene|gene]] for it, containing all of its elements and capturing all of its symmetry without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring that triggers the reflections.|name=Characteristic orthoscheme}} It is the gene for the polytope, which can be replicated to construct the polytope.

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its &lt;math&gt;A_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the regular compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;small&gt;&lt;math&gt;120\times 120 = 14400&lt;/math&gt;&lt;/small&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[16--cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its &lt;math&gt;B_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 384. The 120-cell is the convex hull of the regular compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 24-point (24-cell) is subdivided into 1152 instances of its [[24-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;F_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 1152. The 120-cell is the convex hull of the regular compound of 25 disjoint 24-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;25\times 1152 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has three equivalent origins in distinct root systems, though most are obviously constructed from one root, and not so obviously from the others. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything larger than the pentad can be constructed by compounding hexads, and we shall see that everything as large as a 24-cell has a construction from heptads which are a product of a pentad and a hexad. Surely every uniform polytope can be constructed by some function of expansions and contractions from any of three root systems, however indirect the recipe. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes, related by an equals sign between them, then give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics by [[W:Noether's theorem|Noether's theorem]].

=== Pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 octahedral cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell contains the 600 octahedral cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It also contains the 600 cubical cells of its 75 8-cells. It is a 600-cell in three ways, although the octahedra and cubes are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell and the 8-cell, respectively.|name=600 cells three ways}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (1-pentad 5-cell), out of the 8-point (2-hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (1-pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope that contains 5-point (1-pentad 5-cell) building blocks beside the largest one, it just isn't a ''regular'' 4-polytope. It is semi-regular with two kinds of cells, the 4-point (regular tetrahedron) and the 12-point (truncated cuboctahedron), which have just one kind of regular face between them, a 5-cell face. That semi-regular 4-polytope is the 11-cell.

We said earlier that the 11-cell had 11 disjoint 5-cells and 11 hemi-icosahedral cells in it, and that its construction from them might be hard to see, but now we are going to account numerically, at least, for how it can have 6 pentad cells plus 5 hexad cells magically adding to 11 of the same kind of elevenad cell. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, 11. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (5-cell regular decahedron) cells, and the 12 6-point (hexad) building blocks into 100 12-point (regular icosahedron) cells, and then magically adding one whole 5-point (5-cell regular decahedron) cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 12-point (regular icosahedron) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such (cell bundles of 6 + 5 = 11 cells), with each 11-cell bundle containing 11 of both kinds of cell. Very, very strange!{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden'': §II To Add is to Multiply|p=38|ps=; the sequence of regular polygons with harmonious proportions beginning {5}, {7}, {9}, {11} as a family of natural numbers; &quot;The numbers in this optimal family have remarkable properties [that allow] similar figures to be arranged easily or allow a figure to be dissected into a set of similar figures. This is the origin of the Greek idea of geometric progression — that ''multiplication can be accomplished instead by addition''.... For instance, it is well known that the square of the golden ratio is one more than itself (written above as 𝜙&lt;sup&gt;2&lt;/sup&gt; = 𝜙 + 1) and one less than 𝜙 is its reciprocal (1/𝜙 = 𝜙 - 1). The first relation equates multiplication with addition, while the second accomplishes division through subtraction. [These are pentagon {5} bisectional golden ratios, but] the heptagonal [{7} tri-sectional] ratios also behave in surprising ways — adding to multiply, subtracting to divide.... There are analogous constructions for [nonagon {9}] quadrisectional and [hendecagon {11}] pentasectional systems.&quot;}}

The 120-cell has three interchangeable cyclic expansion-contraction cycles, the pentad, hexad and heptad cyclic designs, rooted in the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems, respectively. Each of them constructs the whole sequence of regular 4-polytopes from a single building block. The key to all three cycles is that 5 + 6 = 11, and there is an 11-point quasi-regular 4-polytope in the sequence of convex 4-polytopes. It lies between the 8-point (16-cell) and the 16-point (8-cell), and is itself a &lt;math&gt;B_4&lt;/math&gt; polytope. Its role is to be a unified single building block common to all three cyclic expansion-contraction cycles. It is involved with the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems and expresses in one building block their relations to each other.

We have already seen how the hexad building cycle works. It is plain compounding of 4-orthoplexes at the beginning, increasing to densely intricate non-disjoint symmetry toward the end, making it difficult to see why it works once past the 24-cell. This is how the build works, in terms of sixes. The 16-cell is a 2-hexad (3 pairs of completely orthogonal great square planes). The 24-cell is 3 16-cells and a 4-hexad, 4 different ways: 16 non-disjoint hexads (great hexagon planes). The 120-point (600-cell) is 25 24-cells and a 20-hexad, 10 different ways: 200 non-disjoint hexads (great hexagon planes). The 600-point (120-cell) is 225 (9 times 25) 24-cells. ....

The pentad building cycle doesn't do plain compounding to regular 4-polytopes until the last one, the 120-cell. Fives don't even appear as pentagons until after the 24-cell. We have seen how there are 11 5-cells in the quasi-regular 11-cell, which occurs before the 8-cell, and also how everything can be compounded from irregular 5-cells (4-orthoschemes specialized for each 4-polytope). It is still not evident how a pentad cycle can work to build all the regular 4-polytopes from the regular 4-simplex. This is how is how the build works, in terms of fives. ....

This is how the build works, in terms of elevens. The 11-cell is 4 orthogonal heptads (intersecting in their skew hexagon Petrie polygon). The 24-cell is also 4 heptads (intersecting in 4 orthogonal digon axes, so each heptad contributes 6 unique points to a single fibration of 24 points). Heptads are quarter elevenads, so the 11-cell and 24-cell are both 1 elevenad. ....

See the tabular sequence of convex 4-polytopes in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Build with the blocks|§Build with the blocks]]'', below: table rows ''Inscribed, Pentad, Hexad, Heptad''.

=== 11 of them ===
There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of eleven 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum eleven of them. There is only a single Hopf fibration of the 11 great circles of an 11-cell. 

The 5-cell and the 11-cell are limit cases. The 5-cells are completely disjoint, but the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 10 other instances of itself.

== Cell rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer &amp; Segerman|2013|loc=''Puzzling the 120-cell''}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his section of the Wikipedia article on the 120-cell entitled ''[[120-cell#Visualization|§Visualization]]''.{{Sfn|Ruen &amp; Goucher et al. eds. 120-cell|2024|loc=Goucher's ''[[120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious place to look for the 11-cell's Hopf map is in Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other place it could be hiding. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The quasi-regular convex 11-cell|§The quasi-regular convex 11-cell]]'', and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen: triacontagon|2011|loc=''Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them except his illustrations. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Four orthogonal projections of the 120-cell to a central plane, selected from the table of 3 regular and 11 compound 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|100 icosahedra&lt;br&gt;120 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|5 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 icosahedra&lt;br&gt;6 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|120
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|4
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|50
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|7.2°
|align=center|36°
|align=center|6°
|align=center|7.2°
|}

....

Here we summarize our findings on the 11-cells so far, including in advance some results to be obtained in the next section two sections.

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section....

== The 15 major chords ring everything ==
The natural numbers are each a distinct flavor.

Each integer has its own distinct taste. The first six integers {0..5} are the six strongest, most familiar tastes in nature, instantly distinguished and utterly unique.

The seventh flavor {6} is the first ratio of two of the others. Thereafter follows in strange sequence all the ratios of all the flavors, each marriage of every two flavors, most of which are fractions &lt;math&gt;\{k/d\}&lt;/math&gt;, but some of which are integers &lt;math&gt;\{k\}&lt;/math&gt;.

Some of these ratios are successful flavors in their own right, as distinct and inimitable as the marriage of chocolate and oranges, but the marriage of many flavor pairs is mud. It is a rule of thumb among flavor specialists that nearly any taste can be successfully alloyed with either onions or chocolate, but the direct marriage of these two flavors is notably unsuccessful. Some flavors make famous pairings or infamous ones, and subtle new pairings are always to be discovered that work surprisingly well for some purpose, but the majority of flavor marriages are undistinguished compounds.

In polytope geometry, each chord of a polytope is both is a distinct 1-dimensional object, a chord of the sphere of a distinct length &lt;math&gt;l&lt;/math&gt;, and a distinct natural number &lt;math&gt;h&lt;/math&gt;, a unique flavor. If the polytope is regular, it is a noteworthy and successful flavor. The chord's length &lt;math&gt;l&lt;/math&gt; is a square root, related to the natural number &lt;math&gt;h = k/d&lt;/math&gt; and to a set of polytopes they both represent, by a formula discovered by Steinbach.{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden''; Figure 5. Optimal sections and proportions|p=37|ps=; the regular polygons {5}, {7}, {9} and {11} with their diagonals define respectively: {5} the golden bisection proportional to 𝜙; {7} an analogous trisection; {9} an analogous quadrasection; {11} an analogous pentasection.}} The chord length &lt;math&gt;l&lt;/math&gt; is related to the number of sides of the regular polygon formed by the chord (its numerator &lt;math&gt;k&lt;/math&gt;), and to the winding number or density of its regular skew Petrie polygon (its denominator &lt;math&gt;d&lt;/math&gt;).

The regular 4-polytopes do not contain an infinite variety of these chord flavors, but only 30 of them: only 15 unique pairs of 180° degree complementary chords, which combine to make every regular ''n''-dimensional thing found in nature.{{Efn|The 15 ''major chords'' suffice to construct all the regular ''n''-polytopes. Their 180° complements include 15 ''minor chords'' which occur only in the 120-cell, not in any smaller regular polytope.{{Efn|name=Major and minor chords}} Including the complement of the 180° {{radic|4}} digon {2} chord (the {0} chord {{radic|0}}), and the skew variant of the 90° {{radic|2}} square {4} chord (the {30/7} chord), there are 32 distinct natural numbers which are all the chords of all the regular polytopes in all dimensions.|name=32 chords}} All the regular polytopes (including the largest one on the cover of the box) can be built not only from a box containing a number of blocks of 3 shapes glued together by gravity, they can be also be built from a box containing a number of sticks of 15 lengths glued together by rubber joints.

[[File:15 major chords.png|500px|thumb|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the others occur only in the interior of 120-cell, not as edges of any of the regular 4-polytopes. There are 30 distinct 4-space chordal distances between vertices within the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to identify any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=Major and minor chords}} chords are properly labelled #1 — #15 in the order they fan out by size.{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''; Figure 3|ps=; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with a &quot;fan of chords&quot; diagram like the one here.|p=23}} They join vertex pairs which are 1 — 15 edges apart on a &lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon. A #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; chord forms &lt;small&gt;&lt;math&gt;f+1&lt;/math&gt;&lt;/small&gt; regular geodesic polygons which include a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; disjoint regular {{nowrap|&lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt;-gons}} (e.g. &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; a compound of 5 disjoint hexagons) plus a single [[W:Triacontagon#Triacontagram|regular skew {{nowrap|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram}}]] that wraps around the sphere &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times before closing its circle. Notice the maverick #11 chord, with its non-conforming origin point for its {{nowrap|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;}} polygram.]]

These 15 chords (natural numbers) include the integers 2, 3, 4, 5, 6, 10, 11, 15 and 30. The others are the fractions 5/2, 10/3, 13/7, 15/2, 15/4, and 15/7. Represented among the chord lengths (the square roots) measured in ''unit radii'' (not in {{radic|2}} radii, as metrics are often given elsewhere in this article) are the {{background color|paleturquoise|roots of the first 4 positive integers {{radic|1}}, {{radic|2}}, {{radic|3}} and {{radic|4}}}}, the {{background color|#FFCCCC|root of the very distinguished rational fraction {{radic|5/2}}}}, and the roots of various distinguished irrational fractions as Steinbach discovered: for example, 4 of them are {{background color|yellow|roots of golden numbers (a small integer ± 𝜙)}}. The 15 chords each form their regular polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, which is just &lt;small&gt;&lt;math&gt;\{k\}&lt;/math&gt;&lt;/small&gt; in 8 cases. Arranged in order #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; from shortest edge-length square root to longest, the #1 — #15 chords bridge vertex pairs which are &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; vertices apart on a [[W:Triacontagon#Petrie polygons|&lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon]]. Each chord makes a [[W:Triacontagon#Triacontagram|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; polygram]] of its edge length, a compound of the smaller &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. The 15 distinct polygons form a sequence you would not anticipate exactly unless you have Steinbach's formula:

{| style=&quot;background-color:transparent; white-space:nowrap;&quot;
|-
| {30/1} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|.073}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{30} 
| 120-cell edges
|-
| {30/2}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.191}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15}
|- style=&quot;background: yellow;&quot;|
| {30/3}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.382}} 
| &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi}&lt;/math&gt;&lt;/small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{10}''' 
| 600-cell edges
|-
| {30/4} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.573}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/2}
|- style=&quot;background: paleturquoise;&quot;|
| {30/5}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|1}}'''
| &lt;math&gt;1&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{6}''' 
| 24-cell, 8-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/6}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|1.382}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{5}'''
|- style=&quot;background: paleturquoise;&quot;|
| {30/7}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|2}}'''
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{4}''' 
| 16-cell edges
|- style=&quot;background: #FFCCCC;&quot;|
| {30/8}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.5}} 
| &lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;&lt;small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/4} 
| 5-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/9}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.618}} 
| &lt;math&gt;\phi&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{10/3}
|- style=&quot;background: paleturquoise;&quot;|
| {30/10}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|3}}''' 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{3}'''
|-
| {30/11}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.426}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{11/11}
|- style=&quot;background: yellow;&quot;|
| {30/12}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.618}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{5/2}
|-
| {30/13}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.810}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{13/7}
|-
| {30/14}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.928}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/7}
|- style=&quot;background: paleturquoise;&quot;|
| {30/15}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|4}}''' 
| &lt;math&gt;2&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{2}'''
|}

{| class=wikitable style=&quot;white-space:nowrap; float:right;&quot;
!colspan=6|Natural number &lt;small&gt;&lt;math&gt;h=\{k/d\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Fibers&lt;br&gt;&lt;small&gt;&lt;math&gt;f\{h\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Cells&lt;br&gt;&lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|4-tope&lt;br&gt;&lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Verts&lt;br&gt;&lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|[[W:Triacontagon#Triacontagram|30-gram]]&lt;br&gt;&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;
|-
!Chord
!Arc
!colspan=2|&lt;small&gt;&lt;math&gt;l/\sqrt{1}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;l/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;h&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#1
|15.5~°
|{{radic|0.073}}
|0.270
|{{radic|0.146}}
|&lt;small&gt;&lt;math&gt;30&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1\{30\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;5\{3,3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/1\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#2
|25.2~°
|{{radic|0.191}}
|0.437
|{{radic|0.382}}
|&lt;small&gt;&lt;math&gt;15&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;..\{3,3\}..&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/2\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#3
|36°
|{{radic|0.382}}
|0.618
|{{radic|0.764}}
|&lt;small&gt;&lt;math&gt;10&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;30\{3,3\}20&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/3\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#4
|44.5~°
|{{radic|0.573}}
|0.757
|{{radic|1.146}}
|&lt;small&gt;&lt;math&gt;15/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/4\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5
|60°
|{{radic|1}}
|1
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;6&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/5\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#6
|72°
|{{radic|1.382}}
|1.175
|{{radic|2.624}}
|&lt;small&gt;&lt;math&gt;5&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}} 
|&lt;small&gt;&lt;math&gt;10\{5,3\}12&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/6\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#7
|90°
|{{radic|2}}
|1.414
|{{radic|4}}
|&lt;small&gt;&lt;math&gt;4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7\{4\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;4\{4,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;9\{3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/7\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: #FFCCCC;&quot;|
|#8
|104.5~°
|{{radic|2.5}}
|1.581
|{{radic|5}}
|&lt;small&gt;&lt;math&gt;15/4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/4\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}} 
|&lt;small&gt;&lt;math&gt;15\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/8\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#9
|108°
|{{radic|2.618}}
|1.618
|{{radic|5.236}}
|&lt;small&gt;&lt;math&gt;10/3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10/3\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/9\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#10
|120°
|{{radic|3}}
|1.732
|{{radic|6}}
|&lt;small&gt;&lt;math&gt;3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;10\{3\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/10\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#11
|135.5~°
|{{radic|3.426}}
|1.851~
|{{radic|6.852}}
|&lt;small&gt;&lt;math&gt;11/11&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;11\{11/11\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;11\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#12
|144°
|{{radic|3.618}}
|1.902
|{{radic|7.235}}
|&lt;small&gt;&lt;math&gt;5/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2},3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/12\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#13
|154.8~°
|{{radic|3.810}}
|1.952
|{{radic|7.621}}
|&lt;small&gt;&lt;math&gt;13/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;13\{13/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;7\{5,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/13\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#14
|164.5~°
|{{radic|3.928}}
|1.982
|{{radic|7.857}}
|&lt;small&gt;&lt;math&gt;15/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;137\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/14\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{radic|8}}
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15\{2\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐𝜙&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;8\{3,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/15\}&lt;/math&gt;&lt;/small&gt;
|}

The polygons '''{10}''', '''{6}''', '''{5}''', '''{4}''', '''{3}''', and '''{2}''' are flat polygons that each lie on a great circle in a central plane of the 3-sphere. The polygons {30}, {15}, {15/2}, {15/4}, {10/3}, {11/11}, {5/2}, {13/7}, {15/7} and all the {30/''n''} polygons are skew polygrams spiraling through curved 3-space, each successive edge in a different central plane of the 3-sphere. Projected to the 1-sphere (a circle in the plane), they are [[W:Star polygon#Regular star polygon|regular star polygrams]], the simplest being the pentagram {5/2}. In 4-space they wind around the 3-sphere more than once ''without self-intersecting'', before they close their circuit. They lie on helical geodesic circles or ''isoclines'' of circumference greater than 2𝝅''r''.

Even the 6 integer chords which form flat polygons also form skew polygrams. Each of the 15 chords occurs as a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, a fiber bundle of Clifford parallel polygons that totals 30 vertices per bundle. Each chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; also forms a single skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram that winds through another, shorter chord's bundle of parallel polygons, intersecting them in succession multiple times each, and weaving that whole bundle of parallel fibers together as the cross-thread does on a loom. The polygram cross-thread has to wind multiple times through the whole bundle (&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times around the [[W:Triacontagon#Triacontagram|30-gram]] in which each edge connects every &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt;th vertex) to complete an enumeration of all 30 vertices in the bundle. For example, the #5 chord forms hexagon &lt;small&gt;&lt;math&gt;\{6\}&lt;/math&gt;&lt;/small&gt; flat planes in parallel bundles of 5, written &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt;, and also a skew &lt;small&gt;&lt;math&gt;\{15/2\}&lt;/math&gt;&lt;/small&gt;-gram of #5 edges which winds 2 times through the #3 chord's bundle of 3 decagon &lt;small&gt;&lt;math&gt;\{10\}&lt;/math&gt;&lt;/small&gt; flat planes. Every chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; forms both its &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram and its fiber bundle of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, and the edges of one chord's parallel polygons are also the edges of its skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram weaving through some shorter chord's bundle of parallel polygons. Each numbered chord is a natural number, a ratio of two integers and their polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, either flat or skew. Each chord number defines a &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; skew polygon that is a compound bundle of the &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. All these flat or skew polygons are regular polygons made of their numerator instances of a single chord #, whose length is that square root, Steinbach's function of the natural number.

The relationship between each bundle of flat Clifford parallel great circle polygons of a distinct length, and its corresponding bundle of skew polygrams of a longer length, is precisely that of a discrete simple rotation and its corresponding discrete isoclinic rotation. For example, the flat great circle hexagons of edge length {{radic|1}} lie in the invariant rotation planes of the 24-cell's characteristic 60° isoclinic rotation, in which the vertices move on skew hexagram rotation paths of edge length {{radic|3}}. The 24-cell contains 4 bundles of 4 Clifford parallel (disjoint) hexagon central planes each. The 24 vertices of the 24-cell in each bundle are linked by rotation in the 4 hexagon invariant planes in two different ways: simply and isoclinically. In a simple rotation the vertices of a single invariant hexagon exchange places on that great circle hexagon of edge {{radic|1}}. In an isoclinic rotation in which any one of these 4 disjoint hexagon central planes is invariant, all 4 of them are necessarily invariant planes in the rotation, because they rotate in parallel. In that isoclinic rotation the vertices of all 4 hexagon invariant planes (all 24 vertices of the 24-cell) exchange places concurrently, moving along 4 disjoint helical hexagrams of edge {{radic|3}} in parallel. It is a characteristic only of this distinct rotation in hexagon invariant planes that the helical vertex path or ''isocline'' is a hexagram of period 6, edge length {{radic|3}}, and circumference 720°. Rotations in other kinds of polygon invariant planes have different characteristic isoclines. The period of their isoclinic rotation must be ''the same or greater'' than the period of their simple rotation; in most cases (unlike this one) it is greater.

The 15 chords form not only distinct fiber bundles of polygons, but also distinct bundles of polyhedra: cell rings &lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt; and compound vertex figures &lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;. These combine to form their characteristic regular 4-polytope &lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Coxeter|1973|p=269|loc=Compounds|ps=; &quot;It is remarkable that the vertices of {5, 3, 3} include the vertices of all the other fifteen regular polytopes in four dimensions.&quot;}} All these elements of a regular 4-polytope constructed from one chord are themselves regular polytopes, and each of the 16 regular 4-polytopes is a function of one characteristic chord. If the regular 4-polytope is convex, its characteristic chord is its unit-radius edge. For a non-convex regular 4-polytope, the characteristic chord is its unit-radius ''mid-edge diameter'', the distance from the midpoint of each edge to the midpoint of the completely orthogonal edge. Each of the characteristic chords (each distinct natural number) has its expression as one of the 16 regular 4-polytopes and all the products of the distinct regular 1-polytopes, 2-polytopes and 3-polytopes that construct it, and the whole distinct construction from the natural number to the 4-polytope is an [[W:SO(4)|SO(4) symmetry group]] expression of that 4-polytope's characteristic discrete [[W:SO(4)#Isoclinic rotations|isoclinic rotation]], in the invariant central planes of its characteristic chord, by the characteristic chord arc.

...demonstrate this theory by proper enumeration of the chord lengths in the table...

...color-code the △ ☐ 𝜙 ✩ symbols ... chord colors in diagram should match them ... some of the diagram colors are wrong and must be changed, but otherwise diagram is pretty good ... i think in a colored diagram all chords of the same length should have the same size line dash, the polygons are not obvious enough and the 15 fanning-out chords are readily identified by their labels...

... add back the counts of chords in the box (very small font number under each table entry), and present these two images as the front and back a box of building sticks and rubber joints...What's in the other box...

...say the rational-root major chords {{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|5/2}} are the ''n''-edges: their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge...

...abbreviated text of the [[120-cell#Relationships among interior polytopes|''§Relationships among interior polytopes'']] section from the 120-cell article...to explain the color-coding

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius=1}}

See also the elements, properties and metrics of the sequence of regular 4-polytopes summarized in tabular form in the section ''[[#Build with the blocks|§Build with the blocks]]'', below.

== Alicia Boole Stott's original formulation of dimensional analogy ==
&lt;blockquote&gt;&quot;...a method by which bodies having a certain kind of semi-regularity may be derived from regular bodies in an Euclidean space of any number of dimensions...&quot;{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''; presents cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues|pp=12-45}}&lt;/blockquote&gt;

Alica Boole Stott's 1910 paper reads like an exceptionally well-written patent application. It is a comprehensive stepwise description of her original invention, a general method of dimensional analogy between regular and semiregular polytopes of the same edge length.

[[File:The Roots of Three.png|thumb|An illustration of Boole Stott's expansion of unit-edge triangle to hexagon captures the geometric basis in 3 of the &lt;math&gt;A_n&lt;/math&gt;root system.]]

The engineering-like articulation of her new principles is entirely discrete and group-theoretic. Boole Stott repeatedly refers to sets of polytope elements as ''groups''. All her transformations are discrete instances of one or another special case of her &quot;two inverse operations which may be called ''expansion'' and ''contraction''.&quot; Hers is an exploration of ''n''-dimensional space at its most basic: space defined by the discrete objects which fill it, themselves defined directly as groups of elements with a few operations on them, which transform them into each other. This is geometry which seems to precede Euclid's postulates, and does not depend on them. Instead it corresponds directly to the deeper group theory mathematics which underlies geometry, even though Boole Stott, like the ancients, did not have that mathematics.

Boole Stott enumerated her expansion and contraction operations as transformations between regular and semi-regular polytopes of a single edge length. In her 1910 paper she did not consider any polytopes of multiple edge lengths, or expansions of the subject polytope by a distance other than that exactly required to produce new edges of the original edge length, so of course she did not enumerate Moxness's non-uniform rhombicosidodecahedron (Hull #8). This may be why section 8&lt;sub&gt;3&lt;/sub&gt; is missing from her earlier drawings of the sections of {5,3,3}; she may not have had a construction for it.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The Boole Stott expansion (or contraction) operation is as precisely defined a transformation as Coxeter's reflection, rotation and translation operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions). Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;
where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;
where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;
For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} An instance of the expansion (or contraction) operation is an application of the general operation to a particular subject polytope, operating on all of its elements symmetrically at once. The general operation is restricted, such that one may only expand (or contract) the subject polytope by pushing all its edges, faces or cells directly away from (or toward) its center, without changing the size of the moved elements. This splits each vertex into several (or coalesces several vertices into one), and has corresponding effects on the other elements, producing new edges ''of the same size'' between pairs of separated vertices (or reducing some edges to single vertices). The distance all the elements are moved away from (or toward) the center must be such that the vertices separate producing new edges of only the original edge length (or come together reducing edges to single vertices). It is permitted to combine these three operations, e.g. by pushing the edges ''and'' the faces, or by pushing all three element sets, provided the combination results in a semi-regular polytope of only one edge length. The three operations are all commutative with one another, so combined operations can be considered either as a sequence of expansions (or contractions) linking several related polytopes, or as their concurrent movements, a single transformation of the subject polytope. Since there are 8 combinations of one to three operations there are 8 polytope variants (including the original regular polytope), and all are adjacent to one another by a single transformation. The variants of the original regular polytope are named for which element set(s) are moved. By enumerating all 8 variants of expansion (and contraction) for each of the six regular convex 4-polytopes, Boole Stott produced a table of regular and semi-regular 4-polytopes divided into 12 sets of 8: six expansion sets and six contraction sets.{{Sfn|Boole Stott|1910|loc=''Table of Polytopes in S&lt;sub&gt;4&lt;/sub&gt;''; note that Boole Stott's usage of ''S&lt;sub&gt;4&lt;/sub&gt;'' denotes a Euclidean 4-space ''E&lt;sub&gt;4&lt;/sub&gt;'', or a 3-sphere space embedded in it, ''not'' a 4-sphere space embedded in a Euclidean 5-space|p=30}} Each set includes just one of the six regular convex 4-polytopes, with its distinct unit-radius edge length, and the 7 possible semi-regular expansions (or contractions) of it, which all preserve that characteristic edge length. The polytopes in each set of 8 all have the same total number of elements as their original regular polytope, but each polytope in the set has a distinct combination of cell, face, and vertex figures. The 12 sets are not completely disjoint, and even within one set the 8 operations do not necessarily produce 8 unique polytopes. For example, the regular 5-cell (the 4-simplex polytope) has 5 cells, 10 faces, 10 edges, and 5 vertices (30 elements). Expanding the 5-cell produces 7 related semi-regular 4-polytopes of 30 elements each (two of which happen to be the same polytope). In each expansion, the 5 cells have expanded into semi-regular or regular polyhedra, the 10 faces and 10 edges ''may'' have expanded into prisms, and the 5 original vertices have expanded into regular or semi-regular cells.

== The regular 11-cell 4-polytope ==

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosadodecahedron). Each non-convex Jessen's icosahedron is a set of 6 reflex edges that are {{radic|5}} 5-cell edges, each set symmetrically arranged in 3 parallel pairs, not as far apart ({{radic|3}}) as they are long, in 3 orthogonal central planes. The 12-point (Jessen's sets of 6 reflex 5-cell edges) are inscribed in the 60-point with their endpoints as a vertex of a pentagon face. The 12 pentagon faces are joined to 5 other pentagon faces by the 5-cell edges; not to their nearest neighbor pentagons but to their next-to-nearest neighbor pentagons; and since there are 5 vertices to each pentagon face, there are 10 12-point (Jessen's sets of 6 reflex 5-cell edges). Each pentagon face is joined to another pentagon face by 5 reflex edges of five distinct Jessen's. The 5 reflex edges are Clifford parallel, the 5 edges of a pentagonal prism column that has a twist to it. The contraction of those 5 edges into 1, and those 10 vertices (two pentagons) into 2, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing all their edges as a single Jessen's. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (octahedron) by conflating its parallel pairs of edges, leaving the 11-cell's abstract 6-point (hexad cell). The hexad's situation is that it occurs in bundles of 5, disjoint but close together in the sense of adjacent, like the great circle rings in a Hopf fiber bundle. 

To summarize, the 12-point (2-hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell). We shall now be able to see how Fuller's perfect hexads combine with Todd's perfect pentads to form the perfect heptads of the 11-cell and 137-cell.  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=alias the hemi-cuboctahedron}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces. (The ''heptad'' arrives at the party, anonymously; they are our distinguished anti-celebrity, who most of the other guests have never even heard of.) The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. This 12-point (truncated tetrahedron) is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 100 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp: Truncated tetrahedron|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp: Truncated tetrahedron|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}} Of course Moxness has been there, seen that, already with his graphics software;{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} see his parameterized illustration of it exactly as it occurs in the 11-cell, below in his bestiary of A3, B3, and H3 Archimedian and Catalan polyhedra in the 120-cell.]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:Archimedean_and_Catalan_solid_hulls_with_their_Weyl_orbit_definitions.svg|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of the 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. In this catalog of Archimedean and Catalan polyhedra sighted in the 120-cell by Moxness's software,{{Sfn|Moxness: Archimedean and Catalan hulls|2023|loc=Hull #1 Archimedean Name A3 110 Truncated Tetrahedron A (upper left)}} it is the first entry (upper left), Moxness's Hull # = 1 with 12 vertices of 3D norm {{radic|11}}/2. On a {{radic|2}}-radius 3-sphere, 5 of these with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell. 20 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges are cells in the [[#The rings of the 11-cells|cell rings of the quasi-regular 11-cell]], which also has 12 regular 5-cells as cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

...

Both kinds of cells in the 11-cell have expressions in two different dimensions (on the 2-sphere and 3-sphere), because they are all centered on the center of the 3-sphere, unlike the cells of most 4-polytopes, which are centered on various off-center points within the 4-polytope's interior. 

One kind of 11-cell cell appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (semi-regular truncated cuboctahedron), a.k.a. the truncated tetrahedron. The other kind of cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now, except in the 120-cell. But this is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron, and a regular tetrahedron behind each face. The 11-cell is thus a honeycomb of mixed regular tetrahedra and truncated cuboctahedra (a.k.a. truncated tetrahedra) that meet at their shared 5-cell faces. 

Both the 5-cells (decahedra) and the truncated cuboctahedra (truncated tetrahedra) are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks in their 4-polytopes (off-center).{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} Therefore all the 11-cell's cells are concentric, which indeed makes a different kind of honeycomb, an inside-out cellular honeycomb with all the cells piled on top of each other overlapping in 4-space. They do not overlap in 3-sphere space however; their 3-volumes (but not their 4-contents) are disjoint. They make a proper 4D honeycomb like the cells of any 4-polytope, with volumetrically disjoint 3D cells that meet each other in the proper way at their shared 2D faces.

An 11-cell contains 6 completely disjoint 5-point (1-pentad 5-cell decahedron cells), and 5 volumetrically disjoint 12-point (2-hexad truncated cuboctahedron cells), 11 volumetrically disjoint cells altogether. Of course it is supposed to have 11 identical 11-point (elevenad hemi-icosahedral cells), and that is also true in its abstract sense, which is not the same sense as disjoint cellular containment: each pentad or hexad cell is inscribed in one of 11 larger 60-point (Moxness's rhombicosadodecahedron cells). The abstract pentads and hexads are only hemi-polyhedra, so the 11-cell's pentad cells are 12 (out of 120) disjoint instances of a pentad 5-point, and its hexad cells are 10 (out of 100) disjoint instances of a hexad 6-point, 1/10th of the 120-cell's cellular volume in both cases. The 11-cell's hemi-icosahedral cells are 11 (out of 120) non-disjoint instances of an elevenad 60-point. Each 11-cell shares each of its hexad cells with 9 other 11-cells, and each of its 11 vertices (out of 600) occurs in 9 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.|name=Tetrahedral vertex figures}}

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

...insert image thumb of 11-cell...

...Construct the 11-cell from the heptad....

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...Four concentric 4-polytopes may be mutually orthogonal. In any set of 4 mutually orthogonal regular 5-cells, one pair of faces in each orthogonal pair are completely orthogonal to each other. Thus three edges in one 5-cell in an orthogonal pair are opposite (completely orthogonal) to a face of the orthogonal 5-cell. .... A 5-cell has 5 vertices in its abstract hemi-icosahedral cells, but only 4 vertices in one concrete icosahedral cell; the 5th lies in an icosahedral cell of another 11-cell (not in another icosahedral cell of the same 11-cell).

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

....

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 11-cell ===

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon.... (wrong, 11-cell's Hopf map is the truncated icosahedron)

....

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk: Truncated icosahedron|2018}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

...

Since the rhombicosadodecahedron has 120 5-cell edges, the 120-cell has 120 completely disjoint 5-cells, and the rhombicosadodecahedron shares each vertex with 11 other rhombicosadodecahedra, how a rhombicosadodecahedron is bound to its neighbor 5-cells is constrained. A further constraint is that the abstract hemi-icosahedron has only 15 5-cell edges, so the 120 actual 5-cell edges of the rhombicosadodecahedron are conflated 8-to-1 in the hemi-icosahedron. To be conflated by contraction into a single edge, those 8 edges must lie parallel to each other in the 120-cell; either actually parallel in the same plane, or Clifford parallel in separate planes, as two edges in completely orthogonal planes may be parallel and orthogonal at the same time. Each 5-cell inscribed in the 120-cell contains 5 such tetrahedra occur in parallel pairs in 4 orthogonal planes (none of them completely orthogonal planes). 

.... 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
Each conflated vertex is the 10 vertices of two orthogonal regular 5-cells...
The 60-point contains 6 pairs of these orthogonal regular 5-cell cells...
...its 20 hexad cells are ..., a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere.

== The regular 137-cell 4-polytope ==

(description originally written for the 11-cell, which applies only to a fibration of 11 11-cells)

It is quasi-regular with two kinds of cells, the 12-point (regular icosahedron) and the 5-point (regular 5-cell), which have just one kind of regular face between them. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of regular facets (it is semi-regular with regular facets), it also means that its facets have expressions of two different dimensionalities because they are concentric in the 3-sphere. The 11-cell has two kinds of regular cells. One appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (regular icosahedron), but since it is embedded at the center of the 3-sphere (unlike the cells of most 4-polytopes) it also acts as an inscribed 4-polytope.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} The other cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now except in the 120-cell. This is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron. The 11-cell is a honeycomb of mixed icosahedron and half-icosahedron cells. Like the icosahedra, the decahedra are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks, off-center in their 4-polytopes. All the 11-cell's cells are concentric, which indeed makes an unusual honeycomb of cells, an inside-out honeycomb. It is a proper 4D honeycomb however, with volumetrically disjoint 3D cells that meet each other in the proper way at shared 2D faces. The embedded 12-point (regular icosahedron) cell is an icosahedral pyramid (as a 4-polytope), which is filled by 20 4-point (irregular tetrahedra) that meet at its center; the center point of the 11-cell acts as a 12th vertex, just as the center point of the 24-cell acts as a 25th vertex in some honeycombs. The 5-point (regular 5-cell) cell contains five 4-point (regular tetrahedra) which meet the 12-point (regular icosahedra)'s irregular tetrahedral cells face-to-face, so the 11-cell is quasi-regular in the usual sense of having two varieties of regular 3-polytope cells, but additionally both varieties of of its cells are bi-dimensional: another clue that the 11-cell has something to do with dimensional analogy

....

The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

....

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur to make an 11-cell. 11 of them occur in a 137-cell, all 120 of them occur in the 120-cell, and they occur nowhere else.

How can this regular 4-polytope possibly exist? By Schläfli's criterion {3, 3, ..} exists, but not as a finite polytope.{{Sfn|Coxeter|1973|loc=§7.7 Schläfli's criterion}} Coxeter found {3, 5, 3} as a finite hyperbolic honeycomb, but not in Euclidean 4-space.{{Sfn|Coxeter|1968|title=The Beauty of Geometry: Twelve Geometric Essays|loc=''Regular Honeycombs in Hyperbolic Space''|ps=; &quot;The honeycombs considered by the above authors [Schlegel, Schläfli, Klein, Sommerville] have finite cells and finite vertex figures. ... A further extension allows a cell or vertex figure to be a ''star''-polytope, so that the honeycomb covers the space several times. .... We shall find that there are four regular star-honeycombs in hyperbolic 4-space, as well as two infinite families of them in the hyperbolic plane.&quot;}} We find that {3, 3, ..} is a central expression of the &lt;math&gt;H_4&lt;/math&gt; root system, being the regular convex hull in 4 dimensions of 137 points, 96 pentads and 600 hexads. The minimal expression of &lt;math&gt;H_4&lt;/math&gt; is the regular convex hull {3, 3, 5} of 120 points, 600 tetrads and 75 hexads, and its maximal expression is the regular convex hull {5, 3, 3} of 600 points, 120 pentads and 675 hexads. {3, 3, ..} is a third regular &lt;math&gt;H_4&lt;/math&gt; polytope which fits between them. The properties and metrics of this regular sequence are presented in tabular form below in ''[[#Build with the blocks|§Build with the blocks]]''.

{|class=&quot;wikitable&quot;
!colspan=3|Regular &lt;math&gt;H_4&lt;/math&gt; polytopes
|-
|[[File:600-cell.gif|200px]]
|[[File:5-cell.gif|200px]]
|[[File:120-cell.gif|200px]]
|-
!120-point 600-cell
!137-point 137-cell
!600-point 120-cell
|}

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 137-cell ===
....

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=Apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|2022|loc=''[[Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]''}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie: On Fuller's use of language|2024|loc=''[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry'']]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center, 1948-1949|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with their mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the gender neutral shad can open their six beaks all the way, until they become flat squares and they becomes a cuboctahedron, or they can shut them all tight like a turtle retracting into their octahedron shell. The six mouths always move in unison. This is [[Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do their ''really'' odd trick -- where they flip their 6 jaws 90 degrees in their 6 faces and shut their 6 beaks on the opposite axis of their squares than the one they opened them on -- or whether they will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than their normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make their edge length exactly the same size as their radius, when they open their mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing their mouths in spherical synchrony, their 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron (the hexad), then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=''[[W:SO(4)|SO(4)]]''}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=Isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its orthogonal hexad's planes (wz, wx, wy).{{Efn|name=Six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of (expanded octahederon 16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh &amp; Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== Legendre's 12-point hexad binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a 12-point double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's perfectly realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, this 2-hexad expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre hexad) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction.{{Efn|name=Concentric polyhedra}} In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== ''n''-simplexes ===
....

The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== ''n''-orthoplexes ===
....
[[File:Great 5-cell √5 digons rectangle.png|thumb|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} ''regular'' great hexagons. These are the same 200 dodecagon central planes illustrated above, which also contain 6 120-cell edges and two opposing ''irregular'' great hexagons (truncated {{radic|6}} triangles). The 1200 {{radic|5}} 5-cell face triangles occupy 1200 other, non-central planes.]]

...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's Hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[16--cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[16--cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=Concentric polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[16--cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== ''n''-taliesins ===
The 4-simplex pentad building block combines with the 4-orthoplex hexad building block to make the 4-taliesin heptad building block. The heptad is the essential building block of the ''taliesin'' family of polytopes. The 11-cell and 57-cell hemi-polytopes and the 137-cell and 120-cell regular 4-polytopes are nuclear members of this trans-dimensional family. The 24-cell and 600-cell are extended members of it, because they also contain instances of the heptad building block.

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its spherical symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* [[W:Taliesin Myrddin Namkai-Meche|Taliesin Myrddin Namkai-Meche]], a young Reed College scholar who was stabbed to death by a crazy man in 2017 on a Portland, Oregon light rail train, when he rose to the nonviolent defense of two black teenagers who were being violently assaulted in racist and anti-Muslim language by a right-wing extremist.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in orthogonal [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].
* '''Taliesan polytope''', an (''n+1)''-polytope whose ''n''-polytope dimensional analogue inscribed concentrically in the ''n''-sphere forms a convex honeycomb of itself which is the (''n+1)''-polytope.{{Efn|name=Embedding point}}
* The taliesan polytope's honeycomb, a Hopf fibration of its great circles.
* The ''n''-dimensional family of dimensionally analogous polytopes related to a taliesan polytope by expansion and contraction operations.
* The characteristic [[w:Orthoscheme|orthoscheme]] of a teliesan polytope, an irregular simplex that is the complete gene for the entire expression of trans-dimensional symmetry relationships among the polytopes of the ''n''-dimensional family, as well as the complete gene for the teliesan polytope's rotational symmetry.{{Efn|name=Characteristic orthoscheme}}

=== ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== ''n''-equilaterals ===
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed at four different [[#n-leviathon|chord]] lengths from each other. In the {{radic|2}}-radius polytopes we have using as examples in this paper they are {{radic|2}}, {{radic|4}}, {{radic|6}} and {{radic|8}}, but in a ''unit-radius'' polytope they are: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}.

Each vertex is joined to 8 others by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}. The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== ''n''-pentagonals ===

[[File:600-cell.gif|thumb|The 120-point (600-cell), &lt;math&gt;H^4&lt;/math&gt;.]]
...

[[File:600-cell vertex geometry.png|thumb|Vertex geometry of the 600-cell, showing the 5 regular great circle polygons and the 8 vertex-to-vertex chord lengths with angles of arc. The golden ratio{{Efn|name=golden chords|group=}} governs the fractional roots of every other chord,{{Efn|name=fractional root chords}} and the radial golden triangles which meet at the center.|alt=|400x400px]]
The 120 vertices of the 600-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏&lt;sup&gt;−1&lt;/sup&gt;) beginning with a vertex; see column ''a''}} at eight different [[#n-leviathon|chord]] lengths from each other. These edges and chords of the 600-cell are simply the edges and chords of its five great circle polygons. In ascending order of length in a ''unit-radius'' polytope (not in a {{radic|2}} radius polytopes like our usual examples), they are {{radic|0.𝚫}}, {{radic|1}}, {{radic|1.𝚫}}, {{radic|2}}, {{radic|2.𝚽}}, {{radic|3}}, {{radic|3.𝚽}}, and {{radic|4}}.{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=Decimal fractions of square roots|group=}}

Notice that the four [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#''n-''equilaterals|hypercubic chords]] of the 24-cell ({{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|4}}) alternate with the four new chords of the 600-cell's additional great circles, the decagons and pentagons. The new ''golden chord'' lengths are necessarily square roots of fractions, but very special fractions related to the golden ratio{{Efn|1=The fractional-root ''golden chords'' are irrational fractions that are functions of {{radic|5}}. They exemplify that the [[W:golden ratio|golden ratio]] &lt;big&gt;𝜙&lt;/big&gt; {{=}} {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618 is a circle ratio related to &lt;big&gt;𝜋&lt;/big&gt;:&lt;br&gt;
: {{sfrac|𝜋|5}} = arccos (𝜙/2)
is one decagon edge, the 𝚽 = {{radic|0.𝚫}} = {{radic|0.382~}} ≈ 0.618 chord.
Reciprocally, in this function discovered by Robert Everest expressing &lt;big&gt;𝜙&lt;/big&gt; as a function of &lt;big&gt;𝜋&lt;/big&gt; and the numbers 1, 2, 3 and 5 of the Fibonacci series:&lt;br&gt;
: &lt;big&gt;𝜙&lt;/big&gt; = 1 – 2 cos ({{sfrac|3𝜋|5}})
{{sfrac|3𝜋|5}} is the arc length of the &lt;big&gt;𝜙&lt;/big&gt; = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618 chord.|name=golden chords|group=}} including the two [[W:golden section|golden section]]s of {{radic|5}}, as shown in the diagram.{{Efn|The 600-cell edges are decagon edges of length {{radic|0.𝚫}}, which is 𝚽, the ''smaller'' golden section of {{radic|5}}; the edges are in the inverse [[W:golden ratio|golden ratio]] {{sfrac|1|φ}} to the {{radic|1}} hexagon chords (the 24-cell edges).
The other fractional-root chords exhibit golden relationships as well. The chord of length {{radic|1.𝚫}} is a pentagon edge.
The next fractional-root chord is a decagon diagonal of length {{radic|2.𝚽}} which is &lt;big&gt;𝜙&lt;/big&gt;, the ''larger'' golden section of {{radic|5}}; it is in the golden ratio{{Efn|name=golden chords|group=}} to the {{radic|1}} chord (and the radius).{{Efn|Notice in the diagram how the &lt;big&gt;𝜙&lt;/big&gt; chord (the ''larger'' golden section) sums with the adjacent 𝚽 edge (the ''smaller'' golden section) to {{radic|5}}, as if together they were a {{radic|5}} chord bent to fit inside the {{radic|4}} diameter.}}
The last fractional-root chord is the pentagon diagonal of length {{radic|3.𝚽}}.
The [[W:Pentagon#Side length is given|diagonal of a regular pentagon]] is always in the golden ratio to its edge, and indeed &lt;big&gt;𝜙&lt;/big&gt;{{radic|1.𝚫}} is {{radic|3.𝚽}}.|name=fractional root chords|group=}}

....


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== ''n''-leviathon ===
...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of its &lt;math&gt;10^2&lt;/math&gt; disjoint instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction, before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]].{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}} .... It is only because she did not have the 11-cell to start with that she did not discover the regular 137-cell {3, 3, ..} by her method.

Between Alicia Boole Stott and her identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and her identification of symmetry groups with conservation laws (in her 1st theorem), we owe the deepest mathematics of geometry and physics originally to two pioneering women, colleagues of their contemporaries the greatest men of the academy, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just an [[W:abstract polytope|abstract 4-polytope]], not just a singleton convex 4-polytope, and not just a honeycomb.{{Sfn|Coxeter|1970|loc=''Twisted Honeycombs''}} The 11-point (11-cell) has a concrete ...regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}&lt;nowiki/&gt;} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, 3, 3} as the 137-point (137-cell).

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 11-cell, 137-cell and 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=''Clifford Spinors and Root System Induction: H4 and the Grand Antiprism''}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=Apology}}

== Build with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Duveneck|1978|loc=Carl Jung, quoted in ''Life on Two Levels''|p=ii|ps=.{{Sfn|Jung|1961|ps=: &quot;The best of truths is of no use unless it has become one's most personal inner experience. It is the duty of everyone who takes a solitary path to share with society what he finds on his journey of discovery.&quot;}}}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1954|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius={{radic|2}}}}

== Acknowledgements ==
...

== Notes ==
{{Regular convex 4-polytopes Notelist}}

== Citations ==
{{Reflist}}

== References ==
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      <text bytes="208610" sha1="q7al2ey0965xodnashv19o38mkots34" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|June 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a regular 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the &lt;s&gt;120-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 5, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 11 [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] cells. Eleven 11-cells (plural) are the 137-cell, the &lt;s&gt;600-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 3, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 137 [[W:Rhombic triacontahedron|triacontahedron]] cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=''Regularity of Graphs, Complexes and Designs''}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=''A Symmetrical Arrangement of Eleven Hemi-Icosahedra''}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]]
The 4-dimensional regular polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:Simplex|simplex]] (called the [[5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:Orthoplex|orthoplex]] (called the [[16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular convex 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the [[w:Tesseract|16-point (8-cell) tesseract]], the 4-cubic, which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found within any of the other regular convex 4-polytopes, except in the largest and most complex one, the [[120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). However, we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the 16-cell is their very starting point, and the most frequently used tool in the box.

== 5-cells and hemi-icosahedra in the 11-cell ==
[[File:Symmetrical_5-set_Venn_diagram.svg|thumb|The 5-point (10-face) regular 5-cell (the regular 4-simplex). Grünbaum's rotationally symmetrical 5-set Venn diagram{{Sfn|Grünbaum|1975|loc=''Rotationally symmetrical 5-set Venn diagram'', Fig 1 (e)|ps=; partitions the individual elements of the 5-cell.}} is an illustration of the 5-cell labeling each of its &lt;math&gt;2^5&lt;/math&gt; elements.{{Sfn|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled; image includes the Python code to render it, optimising for maximum area of the smallest regions.}}]]
[[File:Hemi-icosahedron.png|thumb|The 6-point (10 face) [[W:hemi-icosahedron|hemi-icosahedron]], an abstraction of the regular icosahedron, has half as many faces, edges and vertices. Each element of the abstract polyhedron represents two real elements found in different places in the 11-cell's realization.{{Sfn|Ruen: hemi-icosahedron|2007}}]]
The most apparent relationship between the pentad 5-cell and the hexad hemi-icosahedron is that they both have 10 triangular faces. When we see a facet congruence between a 4-polytope and a 3-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 6-point 5-simplex. It is also the same size as the 6-point 3-orthoplex; thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no chord congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals, pentad and hexad, who are famously stand-offish with each other. Interesting!

The cell of the 11-cell is an abstract hexad hemi-icosahedron with 5 central planes, most handsomely illustrated by Séquin.{{Sfn|Séquin &amp; Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing rings of polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The 11 hemi-icosahedral cells have 10 triangle faces each; the 5-cell's 5 tetrahedral cells have 10 faces and 10 edges all together. Cells meet face-to-face. The only way 11-cells and 5-cells can have the same faces, is if two hemi-icosahedral cells and two tetrahedral cells all meet at each face. In that case 3 of each 5-cell's 5 vertices are a hemi-icosahedron face, and its other 2 vertices must be some 11-cell edge lying opposite the face. Coxeter determined that the 11-cell does indeed have an edge opposite each face, but it does not belong to the same hemi-icosahedral cell as the opposing face. He found that the 10 edges opposite each hemi-icosahedron's faces are the 10 edges of a single 5-cell, which does not share any vertices, edges or faces with the hemi-icosahedron. Of course, there is just one such 5-point 5-cell that is completely disjoint from each 6-point hemi-icosahedron in the 11-point 11-cell.{{Sfn|Coxeter|1984|p=110|loc=§6. The Petrie polygon [of the 11-cell]|ps=; &quot;We may reasonably call this edge and face ''opposites''. It is easy to find the face opposite to a given edge by looking at the faces to which a given edge belongs. ... Conversely, given a face, we can find the opposite edge by seeing which vertices belong to neither of the hemi-icosahedra which share that face. The ten edges opposite to the ten faces of one hemi-icosahedron are the edges of the complementary &lt;math&gt;a_4&lt;/math&gt; [4-simplex], that is, the joins of all pairs of the five vertices [of the 11-cell] not belonging to the given hemi-icosahedron.&quot;}}

There are 11 disjoint 5-cells in each 11-cell, which is comprised of 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. The real 11-cell is harder to see than the abstract 11-cell that represents it, because the real hemi-icosahedron that the abstract hemi-icosahedron represents is hard to see, but seeing these objects will be easier once we have identified the real hemi-icosahedron and exactly where the 11-cell's real elements reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each hemi-icosahedron's 10 faces come from 10 disjoint 5-cells, and there is just one 5-cell with which it does not share a face.{{Efn|As Coxeter observes (in the previous citation), that unrepresented 5-point (5-cell) is the other 5 vertices of the 11-point (11-cell) that are not vertices of this 6-point (hemi-icosahedron), the hemi-icosahedron's disjoint complement.}} Each 5-cell has 10 faces from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share a face.

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in two different planes which are completely orthogonal to each other. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must be sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness: 8 concentric hulls|2022|loc=Hull #8 (lower right)|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell of the 11-cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Compounds in the 120-cell|§Compounds in the 120-cell]]'' that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software which uses Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s to render the polyhedra which are found in the interior of ''n''-dimensional polytopes.{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the tetrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is a real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (Hull #8) is a concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this {12} irregular great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing {6} irregular great hexagons (truncated triangles) of alternating 120-cell edges and 5-cell edges. The great dodecagon is a great circle of the hemi-icosahedral cell, and also a great circle of the 120-cell, because unlike the cells of most 4-polytopes, the 11-cell's cells are embedded in 4-space at the exact center of the 4-polytope, making them inscribed 4-polytopes themselves.{{Efn|name=Embedding point}} There are 25 great dodecagons and 120 5-cell edges in Moxness's 60-point (Hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (Hull #8) polyhedron altogether.

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point (Hull #8) is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] similar to the one from the catalog shown here,{{Sfn|Piesk: Rhombicosidodecahedron|2018}} but a slightly shallower truncation with smaller red pentagons (120-cell faces) and narrower rhombs. Rhombicosidodecahedra are made by truncating the [[W:Rhombic triacontahedron|rhombic triacontahedron]], which is the 30-sided polyhedron with only one kind of face, the dual of the 30-point icosidodecahedron. The 120-cell contains 120 of these 11-cell cells.]]
Moxness's 60-point (Hull #8) is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (Hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|The 5-point (regular 5-cell) can be another abstraction of Moxness's 60-point (Hull #8), 12-vertices-into-1, but it is not strictly the result of a contraction operation as described by Boole Stott, since she did not apply her expansion and contraction operations to uniform polytopes with more than one edge length.}} In fact Moxness was not the first to find a rhombicosidodecahedron in the 120-cell; Alicia Boole Stott was. It is the 6th section of {5,3,3} beginning with a cell, which Boole Stott identified as the ''e&lt;sub&gt;2&lt;/sub&gt; expansion'' of the icosahedron (or equivalently of its dual polyhedron the dodecahedron).{{Sfn|Boole Stott|1910|loc=§Examples of the e&lt;sub&gt;2&lt;/sub&gt; expansion|p=7}} But that rhombicosidodecahedron identified by Boole Stott is not Moxness's Hull #8: rather, it is the semi-regular Archimedean solid, with a single edge length and square faces. Moxness's Hull #8 with its two distinct edge lengths and rectangular faces is Coxeter's 8th section of {5,3,3} beginning with a cell, which is missing from the sections illustrated by Boole Stott.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than tetrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. The 120-cell contains 120 of Moxness's 60-point (11-cell cells). They are non-disjoint, with 12 60-points sharing each vertex.

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (Hull #8) is the real hemi-icosahedron cell, with 120 5-cell edges in it as chords lurking under its surface. The rhombicosadodecahedron has enough of those 5-cell edges in its interior to make 20 disjoint tetrahedral cells which belong to 20 distinct 5-cells.

The 600-point (120-cell) may be constructed as a semi-regular honeycomb of two kinds of cells, the 120 non-disjoint 60-point (Moxness's rhombicosadodecahedra) and their neighbors the 120 disjoint 5-point (5-cells), cell-bonded together by shared tetrahedral cells of edge-length {{radic|5}}. Each rhombicosadodecahedron is bonded to 20 islanded 5-cells, sharing with each a 5-cell tetrahedron (4 of its interior 5-cell faces, and 6 of its interior 5-cell edges). The rhombicosadodecahedron is also bonded directly to 10 other rhombicosadodecahedra (since each 11-cell has 11 rhombicosadodecahedron cells), again not by sharing one of its exterior faces, but this time by sharing an interior face, a {{radic|6}} triangle which lies in a central plane.{{Efn|The {{radic|6}} chord is another hidden chord lurking beneath the 60-point rhombicosadodecahedron's surface, like the {{radic|5}} 5-cell edge. Both these hidden chords lie on the same great circle with the 120-cell edges (the pentagon face edges). Three {{radic|6}} chords make a triangle which lies in a central plane. This {{radic|6}} great triangle must not be confused with the {{radic|5}} face triangle of the 5-cell, which does not lie in a central plane.}} The 120 rhombicosadodecahedra are also bonded directly together at ''all'' their exterior faces: at their pentagon faces (which are 120-cell faces of dodecahedral cells); at their small triangular faces (which are 600-cell faces of small tetrahedral cells, 10 of which are inscribed in each dodecahedral cell); and at their rectangular faces (which are faces of 10-point [[w:Pentagonal_prism|pentagonal prism]] cells, where the pentagonal prism is a column inscribed in a dodecahedral cell, joining its opposite pentagon faces). The 120 rhombicosadodecahedra are bonded directly to each other at their own exterior faces; no dodecahedra, small tetrahedra, or pentagonal prisms lie between them, and those objects are not cells of this honeycomb; we mention them here only to help locate the rhombicosadodecahedra within the 120-cell. The only objects which fill holes between the rhombicosadodecahedra are the 120 islanded 5-cells.{{Efn|This honeycomb of the 120-cell resembles the [[600-cell#Icosahedra|honeycomb of the 600-cell consisting of 24 face-bonded regular icosahedra]] with 24 clusters of 5 tetrahedra filling the holes between them. In the 120-cell there are 5 times as many of everything as in the 600-cell, so there are 120 (rhombicosadodecahedron) icosahedral cells with 600 tetrahedra filling the holes between them.}}

It is possible for rhombicosadodecahedra to meet each other at more than one kind of exterior face because they are not meeting there as cells of the same 11-cell; they belong to different 11-cells. The 11 rhombicosadodecahedra in the same 11-cell do not meet at their own exterior faces, they meet at their interior {{radic|6}} triangle faces. As a cellular 4-polytope, the 11-cell is inside-out. Its cells meet each other at interior planes, sharing interior {{radic|6}} triangle faces, instead of in the usual way that 4-polytopes' cells share an exterior face. It is a deeper kind of intertwining to be sure, but the 11 rhombicosadodecahedra in the same 11-cell are still volumetrically disjoint cells, and the 11-cell is a perfectly real 4-polytope with real cells. It is a semi-regular 4-polytope, because it has two different kinds of cells: 11 rhombicosadodecahedra and 11 5-cells. The 11 5-cells are islanded (not bound to each other), and the 11 rhombicosadodecahedra are bound both to 5-cells (at {{radic|5}} 5-cell faces) and to each other (at {{radic|6}} central triangle faces). In the section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#What's in the box|§What's in the box]]'', below, we will look closely at the relation between the {{radic|5}} chord and the {{radic|6}} chord, which is the relation between the pentad and the hexad, and at the general phenomenon of hemi-polytopes which meet each other at a central plane. Then in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Cell rings of the 11-cells|§Cell rings of the 11-cells]]'' we will look at the two-cell cellular structure of the 11-cell in more detail.

Considered as a single complex of overlapping cells rather than as distinct honeycombs of volumetrically disjoint cells, the 120-cell contains 60-point (rhombicosadodecahedral cells), islanded 5-point (5-cells), 20-point (dodecahedral cells), and 10-point ([[w:Pentagonal_prism|pentagonal prism]] cells). The 60-points meet each other in pairs at their pentagon faces, meet regular dodecahedra at those pentagon faces also, meet each other in quads at their 600-cell faces where they share an interior 4-point (600-cell tetrahedral cell) and its four faces (just as they share a larger interior 5-cell tetrahedron and its faces), and meet each other and the 10-point (pentagonal prisms) at their rectangle faces. We might try to imagine such a complex with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with parts truncated from 30 neighboring truncated dodecahedra pentagonal prisms). There are 120 of each of these 3 kinds of dodecahedron. 

Moxness's 60-point (rhombicosadodecahedron cell) is a central object of the 120-cell, and contains many wonders. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra by expansion and contraction operations, as Alicia Boole Stott discovered before 1910,{{Sfn|Polo-Blanco: ''Theory and history of geometric models of Alicia Boole Stott''|2007|loc=§5.3.2 1910 paper on semi-regular polytopes|pp=152-158|ps=; summarizes Boole Stott's method and results from {{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}}, including her tables of expansion-contraction dimensional analogies and a few of her illustrations.}} and those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, rediscovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter &amp; Petrie|1938|p=4|loc=''[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]''}}|name=Concentric polyhedra}} Before we move on from the 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice one more thing about the hidden {{radic|5}} chords lurking below its surface. The 12 little pentagon faces are spanned by sets of 6 {{radic|5}} chords (5-cell edges). The 6 chords of each set are disjoint (they don't touch or form 5-cell faces), but they are symmetrically arranged in 3 parallel pairs, {{radic|3}} apart, which lie in 3 orthogonal central planes. It turns out they are the 6 reflex edges of a 12-point non-convex polyhedron called the [[w:Jessen's_icosahedron|Jessen's icosahedron]], inscribed in the 60-point cell. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again and work our way outwards, before eventually in ''[[#The perfection of Fuller's cyclic design|§The perfection of Fuller's cyclic design]]'' we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cell)s ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks, showing its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Pentahemidemicube.png|120px]]
|[[File:Tetrahemihexahedron_rotation.gif|120px]]
|[[File:Pentahemicosahedron.png|120px]]
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts needed to build the 120-cell (the picture on the cover of the box), and everything else that's invisible in that picture which fits inside the 120-cell: all the astonishing regular 4-polytopes. The pentad building block is half a pair of orthogonal 5-point (regular 5-cells), the hexad building block is half an 8-point (16-cell), and the heptad building block is a quarter of an 11-point (11-cell). Everything can be built from these blocks by embedding them at the center of the 3-sphere, inscribing them as 4-polytopes to make compounds as with the 16-cell building block, rather than by assembling them face-to-face as 3-polytopes to make honeycombs in which they are cells or vertex figures.{{Efn|name=Embedding point}}

The pentad building block is the 5-point '''pentahemidemicube''', an abstract polyhedron with 10 equilateral triangle faces, 10 {{radic|5}} edges, and 5 vertices.{{Sfn|Christie: Pentahemidemicube|2024|loc=File:Pentahemidemicube.png|ps=; &quot;The 5-point ''pentahemidemicube'' is an abstract polyhedron with 10 equilateral triangle faces, 10 edges, and 5 vertices. It has a real presentation as the regular 5-cell.&quot;}} It has a real presentation as the 5-point (regular 5-cell 4-polytope), a regular tetrahedron expanded by addition of a fifth vertex equidistant from all the others in a fourth dimension. The pentad block is a tetrahedral pyramid (a 5-cell) inscribed in a cube: a tetrahedron inscribed in a cube (a demicube) expanded by addition of a fifth vertex, one of the cube's other 4 vertices. Each pentad (5-cell) building block lies in the 120-cell orthogonal to three other pentad (5-cell) building blocks. They are four completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges. Each regular 5-cell in the 120-cell is a pentad (pentahemidemicube) building block, in one of 30 orthogonal sets of 4. There are 120 pentad building blocks in the box.

Every vertex of the 16-cell, and therefore every vertex of the 8-cell, 24-cell, 600-cell and 120-cell, is surrounded by a 6-point (octahedral vertex figure). The hexad building block is that 6-point octahedron embedded at the center of the 4-polytope instead of at a vertex, that is, inscribed in the 3-sphere as a 4-polytope, the 6-point '''[[w:Tetrahemihexahedron|tetrahemihexahedron]]''' or '''hemi-cuboctahedron'''.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; &quot;The tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U&lt;sub&gt;4&lt;/sub&gt;. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices.&quot;}} The hexads have 4 red triangle 5-cell faces, bounded by 12 {{radic|5}} edges. Their 3 yellow square faces lie in central planes, because the building block is an abstract 4-polytope. The hexad is an irregular abstract polyhedron that also has 6 {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

Two orthogonal 5-point (pentad) building blocks intersect in a 4-point (inscribed tetrahedron) to form a 6-point (hexad) building block. Two orthogonal 6-point (hexad) building blocks intersect in a 4-point (central square) to form an 8-point (16-cell) building block, the most usable building block, from which everything larger can be built by compounding.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 120 pentads to pair with a completely orthogonal hexad, and although there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed) they only result in 120 pairs two different ways, so there are 120 11-cells. The 11-cell is 6 pentads' and 5 hexads' smallest joint compound, and the 120-cell is their largest, 120 pentads and 100 hexads.

The heptad{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''|pages=22–31}} building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|4}} edges and 9 {{radic|5}} edges), and 7 vertices.{{Sfn|Christie: Pentahemicosahedron|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;}} It is a compound of a 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]) and a 6-point (hexad [[W:hemi-icosahedron|hemi-icosahedron]]) intersecting in a 4-point (tetrahedron), that is, of three orthogonal 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]&lt;nowiki/&gt;s) intersecting in their common 4-point (tetrahedron). It is also the abstract cube contracted by removal of one vertex, which reduces one of the cube's two inscribed tetrahedra (demicubes) to a single remaining triangular face. As with all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Four &lt;small&gt;&lt;math&gt;(11-7)&lt;/math&gt;&lt;/small&gt; orthogonal 7-point (pentahemicosahedron) blocks can be joined at a common Petrie polygon (a skew hexagon which contains their orthogonal 6-edge paths) to construct a real 11-point ([[W:11-cell|11-cell]] semi-regular 4-polytope), which magically contains six &lt;small&gt;&lt;math&gt;(11-5)&lt;/math&gt;&lt;/small&gt; 5-point (1-pentad regular decahedra) as cells and five &lt;small&gt;&lt;math&gt;(11-6)&lt;/math&gt;&lt;/small&gt; 12-point (2-hexad semi-regular cuboctahedra) as cells. There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like [[W:Euclid's postulates|Euclid's postulates]], and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as honeycombs of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles. It possesses the complete symmetry of the polytope without any redundancy, because it contains one of each of the polytope's characteristic root elements.{{Efn|A [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be partitioned radially by its planes of symmetry (Coxeter's &quot;reflecting circles&quot;) into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated rotationally to generate its regular 4-polytope because it is the complete [[W:gene|gene]] for it, containing all of its elements and capturing all of its symmetry without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring that triggers the reflections.|name=Characteristic orthoscheme}} It is the gene for the polytope, which can be replicated to construct the polytope.

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its &lt;math&gt;A_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the regular compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;small&gt;&lt;math&gt;120\times 120 = 14400&lt;/math&gt;&lt;/small&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[16--cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its &lt;math&gt;B_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 384. The 120-cell is the convex hull of the regular compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 24-point (24-cell) is subdivided into 1152 instances of its [[24-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;F_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 1152. The 120-cell is the convex hull of the regular compound of 25 disjoint 24-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;25\times 1152 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has three equivalent origins in distinct root systems, though most are obviously constructed from one root, and not so obviously from the others. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything larger than the pentad can be constructed by compounding hexads, and we shall see that everything as large as a 24-cell has a construction from heptads which are a product of a pentad and a hexad. Surely every uniform polytope can be constructed by some function of expansions and contractions from any of three root systems, however indirect the recipe. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes, related by an equals sign between them, then give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics by [[W:Noether's theorem|Noether's theorem]].

=== Pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 octahedral cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell contains the 600 octahedral cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It also contains the 600 cubical cells of its 75 8-cells. It is a 600-cell in three ways, although the octahedra and cubes are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell and the 8-cell, respectively.|name=600 cells three ways}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (1-pentad 5-cell), out of the 8-point (2-hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (1-pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope that contains 5-point (1-pentad 5-cell) building blocks beside the largest one, it just isn't a ''regular'' 4-polytope. It is semi-regular with two kinds of cells, the 4-point (regular tetrahedron) and the 12-point (truncated cuboctahedron), which have just one kind of regular face between them, a 5-cell face. That semi-regular 4-polytope is the 11-cell.

We said earlier that the 11-cell had 11 disjoint 5-cells and 11 hemi-icosahedral cells in it, and that its construction from them might be hard to see, but now we are going to account numerically, at least, for how it can have 6 pentad cells plus 5 hexad cells magically adding to 11 of the same kind of elevenad cell. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, 11. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (5-cell regular decahedron) cells, and the 12 6-point (hexad) building blocks into 100 12-point (regular icosahedron) cells, and then magically adding one whole 5-point (5-cell regular decahedron) cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 12-point (regular icosahedron) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such (cell bundles of 6 + 5 = 11 cells), with each 11-cell bundle containing 11 of both kinds of cell. Very, very strange!{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden'': §II To Add is to Multiply|p=38|ps=; the sequence of regular polygons with harmonious proportions beginning {5}, {7}, {9}, {11} as a family of natural numbers; &quot;The numbers in this optimal family have remarkable properties [that allow] similar figures to be arranged easily or allow a figure to be dissected into a set of similar figures. This is the origin of the Greek idea of geometric progression — that ''multiplication can be accomplished instead by addition''.... For instance, it is well known that the square of the golden ratio is one more than itself (written above as 𝜙&lt;sup&gt;2&lt;/sup&gt; = 𝜙 + 1) and one less than 𝜙 is its reciprocal (1/𝜙 = 𝜙 - 1). The first relation equates multiplication with addition, while the second accomplishes division through subtraction. [These are pentagon {5} bisectional golden ratios, but] the heptagonal [{7} tri-sectional] ratios also behave in surprising ways — adding to multiply, subtracting to divide.... There are analogous constructions for [nonagon {9}] quadrisectional and [hendecagon {11}] pentasectional systems.&quot;}}

The 120-cell has three interchangeable cyclic expansion-contraction cycles, the pentad, hexad and heptad cyclic designs, rooted in the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems, respectively. Each of them constructs the whole sequence of regular 4-polytopes from a single building block. The key to all three cycles is that 5 + 6 = 11, and there is an 11-point quasi-regular 4-polytope in the sequence of convex 4-polytopes. It lies between the 8-point (16-cell) and the 16-point (8-cell), and is itself a &lt;math&gt;B_4&lt;/math&gt; polytope. Its role is to be a unified single building block common to all three cyclic expansion-contraction cycles. It is involved with the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems and expresses in one building block their relations to each other.

We have already seen how the hexad building cycle works. It is plain compounding of 4-orthoplexes at the beginning, increasing to densely intricate non-disjoint symmetry toward the end, making it difficult to see why it works once past the 24-cell. This is how the build works, in terms of sixes. The 16-cell is a 2-hexad (3 pairs of completely orthogonal great square planes). The 24-cell is 3 16-cells and a 4-hexad, 4 different ways: 16 non-disjoint hexads (great hexagon planes). The 120-point (600-cell) is 25 24-cells and a 20-hexad, 10 different ways: 200 non-disjoint hexads (great hexagon planes). The 600-point (120-cell) is 225 (9 times 25) 24-cells. ....

The pentad building cycle doesn't do plain compounding to regular 4-polytopes until the last one, the 120-cell. Fives don't even appear as pentagons until after the 24-cell. We have seen how there are 11 5-cells in the quasi-regular 11-cell, which occurs before the 8-cell, and also how everything can be compounded from irregular 5-cells (4-orthoschemes specialized for each 4-polytope). It is still not evident how a pentad cycle can work to build all the regular 4-polytopes from the regular 4-simplex. This is how is how the build works, in terms of fives. ....

This is how the build works, in terms of elevens. The 11-cell is 4 orthogonal heptads (intersecting in their skew hexagon Petrie polygon). The 24-cell is also 4 heptads (intersecting in 4 orthogonal digon axes, so each heptad contributes 6 unique points to a single fibration of 24 points). Heptads are quarter elevenads, so the 11-cell and 24-cell are both 1 elevenad. ....

See the tabular sequence of convex 4-polytopes in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Build with the blocks|§Build with the blocks]]'', below: table rows ''Inscribed, Pentad, Hexad, Heptad''.

=== 11 of them ===
There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of eleven 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum eleven of them. There is only a single Hopf fibration of the 11 great circles of an 11-cell. 

The 5-cell and the 11-cell are limit cases. The 5-cells are completely disjoint, but the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 10 other instances of itself.

== Cell rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer &amp; Segerman|2013|loc=''Puzzling the 120-cell''}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his section of the Wikipedia article on the 120-cell entitled ''[[120-cell#Visualization|§Visualization]]''.{{Sfn|Ruen &amp; Goucher et al. eds. 120-cell|2024|loc=Goucher's ''[[120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious place to look for the 11-cell's Hopf map is in Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other place it could be hiding. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The quasi-regular convex 11-cell|§The quasi-regular convex 11-cell]]'', and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen: triacontagon|2011|loc=''Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them except his illustrations. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Four orthogonal projections of the 120-cell to a central plane, selected from the table of 3 regular and 11 compound 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|100 icosahedra&lt;br&gt;120 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|5 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 icosahedra&lt;br&gt;6 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|120
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|4
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|50
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|7.2°
|align=center|36°
|align=center|6°
|align=center|7.2°
|}

....

Here we summarize our findings on the 11-cells so far, including in advance some results to be obtained in the next section two sections.

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section....

== The 15 major chords ring everything ==
The natural numbers are each a distinct flavor.

Each integer has its own distinct taste. The first six integers {0..5} are the six strongest, most familiar tastes in nature, instantly distinguished and utterly unique.

The seventh flavor {6} is the first ratio of two of the others. Thereafter follows in strange sequence all the ratios of all the flavors, each marriage of every two flavors, most of which are fractions &lt;math&gt;\{k/d\}&lt;/math&gt;, but some of which are integers &lt;math&gt;\{k\}&lt;/math&gt;.

Some of these ratios are successful flavors in their own right, as distinct and inimitable as the marriage of chocolate and oranges, but the marriage of many flavor pairs is mud. It is a rule of thumb among flavor specialists that nearly any taste can be successfully alloyed with either onions or chocolate, but the direct marriage of these two flavors is notably unsuccessful. Some flavors make famous pairings or infamous ones, and subtle new pairings are always to be discovered that work surprisingly well for some purpose, but the majority of flavor marriages are undistinguished compounds.

In polytope geometry, each chord of a polytope is both is a distinct 1-dimensional object, a chord of the sphere of a distinct length &lt;math&gt;l&lt;/math&gt;, and a distinct natural number &lt;math&gt;h&lt;/math&gt;, a unique flavor. If the polytope is regular, it is a noteworthy and successful flavor. The chord's length &lt;math&gt;l&lt;/math&gt; is a square root, related to the natural number &lt;math&gt;h = k/d&lt;/math&gt; and to a set of polytopes they both represent, by a formula discovered by Steinbach.{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden''; Figure 5. Optimal sections and proportions|p=37|ps=; the regular polygons {5}, {7}, {9} and {11} with their diagonals define respectively: {5} the golden bisection proportional to 𝜙; {7} an analogous trisection; {9} an analogous quadrasection; {11} an analogous pentasection.}} The chord length &lt;math&gt;l&lt;/math&gt; is related to the number of sides of the regular polygon formed by the chord (its numerator &lt;math&gt;k&lt;/math&gt;), and to the winding number or density of its regular skew Petrie polygon (its denominator &lt;math&gt;d&lt;/math&gt;).

The regular 4-polytopes do not contain an infinite variety of these chord flavors, but only 30 of them: only 15 unique pairs of 180° degree complementary chords, which combine to make every regular ''n''-dimensional thing found in nature.{{Efn|The 15 ''major chords'' suffice to construct all the regular ''n''-polytopes. Their 180° complements include 15 ''minor chords'' which occur only in the 120-cell, not in any smaller regular polytope.{{Efn|name=Major and minor chords}} Including the complement of the 180° {{radic|4}} digon {2} chord (the {0} chord {{radic|0}}), and the skew variant of the 90° {{radic|2}} square {4} chord (the {30/7} chord), there are 32 distinct natural numbers which are all the chords of all the regular polytopes in all dimensions.|name=32 chords}} All the regular polytopes (including the largest one on the cover of the box) can be built not only from a box containing a number of blocks of 3 shapes glued together by gravity, they can be also be built from a box containing a number of sticks of 15 lengths glued together by rubber joints.

[[File:15 major chords.png|500px|thumb|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the others occur only in the interior of 120-cell, not as edges of any of the regular 4-polytopes. There are 30 distinct 4-space chordal distances between vertices within the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to identify any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=Major and minor chords}} chords are properly labelled #1 — #15 in the order they fan out by size.{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''; Figure 3|ps=; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with a &quot;fan of chords&quot; diagram like the one here.|p=23}} They join vertex pairs which are 1 — 15 edges apart on a &lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon. A #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; chord forms &lt;small&gt;&lt;math&gt;f+1&lt;/math&gt;&lt;/small&gt; regular geodesic polygons which include a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; disjoint regular {{nowrap|&lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt;-gons}} (e.g. &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; a compound of 5 disjoint hexagons) plus a single [[W:Triacontagon#Triacontagram|regular skew {{nowrap|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram}}]] that wraps around the sphere &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times before closing its circle. Notice the maverick #11 chord, with its non-conforming origin point for its {{nowrap|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;}} polygram.]]

These 15 chords (natural numbers) include the integers 2, 3, 4, 5, 6, 10, 11, 15 and 30. The others are the fractions 5/2, 10/3, 13/7, 15/2, 15/4, and 15/7. Represented among the chord lengths (the square roots) measured in ''unit radii'' (not in {{radic|2}} radii, as metrics are often given elsewhere in this article) are the {{background color|paleturquoise|roots of the first 4 positive integers {{radic|1}}, {{radic|2}}, {{radic|3}} and {{radic|4}}}}, the {{background color|#FFCCCC|root of the very distinguished rational fraction {{radic|5/2}}}}, and the roots of various distinguished irrational fractions as Steinbach discovered: for example, 4 of them are {{background color|yellow|roots of golden numbers (a small integer ± 𝜙)}}. The 15 chords each form their regular polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, which is just &lt;small&gt;&lt;math&gt;\{k\}&lt;/math&gt;&lt;/small&gt; in 8 cases. Arranged in order #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; from shortest edge-length square root to longest, the #1 — #15 chords bridge vertex pairs which are &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; vertices apart on a [[W:Triacontagon#Petrie polygons|&lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon]]. Each chord makes a [[W:Triacontagon#Triacontagram|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; polygram]] of its edge length, a compound of the smaller &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. The 15 distinct polygons form a sequence you would not anticipate exactly unless you have Steinbach's formula:

{| style=&quot;background-color:transparent; white-space:nowrap;&quot;
|-
| {30/1} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|.073}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{30} 
| 120-cell edges
|-
| {30/2}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.191}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15}
|- style=&quot;background: yellow;&quot;|
| {30/3}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.382}} 
| &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi}&lt;/math&gt;&lt;/small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{10}''' 
| 600-cell edges
|-
| {30/4} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.573}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/2}
|- style=&quot;background: paleturquoise;&quot;|
| {30/5}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|1}}'''
| &lt;math&gt;1&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{6}''' 
| 24-cell, 8-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/6}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|1.382}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{5}'''
|- style=&quot;background: paleturquoise;&quot;|
| {30/7}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|2}}'''
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{4}''' 
| 16-cell edges
|- style=&quot;background: #FFCCCC;&quot;|
| {30/8}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.5}} 
| &lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;&lt;small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/4} 
| 5-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/9}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.618}} 
| &lt;math&gt;\phi&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{10/3}
|- style=&quot;background: paleturquoise;&quot;|
| {30/10}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|3}}''' 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{3}'''
|-
| {30/11}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.426}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{11/11}
|- style=&quot;background: yellow;&quot;|
| {30/12}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.618}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{5/2}
|-
| {30/13}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.810}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{13/7}
|-
| {30/14}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.928}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/7}
|- style=&quot;background: paleturquoise;&quot;|
| {30/15}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|4}}''' 
| &lt;math&gt;2&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{2}'''
|}

{| class=wikitable style=&quot;white-space:nowrap; float:right;&quot;
!colspan=6|Natural number &lt;small&gt;&lt;math&gt;h=\{k/d\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Fibers&lt;br&gt;&lt;small&gt;&lt;math&gt;f\{h\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Cells&lt;br&gt;&lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|4-tope&lt;br&gt;&lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Verts&lt;br&gt;&lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|[[W:Triacontagon#Triacontagram|30-gram]]&lt;br&gt;&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;
|-
!Chord
!Arc
!colspan=2|&lt;small&gt;&lt;math&gt;l/\sqrt{1}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;l/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;h&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#1
|15.5~°
|{{radic|0.073}}
|0.270
|{{radic|0.146}}
|&lt;small&gt;&lt;math&gt;30&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1\{30\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;5\{3,3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/1\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#2
|25.2~°
|{{radic|0.191}}
|0.437
|{{radic|0.382}}
|&lt;small&gt;&lt;math&gt;15&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;..\{3,3\}..&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/2\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#3
|36°
|{{radic|0.382}}
|0.618
|{{radic|0.764}}
|&lt;small&gt;&lt;math&gt;10&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;30\{3,3\}20&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/3\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#4
|44.5~°
|{{radic|0.573}}
|0.757
|{{radic|1.146}}
|&lt;small&gt;&lt;math&gt;15/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/4\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5
|60°
|{{radic|1}}
|1
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;6&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/5\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#6
|72°
|{{radic|1.382}}
|1.175
|{{radic|2.624}}
|&lt;small&gt;&lt;math&gt;5&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}} 
|&lt;small&gt;&lt;math&gt;10\{5,3\}12&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/6\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#7
|90°
|{{radic|2}}
|1.414
|{{radic|4}}
|&lt;small&gt;&lt;math&gt;4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7\{4\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;4\{4,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;9\{3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/7\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: #FFCCCC;&quot;|
|#8
|104.5~°
|{{radic|2.5}}
|1.581
|{{radic|5}}
|&lt;small&gt;&lt;math&gt;15/4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/4\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}} 
|&lt;small&gt;&lt;math&gt;15\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/8\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#9
|108°
|{{radic|2.618}}
|1.618
|{{radic|5.236}}
|&lt;small&gt;&lt;math&gt;10/3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10/3\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/9\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#10
|120°
|{{radic|3}}
|1.732
|{{radic|6}}
|&lt;small&gt;&lt;math&gt;3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;10\{3\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/10\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#11
|135.5~°
|{{radic|3.426}}
|1.851~
|{{radic|6.852}}
|&lt;small&gt;&lt;math&gt;11/11&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;11\{11/11\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;11\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#12
|144°
|{{radic|3.618}}
|1.902
|{{radic|7.235}}
|&lt;small&gt;&lt;math&gt;5/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2},3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/12\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#13
|154.8~°
|{{radic|3.810}}
|1.952
|{{radic|7.621}}
|&lt;small&gt;&lt;math&gt;13/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;13\{13/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;7\{5,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/13\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#14
|164.5~°
|{{radic|3.928}}
|1.982
|{{radic|7.857}}
|&lt;small&gt;&lt;math&gt;15/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;137\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/14\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{radic|8}}
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15\{2\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐𝜙&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;8\{3,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/15\}&lt;/math&gt;&lt;/small&gt;
|}

The polygons '''{10}''', '''{6}''', '''{5}''', '''{4}''', '''{3}''', and '''{2}''' are flat polygons that each lie on a great circle in a central plane of the 3-sphere. The polygons {30}, {15}, {15/2}, {15/4}, {10/3}, {11/11}, {5/2}, {13/7}, {15/7} and all the {30/''n''} polygons are skew polygrams spiraling through curved 3-space, each successive edge in a different central plane of the 3-sphere. Projected to the 1-sphere (a circle in the plane), they are [[W:Star polygon#Regular star polygon|regular star polygrams]], the simplest being the pentagram {5/2}. In 4-space they wind around the 3-sphere more than once ''without self-intersecting'', before they close their circuit. They lie on helical geodesic circles or ''isoclines'' of circumference greater than 2𝝅''r''.

Even the 6 integer chords which form flat polygons also form skew polygrams. Each of the 15 chords occurs as a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, a fiber bundle of Clifford parallel polygons that totals 30 vertices per bundle. Each chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; also forms a single skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram that winds through another, shorter chord's bundle of parallel polygons, intersecting them in succession multiple times each, and weaving that whole bundle of parallel fibers together as the cross-thread does on a loom. The polygram cross-thread has to wind multiple times through the whole bundle (&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times around the [[W:Triacontagon#Triacontagram|30-gram]] in which each edge connects every &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt;th vertex) to complete an enumeration of all 30 vertices in the bundle. For example, the #5 chord forms hexagon &lt;small&gt;&lt;math&gt;\{6\}&lt;/math&gt;&lt;/small&gt; flat planes in parallel bundles of 5, written &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt;, and also a skew &lt;small&gt;&lt;math&gt;\{15/2\}&lt;/math&gt;&lt;/small&gt;-gram of #5 edges which winds 2 times through the #3 chord's bundle of 3 decagon &lt;small&gt;&lt;math&gt;\{10\}&lt;/math&gt;&lt;/small&gt; flat planes. Every chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; forms both its &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram and its fiber bundle of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, and the edges of one chord's parallel polygons are also the edges of its skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram weaving through some shorter chord's bundle of parallel polygons. Each numbered chord is a natural number, a ratio of two integers and their polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, either flat or skew. Each chord number defines a &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; skew polygon that is a compound bundle of the &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. All these flat or skew polygons are regular polygons made of their numerator instances of a single chord #, whose length is that square root, Steinbach's function of the natural number.

The relationship between each bundle of flat Clifford parallel great circle polygons of a distinct length, and its corresponding bundle of skew polygrams of a longer length, is precisely that of a discrete simple rotation and its corresponding discrete isoclinic rotation. For example, the flat great circle hexagons of edge length {{radic|1}} lie in the invariant rotation planes of the 24-cell's characteristic 60° isoclinic rotation, in which the vertices move on skew hexagram rotation paths of edge length {{radic|3}}. The 24-cell contains 4 bundles of 4 Clifford parallel (disjoint) hexagon central planes each. The 24 vertices of the 24-cell in each bundle are linked by rotation in the 4 hexagon invariant planes in two different ways: simply and isoclinically. In a simple rotation the vertices of a single invariant hexagon exchange places on that great circle hexagon of edge {{radic|1}}. In an isoclinic rotation in which any one of these 4 disjoint hexagon central planes is invariant, all 4 of them are necessarily invariant planes in the rotation, because they rotate in parallel. In that isoclinic rotation the vertices of all 4 hexagon invariant planes (all 24 vertices of the 24-cell) exchange places concurrently, moving along 4 disjoint helical hexagrams of edge {{radic|3}} in parallel. It is a characteristic only of this distinct rotation in hexagon invariant planes that the helical vertex path or ''isocline'' is a hexagram of period 6, edge length {{radic|3}}, and circumference 720°. Rotations in other kinds of polygon invariant planes have different characteristic isoclines. The period of their isoclinic rotation must be ''the same or greater'' than the period of their simple rotation; in most cases (unlike this one) it is greater.

The 15 chords form not only distinct fiber bundles of polygons, but also distinct bundles of polyhedra: cell rings &lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt; and compound vertex figures &lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;. These combine to form their characteristic regular 4-polytope &lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Coxeter|1973|p=269|loc=Compounds|ps=; &quot;It is remarkable that the vertices of {5, 3, 3} include the vertices of all the other fifteen regular polytopes in four dimensions.&quot;}} All these elements of a regular 4-polytope constructed from one chord are themselves regular polytopes, and each of the 16 regular 4-polytopes is a function of one characteristic chord. If the regular 4-polytope is convex, its characteristic chord is its unit-radius edge. For a non-convex regular 4-polytope, the characteristic chord is its unit-radius ''mid-edge diameter'', the distance from the midpoint of each edge to the midpoint of the completely orthogonal edge. Each of the characteristic chords (each distinct natural number) has its expression as one of the 16 regular 4-polytopes and all the products of the distinct regular 1-polytopes, 2-polytopes and 3-polytopes that construct it, and the whole distinct construction from the natural number to the 4-polytope is an [[W:SO(4)|SO(4) symmetry group]] expression of that 4-polytope's characteristic discrete [[W:SO(4)#Isoclinic rotations|isoclinic rotation]], in the invariant central planes of its characteristic chord, by the characteristic chord arc.

...demonstrate this theory by proper enumeration of the chord lengths in the table...

...color-code the △ ☐ 𝜙 ✩ symbols ... chord colors in diagram should match them ... some of the diagram colors are wrong and must be changed, but otherwise diagram is pretty good ... i think in a colored diagram all chords of the same length should have the same size line dash, the polygons are not obvious enough and the 15 fanning-out chords are readily identified by their labels...

... add back the counts of chords in the box (very small font number under each table entry), and present these two images as the front and back a box of building sticks and rubber joints...What's in the other box...

...say the rational-root major chords {{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|5/2}} are the ''n''-edges: their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge...

...abbreviated text of the [[120-cell#Relationships among interior polytopes|''§Relationships among interior polytopes'']] section from the 120-cell article...to explain the color-coding

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius=1}}

See also the elements, properties and metrics of the sequence of regular 4-polytopes summarized in tabular form in the section ''[[#Build with the blocks|§Build with the blocks]]'', below.

== Alicia Boole Stott's original formulation of dimensional analogy ==
&lt;blockquote&gt;&quot;...a method by which bodies having a certain kind of semi-regularity may be derived from regular bodies in an Euclidean space of any number of dimensions...&quot;{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''; presents cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues|pp=12-45}}&lt;/blockquote&gt;

Alica Boole Stott's 1910 paper reads like an exceptionally well-written patent application. It is a comprehensive stepwise description of her original invention, a general method of dimensional analogy between regular and semiregular polytopes of the same edge length.

[[File:The Roots of Three.png|thumb|An illustration of Boole Stott's expansion of unit-edge triangle to hexagon captures the geometric basis in 3 of the &lt;math&gt;A_n&lt;/math&gt;root system.]]

The engineering-like articulation of her new principles is entirely discrete and group-theoretic. Boole Stott repeatedly refers to sets of polytope elements as ''groups''. All her transformations are discrete instances of one or another special case of her &quot;two inverse operations which may be called ''expansion'' and ''contraction''.&quot; Hers is an exploration of ''n''-dimensional space at its most basic: space defined by the discrete objects which fill it, themselves defined directly as groups of elements with a few operations on them, which transform them into each other. This is geometry which seems to precede Euclid's postulates, and does not depend on them. Instead it corresponds directly to the deeper group theory mathematics which underlies geometry, even though Boole Stott, like the ancients, did not have that mathematics.

Boole Stott enumerated her expansion and contraction operations as transformations between regular and semi-regular polytopes of a single edge length. In her 1910 paper she did not consider any polytopes of multiple edge lengths, or expansions of the subject polytope by a distance other than that exactly required to produce new edges of the original edge length, so of course she did not enumerate Moxness's non-uniform rhombicosidodecahedron (Hull #8). This may be why section 8&lt;sub&gt;3&lt;/sub&gt; is missing from her earlier drawings of the sections of {5,3,3}; she may not have had a construction for it.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The Boole Stott expansion (or contraction) operation is as precisely defined a transformation as Coxeter's reflection, rotation and translation operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions). Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;
where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;
where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;
For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} An instance of the expansion (or contraction) operation is an application of the general operation to a particular subject polytope, operating on all of its elements symmetrically at once. The general operation is restricted, such that one may only expand (or contract) the subject polytope by pushing all its edges, faces or cells directly away from (or toward) its center, without changing the size of the moved elements. This splits each vertex into several (or coalesces several vertices into one), and has corresponding effects on the other elements, producing new edges ''of the same size'' between pairs of separated vertices (or reducing some edges to single vertices). The distance all the elements are moved away from (or toward) the center must be such that the vertices separate producing new edges of only the original edge length (or come together reducing edges to single vertices). It is permitted to combine these three operations, e.g. by pushing the edges ''and'' the faces, or by pushing all three element sets, provided the combination results in a semi-regular polytope of only one edge length. The three operations are all commutative with one another, so combined operations can be considered either as a sequence of expansions (or contractions) linking several related polytopes, or as their concurrent movements, a single transformation of the subject polytope. Since there are 8 combinations of one to three operations there are 8 polytope variants (including the original regular polytope), and all are adjacent to one another by a single transformation. The variants of the original regular polytope are named for which element set(s) are moved. By enumerating all 8 variants of expansion (and contraction) for each of the six regular convex 4-polytopes, Boole Stott produced a table of regular and semi-regular 4-polytopes divided into 12 sets of 8: six expansion sets and six contraction sets.{{Sfn|Boole Stott|1910|loc=''Table of Polytopes in S&lt;sub&gt;4&lt;/sub&gt;''; note that Boole Stott's usage of ''S&lt;sub&gt;4&lt;/sub&gt;'' denotes a Euclidean 4-space ''E&lt;sub&gt;4&lt;/sub&gt;'', or a 3-sphere space embedded in it, ''not'' a 4-sphere space embedded in a Euclidean 5-space|p=30}} Each set includes just one of the six regular convex 4-polytopes, with its distinct unit-radius edge length, and the 7 possible semi-regular expansions (or contractions) of it, which all preserve that characteristic edge length. The polytopes in each set of 8 all have the same total number of elements as their original regular polytope, but each polytope in the set has a distinct combination of cell, face, and vertex figures. The 12 sets are not completely disjoint, and even within one set the 8 operations do not necessarily produce 8 unique polytopes. For example, the regular 5-cell (the 4-simplex polytope) has 5 cells, 10 faces, 10 edges, and 5 vertices (30 elements). Expanding the 5-cell produces 7 related semi-regular 4-polytopes of 30 elements each (two of which happen to be the same polytope). In each expansion, the 5 cells have expanded into semi-regular or regular polyhedra, the 10 faces and 10 edges ''may'' have expanded into prisms, and the 5 original vertices have expanded into regular or semi-regular cells.

== The regular 11-cell 4-polytope ==

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosadodecahedron). Each non-convex Jessen's icosahedron is a set of 6 reflex edges that are {{radic|5}} 5-cell edges, each set symmetrically arranged in 3 parallel pairs, not as far apart ({{radic|3}}) as they are long, in 3 orthogonal central planes. The 12-point (Jessen's sets of 6 reflex 5-cell edges) are inscribed in the 60-point with their endpoints as a vertex of a pentagon face. The 12 pentagon faces are joined to 5 other pentagon faces by the 5-cell edges; not to their nearest neighbor pentagons but to their next-to-nearest neighbor pentagons; and since there are 5 vertices to each pentagon face, there are 10 12-point (Jessen's sets of 6 reflex 5-cell edges). Each pentagon face is joined to another pentagon face by 5 reflex edges of five distinct Jessen's. The 5 reflex edges are Clifford parallel, the 5 edges of a pentagonal prism column that has a twist to it. The contraction of those 5 edges into 1, and those 10 vertices (two pentagons) into 2, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing all their edges as a single Jessen's. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (octahedron) by conflating its parallel pairs of edges, leaving the 11-cell's abstract 6-point (hexad cell). The hexad's situation is that it occurs in bundles of 5, disjoint but close together in the sense of adjacent, like the great circle rings in a Hopf fiber bundle. 

To summarize, the 12-point (2-hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell). We shall now be able to see how Fuller's perfect hexads combine with Todd's perfect pentads to form the perfect heptads of the 11-cell and 137-cell.  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=alias the hemi-cuboctahedron}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces. (The ''heptad'' arrives at the party, anonymously; they are our distinguished anti-celebrity, who most of the other guests have never even heard of.) The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. This 12-point (truncated tetrahedron) is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 100 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp: Truncated tetrahedron|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp: Truncated tetrahedron|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}} Of course Moxness has been there, seen that, already with his graphics software;{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} see his parameterized illustration of it exactly as it occurs in the 11-cell, below in his bestiary of A3, B3, and H3 Archimedian and Catalan polyhedra in the 120-cell.]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:Archimedean_and_Catalan_solid_hulls_with_their_Weyl_orbit_definitions.svg|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of the 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. In this catalog of Archimedean and Catalan polyhedra sighted in the 120-cell by Moxness's software,{{Sfn|Moxness: Archimedean and Catalan hulls|2023|loc=Hull #1 Archimedean Name A3 110 Truncated Tetrahedron A (upper left)}} it is the first entry (upper left), Moxness's Hull # = 1 with 12 vertices of 3D norm {{radic|11}}/2. On a {{radic|2}}-radius 3-sphere, 5 of these with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell. 20 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges are cells in the [[#The rings of the 11-cells|cell rings of the quasi-regular 11-cell]], which also has 12 regular 5-cells as cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

...

Both kinds of cells in the 11-cell have expressions in two different dimensions (on the 2-sphere and 3-sphere), because they are all centered on the center of the 3-sphere, unlike the cells of most 4-polytopes, which are centered on various off-center points within the 4-polytope's interior. 

One kind of 11-cell cell appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (semi-regular truncated cuboctahedron), a.k.a. the truncated tetrahedron. The other kind of cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now, except in the 120-cell. But this is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron, and a regular tetrahedron behind each face. The 11-cell is thus a honeycomb of mixed regular tetrahedra and truncated cuboctahedra (a.k.a. truncated tetrahedra) that meet at their shared 5-cell faces. 

Both the 5-cells (decahedra) and the truncated cuboctahedra (truncated tetrahedra) are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks in their 4-polytopes (off-center).{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} Therefore all the 11-cell's cells are concentric, which indeed makes a different kind of honeycomb, an inside-out cellular honeycomb with all the cells piled on top of each other overlapping in 4-space. They do not overlap in 3-sphere space however; their 3-volumes (but not their 4-contents) are disjoint. They make a proper 4D honeycomb like the cells of any 4-polytope, with volumetrically disjoint 3D cells that meet each other in the proper way at their shared 2D faces.

An 11-cell contains 6 completely disjoint 5-point (1-pentad 5-cell decahedron cells), and 5 volumetrically disjoint 12-point (2-hexad truncated cuboctahedron cells), 11 volumetrically disjoint cells altogether. Of course it is supposed to have 11 identical 11-point (elevenad hemi-icosahedral cells), and that is also true in its abstract sense, which is not the same sense as disjoint cellular containment: each pentad or hexad cell is inscribed in one of 11 larger 60-point (Moxness's rhombicosadodecahedron cells). The abstract pentads and hexads are only hemi-polyhedra, so the 11-cell's pentad cells are 12 (out of 120) disjoint instances of a pentad 5-point, and its hexad cells are 10 (out of 100) disjoint instances of a hexad 6-point, 1/10th of the 120-cell's cellular volume in both cases. The 11-cell's hemi-icosahedral cells are 11 (out of 120) non-disjoint instances of an elevenad 60-point. Each 11-cell shares each of its hexad cells with 9 other 11-cells, and each of its 11 vertices (out of 600) occurs in 9 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.|name=Tetrahedral vertex figures}}

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

...insert image thumb of 11-cell...

...Construct the 11-cell from the heptad....

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...Four concentric 4-polytopes may be mutually orthogonal. In any set of 4 mutually orthogonal regular 5-cells, one pair of faces in each orthogonal pair are completely orthogonal to each other. Thus three edges in one 5-cell in an orthogonal pair are opposite (completely orthogonal) to a face of the orthogonal 5-cell. .... A 5-cell has 5 vertices in its abstract hemi-icosahedral cells, but only 4 vertices in one concrete icosahedral cell; the 5th lies in an icosahedral cell of another 11-cell (not in another icosahedral cell of the same 11-cell).

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

....

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 11-cell ===

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon.... (wrong, 11-cell's Hopf map is the truncated icosahedron)

....

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk: Truncated icosahedron|2018}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

...

Since the rhombicosadodecahedron has 120 5-cell edges, the 120-cell has 120 completely disjoint 5-cells, and the rhombicosadodecahedron shares each vertex with 11 other rhombicosadodecahedra, how a rhombicosadodecahedron is bound to its neighbor 5-cells is constrained. A further constraint is that the abstract hemi-icosahedron has only 15 5-cell edges, so the 120 actual 5-cell edges of the rhombicosadodecahedron are conflated 8-to-1 in the hemi-icosahedron. To be conflated by contraction into a single edge, those 8 edges must lie parallel to each other in the 120-cell; either actually parallel in the same plane, or Clifford parallel in separate planes, as two edges in completely orthogonal planes may be parallel and orthogonal at the same time. Each 5-cell inscribed in the 120-cell contains 5 such tetrahedra occur in parallel pairs in 4 orthogonal planes (none of them completely orthogonal planes). 

.... 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
Each conflated vertex is the 10 vertices of two orthogonal regular 5-cells...
The 60-point contains 6 pairs of these orthogonal regular 5-cell cells...
...its 20 hexad cells are ..., a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere.

== The regular 137-cell 4-polytope ==

(description originally written for the 11-cell, which applies only to a fibration of 11 11-cells)

It is quasi-regular with two kinds of cells, the 12-point (regular icosahedron) and the 5-point (regular 5-cell), which have just one kind of regular face between them. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of regular facets (it is semi-regular with regular facets), it also means that its facets have expressions of two different dimensionalities because they are concentric in the 3-sphere. The 11-cell has two kinds of regular cells. One appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (regular icosahedron), but since it is embedded at the center of the 3-sphere (unlike the cells of most 4-polytopes) it also acts as an inscribed 4-polytope.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} The other cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now except in the 120-cell. This is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron. The 11-cell is a honeycomb of mixed icosahedron and half-icosahedron cells. Like the icosahedra, the decahedra are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks, off-center in their 4-polytopes. All the 11-cell's cells are concentric, which indeed makes an unusual honeycomb of cells, an inside-out honeycomb. It is a proper 4D honeycomb however, with volumetrically disjoint 3D cells that meet each other in the proper way at shared 2D faces. The embedded 12-point (regular icosahedron) cell is an icosahedral pyramid (as a 4-polytope), which is filled by 20 4-point (irregular tetrahedra) that meet at its center; the center point of the 11-cell acts as a 12th vertex, just as the center point of the 24-cell acts as a 25th vertex in some honeycombs. The 5-point (regular 5-cell) cell contains five 4-point (regular tetrahedra) which meet the 12-point (regular icosahedra)'s irregular tetrahedral cells face-to-face, so the 11-cell is quasi-regular in the usual sense of having two varieties of regular 3-polytope cells, but additionally both varieties of of its cells are bi-dimensional: another clue that the 11-cell has something to do with dimensional analogy

....

The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

....

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur to make an 11-cell. 11 of them occur in a 137-cell, all 120 of them occur in the 120-cell, and they occur nowhere else.

How can this regular 4-polytope possibly exist? By Schläfli's criterion {3, 3, ..} exists, but not as a finite polytope.{{Sfn|Coxeter|1973|loc=§7.7 Schläfli's criterion}} Coxeter found {3, 5, 3} as a finite hyperbolic honeycomb, but not in Euclidean 4-space.{{Sfn|Coxeter|1968|title=The Beauty of Geometry: Twelve Geometric Essays|loc=''Regular Honeycombs in Hyperbolic Space''|ps=; &quot;The honeycombs considered by the above authors [Schlegel, Schläfli, Klein, Sommerville] have finite cells and finite vertex figures. ... A further extension allows a cell or vertex figure to be a ''star''-polytope, so that the honeycomb covers the space several times. .... We shall find that there are four regular star-honeycombs in hyperbolic 4-space, as well as two infinite families of them in the hyperbolic plane.&quot;}} We find that {3, 3, ..} is a central expression of the &lt;math&gt;H_4&lt;/math&gt; root system, being the regular convex hull in 4 dimensions of 137 points, 96 pentads and 600 hexads. The minimal expression of &lt;math&gt;H_4&lt;/math&gt; is the regular convex hull {3, 3, 5} of 120 points, 600 tetrads and 75 hexads, and its maximal expression is the regular convex hull {5, 3, 3} of 600 points, 120 pentads and 675 hexads. {3, 3, ..} is a third regular &lt;math&gt;H_4&lt;/math&gt; polytope which fits between them. The properties and metrics of this regular sequence are presented in tabular form below in ''[[#Build with the blocks|§Build with the blocks]]''.

{|class=&quot;wikitable&quot;
!colspan=3|Regular &lt;math&gt;H_4&lt;/math&gt; polytopes
|-
|[[File:600-cell.gif|200px]]
|[[File:5-cell.gif|200px]]
|[[File:120-cell.gif|200px]]
|-
!120-point 600-cell
!137-point 137-cell
!600-point 120-cell
|}

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 137-cell ===
....

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=Apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|2022|loc=''[[Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]''}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie: On Fuller's use of language|2024|loc=''[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry'']]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center, 1948-1949|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with their mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the gender neutral shad can open their six beaks all the way, until they become flat squares and they becomes a cuboctahedron, or they can shut them all tight like a turtle retracting into their octahedron shell. The six mouths always move in unison. This is [[Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do their ''really'' odd trick -- where they flip their 6 jaws 90 degrees in their 6 faces and shut their 6 beaks on the opposite axis of their squares than the one they opened them on -- or whether they will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than their normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make their edge length exactly the same size as their radius, when they open their mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing their mouths in spherical synchrony, their 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron (the hexad), then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=''[[W:SO(4)|SO(4)]]''}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=Isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its orthogonal hexad's planes (wz, wx, wy).{{Efn|name=Six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of (expanded octahederon 16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh &amp; Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== Legendre's 12-point hexad binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a 12-point double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's perfectly realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, this 2-hexad expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre hexad) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction.{{Efn|name=Concentric polyhedra}} In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== ''n''-simplexes ===
....

The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== ''n''-orthoplexes ===
....
[[File:Great 5-cell √5 digons rectangle.png|thumb|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} ''regular'' great hexagons. These are the same 200 dodecagon central planes illustrated above, which also contain 6 120-cell edges and two opposing ''irregular'' great hexagons (truncated {{radic|6}} triangles). The 1200 {{radic|5}} 5-cell face triangles occupy 1200 other, non-central planes.]]

...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's Hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[16--cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[16--cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=Concentric polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[16--cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== ''n''-taliesins ===
The 4-simplex pentad building block combines with the 4-orthoplex hexad building block to make the 4-taliesin heptad building block. The heptad is the essential building block of the ''taliesin'' family of polytopes. The 11-cell and 57-cell hemi-polytopes and the 137-cell and 120-cell regular 4-polytopes are nuclear members of this trans-dimensional family. The 24-cell and 600-cell are extended members of it, because they also contain instances of the heptad building block.

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its spherical symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* [[W:Taliesin Myrddin Namkai-Meche|Taliesin Myrddin Namkai-Meche]], a young Reed College scholar who was stabbed to death by a crazy man in 2017 on a Portland, Oregon light rail train, when he rose to the nonviolent defense of two black teenagers who were being violently assaulted in racist and anti-Muslim language by a right-wing extremist.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in orthogonal [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].
* '''Taliesan polytope''', an (''n+1)''-polytope whose ''n''-polytope dimensional analogue inscribed concentrically in the ''n''-sphere forms a convex honeycomb of itself which is the (''n+1)''-polytope.{{Efn|name=Embedding point}}
* The taliesan polytope's honeycomb, a Hopf fibration of its great circles.
* The ''n''-dimensional family of dimensionally analogous polytopes related to a taliesan polytope by expansion and contraction operations.
* The characteristic [[w:Orthoscheme|orthoscheme]] of a teliesan polytope, an irregular simplex that is the complete gene for the entire expression of trans-dimensional symmetry relationships among the polytopes of the ''n''-dimensional family, as well as the complete gene for the teliesan polytope's rotational symmetry.{{Efn|name=Characteristic orthoscheme}}

=== ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== ''n''-equilaterals ===
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed at four different [[#n-leviathon|chord]] lengths from each other. In the {{radic|2}}-radius polytopes we have using as examples in this paper they are {{radic|2}}, {{radic|4}}, {{radic|6}} and {{radic|8}}, but in a ''unit-radius'' polytope they are: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}.

Each vertex is joined to 8 others by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}. The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== ''n''-pentagonals ===

[[File:600-cell.gif|thumb|The 120-point (600-cell), &lt;math&gt;H^4&lt;/math&gt;.]]
...

[[File:600-cell vertex geometry.png|thumb|Vertex geometry of the 600-cell, showing the 5 regular great circle polygons and the 8 vertex-to-vertex chord lengths with angles of arc. The golden ratio{{Efn|name=golden chords|group=}} governs the fractional roots of every other chord,{{Efn|name=fractional root chords}} and the radial golden triangles which meet at the center.|alt=|400x400px]]
The 120 vertices of the 600-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏&lt;sup&gt;−1&lt;/sup&gt;) beginning with a vertex; see column ''a''}} at eight different [[#n-leviathon|chord]] lengths from each other. These edges and chords of the 600-cell are simply the edges and chords of its five great circle polygons. In ascending order of length in a ''unit-radius'' polytope (not in a {{radic|2}} radius polytopes like our usual examples), they are {{radic|0.𝚫}}, {{radic|1}}, {{radic|1.𝚫}}, {{radic|2}}, {{radic|2.𝚽}}, {{radic|3}}, {{radic|3.𝚽}}, and {{radic|4}}.{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=Decimal fractions of square roots|group=}}

Notice that the four [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#''n-''equilaterals|hypercubic chords]] of the 24-cell ({{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|4}}) alternate with the four new chords of the 600-cell's additional great circles, the decagons and pentagons. The new ''golden chord'' lengths are necessarily square roots of fractions, but very special fractions related to the golden ratio{{Efn|1=The fractional-root ''golden chords'' are irrational fractions that are functions of {{radic|5}}. They exemplify that the [[W:golden ratio|golden ratio]] &lt;big&gt;𝜙&lt;/big&gt; {{=}} {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618 is a circle ratio related to &lt;big&gt;𝜋&lt;/big&gt;:&lt;br&gt;
: {{sfrac|𝜋|5}} = arccos (𝜙/2)
is one decagon edge, the 𝚽 = {{radic|0.𝚫}} = {{radic|0.382~}} ≈ 0.618 chord.
Reciprocally, in this function discovered by Robert Everest expressing &lt;big&gt;𝜙&lt;/big&gt; as a function of &lt;big&gt;𝜋&lt;/big&gt; and the numbers 1, 2, 3 and 5 of the Fibonacci series:&lt;br&gt;
: &lt;big&gt;𝜙&lt;/big&gt; = 1 – 2 cos ({{sfrac|3𝜋|5}})
{{sfrac|3𝜋|5}} is the arc length of the &lt;big&gt;𝜙&lt;/big&gt; = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618 chord.|name=golden chords|group=}} including the two [[W:golden section|golden section]]s of {{radic|5}}, as shown in the diagram.{{Efn|The 600-cell edges are decagon edges of length {{radic|0.𝚫}}, which is 𝚽, the ''smaller'' golden section of {{radic|5}}; the edges are in the inverse [[W:golden ratio|golden ratio]] {{sfrac|1|φ}} to the {{radic|1}} hexagon chords (the 24-cell edges).
The other fractional-root chords exhibit golden relationships as well. The chord of length {{radic|1.𝚫}} is a pentagon edge.
The next fractional-root chord is a decagon diagonal of length {{radic|2.𝚽}} which is &lt;big&gt;𝜙&lt;/big&gt;, the ''larger'' golden section of {{radic|5}}; it is in the golden ratio{{Efn|name=golden chords|group=}} to the {{radic|1}} chord (and the radius).{{Efn|Notice in the diagram how the &lt;big&gt;𝜙&lt;/big&gt; chord (the ''larger'' golden section) sums with the adjacent 𝚽 edge (the ''smaller'' golden section) to {{radic|5}}, as if together they were a {{radic|5}} chord bent to fit inside the {{radic|4}} diameter.}}
The last fractional-root chord is the pentagon diagonal of length {{radic|3.𝚽}}.
The [[W:Pentagon#Side length is given|diagonal of a regular pentagon]] is always in the golden ratio to its edge, and indeed &lt;big&gt;𝜙&lt;/big&gt;{{radic|1.𝚫}} is {{radic|3.𝚽}}.|name=fractional root chords|group=}}

....


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== ''n''-leviathon ===
...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of its &lt;math&gt;10^2&lt;/math&gt; disjoint instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction, before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]].{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}} .... It is only because she did not have the 11-cell to start with that she did not discover the regular 137-cell {3, 3, ..} by her method.

Between Alicia Boole Stott and her identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and her identification of symmetry groups with conservation laws (in her 1st theorem), we owe the deepest mathematics of geometry and physics originally to two pioneering women, colleagues of their contemporaries the greatest men of the academy, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just an [[W:abstract polytope|abstract 4-polytope]], not just a singleton convex 4-polytope, and not just a honeycomb.{{Sfn|Coxeter|1970|loc=''Twisted Honeycombs''}} The 11-point (11-cell) has a concrete ...regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}&lt;nowiki/&gt;} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, 3, 3} as the 137-point (137-cell).

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 11-cell, 137-cell and 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=''Clifford Spinors and Root System Induction: H4 and the Grand Antiprism''}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=Apology}}

== Build with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Duveneck|1978|loc=Carl Jung, quoted in ''Life on Two Levels''|p=ii|ps=.{{Sfn|Jung|1961|ps=: &quot;The best of truths is of no use unless it has become one's most personal inner experience. It is the duty of everyone who takes a solitary path to share with society what he finds on his journey of discovery.&quot;}}}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1954|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius={{radic|2}}}}

== Acknowledgements ==
...

== Notes ==
{{Regular convex 4-polytopes Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Citation|last=Boole Stott|first=Alicia|author-link=W:Alicia Boole Stott|title=Geometrical deduction of semiregular from regular polytopes and space fillings|date=1910|pages=12-45|publisher=Johannes Muller|place=Amsterdam|url=https://dwc.knaw.nl/DL/publications/PU00011492.pdf}}
* {{Citation|last=Schoute|first=Pieter Hendrik|author-link=W:Pieter Hendrik Schoute|title=Analytical treatment of the polytopes regularly derived from the regular  polytopes|date=1911|pages=114-197|publisher=Johannes Muller|place=Amsterdam|url=https://www.google.com/books/edition/Analytical_Treatment_of_the_Polytopes_Re/G_oxAQAAMAAJ?hl=en&amp;gbpv=0}}
* {{Citation | last=Grünbaum | first=Branko | author-link=W:Branko Grünbaum | year=1976 | title=Regularity of Graphs, Complexes and Designs | journal=Colloques Internationaux C.N.R.S. | publisher=Orsay | volume=260 | pages=191-197 | url=https://faculty.washington.edu/moishe/branko/BG111.Regularity%20of%20graphs,etc.pdf | ref=}}
* {{Citation | last=Grünbaum | first=Branko | author-link=W:Branko Grünbaum | year=1975 | title=Venn Diagrams and Independent Families of Sets | journal=Mathematics Magazine | volume=48 | issue=1 | url=https://maa.org/sites/default/files/pdf/upload_library/22/Ford/BrankoGrunbaum.pdf | doi=10.1080/0025570X.1975.11976431 }}
* {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1984 | title=A Symmetrical Arrangement of Eleven Hemi-Icosahedra | journal=Annals of Discrete Mathematics (20): Convexity and graph theory | series=North-Holland Mathematics Studies | publisher=North-Holland | volume=87 | pages=103-114 | doi=10.1016/S0304-0208(08)72814-7 | url=https://www.sciencedirect.com/science/article/pii/S0304020808728147
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      <text bytes="208604" sha1="j6x4vqeilt6xuds1nk7529a1lp9mdxm" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|June 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a regular 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the &lt;s&gt;120-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 5, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 11 [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] cells. Eleven 11-cells (plural) are the 137-cell, the &lt;s&gt;600-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 3, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 137 [[W:Rhombic triacontahedron|triacontahedron]] cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=''Regularity of Graphs, Complexes and Designs''}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=''A Symmetrical Arrangement of Eleven Hemi-Icosahedra''}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]]
The 4-dimensional regular polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:Simplex|simplex]] (called the [[5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:Orthoplex|orthoplex]] (called the [[16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular convex 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the [[w:Tesseract|16-point (8-cell) tesseract]], the 4-cubic, which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found within any of the other regular convex 4-polytopes, except in the largest and most complex one, the [[120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). However, we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the 16-cell is their very starting point, and the most frequently used tool in the box.

== 5-cells and hemi-icosahedra in the 11-cell ==
[[File:Symmetrical_5-set_Venn_diagram.svg|thumb|The 5-point (10-face) regular 5-cell (the regular 4-simplex). Grünbaum's rotationally symmetrical 5-set Venn diagram{{Sfn|Grünbaum|1975|loc=''Rotationally symmetrical 5-set Venn diagram'', Fig 1 (e)|ps=; partitions the individual elements of the 5-cell.}} is an illustration of the 5-cell labeling each of its &lt;math&gt;2^5&lt;/math&gt; elements.{{Sfn|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled; image includes the Python code to render it, optimising for maximum area of the smallest regions.}}]]
[[File:Hemi-icosahedron.png|thumb|The 6-point (10 face) [[W:hemi-icosahedron|hemi-icosahedron]], an abstraction of the regular icosahedron, has half as many faces, edges and vertices. Each element of the abstract polyhedron represents two real elements found in different places in the 11-cell's realization.{{Sfn|Ruen: hemi-icosahedron|2007}}]]
The most apparent relationship between the pentad 5-cell and the hexad hemi-icosahedron is that they both have 10 triangular faces. When we see a facet congruence between a 4-polytope and a 3-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 6-point 5-simplex. It is also the same size as the 6-point 3-orthoplex; thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no chord congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals, pentad and hexad, who are famously stand-offish with each other. Interesting!

The cell of the 11-cell is an abstract hexad hemi-icosahedron with 5 central planes, most handsomely illustrated by Séquin.{{Sfn|Séquin &amp; Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing rings of polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The 11 hemi-icosahedral cells have 10 triangle faces each; the 5-cell's 5 tetrahedral cells have 10 faces and 10 edges all together. Cells meet face-to-face, so the only way 11-cells and 5-cells can have the same faces is if two hemi-icosahedral cells and two tetrahedral cells all meet at each face. Then 3 of each 5-cell's 5 vertices are a hemi-icosahedron face, and its other 2 vertices must be some 11-cell edge lying opposite the face. Coxeter determined that the 11-cell does indeed have an edge opposite each face, but it does not belong to the same hemi-icosahedral cell as the opposing face. He found that the 10 edges opposite each hemi-icosahedron's faces are the 10 edges of a single 5-cell, which does not share any vertices, edges or faces with the hemi-icosahedron. Of course, there is just one such 5-point 5-cell that is completely disjoint from each 6-point hemi-icosahedron in the 11-point 11-cell.{{Sfn|Coxeter|1984|p=110|loc=§6. The Petrie polygon [of the 11-cell]|ps=; &quot;We may reasonably call this edge and face ''opposites''. It is easy to find the face opposite to a given edge by looking at the faces to which a given edge belongs. ... Conversely, given a face, we can find the opposite edge by seeing which vertices belong to neither of the hemi-icosahedra which share that face. The ten edges opposite to the ten faces of one hemi-icosahedron are the edges of the complementary &lt;math&gt;a_4&lt;/math&gt; [4-simplex], that is, the joins of all pairs of the five vertices [of the 11-cell] not belonging to the given hemi-icosahedron.&quot;}}

There are 11 disjoint 5-cells in each 11-cell, which is comprised of 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. The real 11-cell is harder to see than the abstract 11-cell that represents it, because the real hemi-icosahedron that the abstract hemi-icosahedron represents is hard to see, but seeing these objects will be easier once we have identified the real hemi-icosahedron and exactly where the 11-cell's real elements reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each hemi-icosahedron's 10 faces come from 10 disjoint 5-cells, and there is just one 5-cell with which it does not share a face.{{Efn|As Coxeter observes (in the previous citation), that unrepresented 5-point (5-cell) is the other 5 vertices of the 11-point (11-cell) that are not vertices of this 6-point (hemi-icosahedron), the hemi-icosahedron's disjoint complement.}} Each 5-cell has 10 faces from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share a face.

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in two different planes which are completely orthogonal to each other. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must be sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness: 8 concentric hulls|2022|loc=Hull #8 (lower right)|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell of the 11-cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Compounds in the 120-cell|§Compounds in the 120-cell]]'' that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software which uses Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s to render the polyhedra which are found in the interior of ''n''-dimensional polytopes.{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the tetrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is a real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (Hull #8) is a concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this {12} irregular great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing {6} irregular great hexagons (truncated triangles) of alternating 120-cell edges and 5-cell edges. The great dodecagon is a great circle of the hemi-icosahedral cell, and also a great circle of the 120-cell, because unlike the cells of most 4-polytopes, the 11-cell's cells are embedded in 4-space at the exact center of the 4-polytope, making them inscribed 4-polytopes themselves.{{Efn|name=Embedding point}} There are 25 great dodecagons and 120 5-cell edges in Moxness's 60-point (Hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (Hull #8) polyhedron altogether.

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point (Hull #8) is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] similar to the one from the catalog shown here,{{Sfn|Piesk: Rhombicosidodecahedron|2018}} but a slightly shallower truncation with smaller red pentagons (120-cell faces) and narrower rhombs. Rhombicosidodecahedra are made by truncating the [[W:Rhombic triacontahedron|rhombic triacontahedron]], which is the 30-sided polyhedron with only one kind of face, the dual of the 30-point icosidodecahedron. The 120-cell contains 120 of these 11-cell cells.]]
Moxness's 60-point (Hull #8) is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (Hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|The 5-point (regular 5-cell) can be another abstraction of Moxness's 60-point (Hull #8), 12-vertices-into-1, but it is not strictly the result of a contraction operation as described by Boole Stott, since she did not apply her expansion and contraction operations to uniform polytopes with more than one edge length.}} In fact Moxness was not the first to find a rhombicosidodecahedron in the 120-cell; Alicia Boole Stott was. It is the 6th section of {5,3,3} beginning with a cell, which Boole Stott identified as the ''e&lt;sub&gt;2&lt;/sub&gt; expansion'' of the icosahedron (or equivalently of its dual polyhedron the dodecahedron).{{Sfn|Boole Stott|1910|loc=§Examples of the e&lt;sub&gt;2&lt;/sub&gt; expansion|p=7}} But that rhombicosidodecahedron identified by Boole Stott is not Moxness's Hull #8: rather, it is the semi-regular Archimedean solid, with a single edge length and square faces. Moxness's Hull #8 with its two distinct edge lengths and rectangular faces is Coxeter's 8th section of {5,3,3} beginning with a cell, which is missing from the sections illustrated by Boole Stott.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than tetrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. The 120-cell contains 120 of Moxness's 60-point (11-cell cells). They are non-disjoint, with 12 60-points sharing each vertex.

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (Hull #8) is the real hemi-icosahedron cell, with 120 5-cell edges in it as chords lurking under its surface. The rhombicosadodecahedron has enough of those 5-cell edges in its interior to make 20 disjoint tetrahedral cells which belong to 20 distinct 5-cells.

The 600-point (120-cell) may be constructed as a semi-regular honeycomb of two kinds of cells, the 120 non-disjoint 60-point (Moxness's rhombicosadodecahedra) and their neighbors the 120 disjoint 5-point (5-cells), cell-bonded together by shared tetrahedral cells of edge-length {{radic|5}}. Each rhombicosadodecahedron is bonded to 20 islanded 5-cells, sharing with each a 5-cell tetrahedron (4 of its interior 5-cell faces, and 6 of its interior 5-cell edges). The rhombicosadodecahedron is also bonded directly to 10 other rhombicosadodecahedra (since each 11-cell has 11 rhombicosadodecahedron cells), again not by sharing one of its exterior faces, but this time by sharing an interior face, a {{radic|6}} triangle which lies in a central plane.{{Efn|The {{radic|6}} chord is another hidden chord lurking beneath the 60-point rhombicosadodecahedron's surface, like the {{radic|5}} 5-cell edge. Both these hidden chords lie on the same great circle with the 120-cell edges (the pentagon face edges). Three {{radic|6}} chords make a triangle which lies in a central plane. This {{radic|6}} great triangle must not be confused with the {{radic|5}} face triangle of the 5-cell, which does not lie in a central plane.}} The 120 rhombicosadodecahedra are also bonded directly together at ''all'' their exterior faces: at their pentagon faces (which are 120-cell faces of dodecahedral cells); at their small triangular faces (which are 600-cell faces of small tetrahedral cells, 10 of which are inscribed in each dodecahedral cell); and at their rectangular faces (which are faces of 10-point [[w:Pentagonal_prism|pentagonal prism]] cells, where the pentagonal prism is a column inscribed in a dodecahedral cell, joining its opposite pentagon faces). The 120 rhombicosadodecahedra are bonded directly to each other at their own exterior faces; no dodecahedra, small tetrahedra, or pentagonal prisms lie between them, and those objects are not cells of this honeycomb; we mention them here only to help locate the rhombicosadodecahedra within the 120-cell. The only objects which fill holes between the rhombicosadodecahedra are the 120 islanded 5-cells.{{Efn|This honeycomb of the 120-cell resembles the [[600-cell#Icosahedra|honeycomb of the 600-cell consisting of 24 face-bonded regular icosahedra]] with 24 clusters of 5 tetrahedra filling the holes between them. In the 120-cell there are 5 times as many of everything as in the 600-cell, so there are 120 (rhombicosadodecahedron) icosahedral cells with 600 tetrahedra filling the holes between them.}}

It is possible for rhombicosadodecahedra to meet each other at more than one kind of exterior face because they are not meeting there as cells of the same 11-cell; they belong to different 11-cells. The 11 rhombicosadodecahedra in the same 11-cell do not meet at their own exterior faces, they meet at their interior {{radic|6}} triangle faces. As a cellular 4-polytope, the 11-cell is inside-out. Its cells meet each other at interior planes, sharing interior {{radic|6}} triangle faces, instead of in the usual way that 4-polytopes' cells share an exterior face. It is a deeper kind of intertwining to be sure, but the 11 rhombicosadodecahedra in the same 11-cell are still volumetrically disjoint cells, and the 11-cell is a perfectly real 4-polytope with real cells. It is a semi-regular 4-polytope, because it has two different kinds of cells: 11 rhombicosadodecahedra and 11 5-cells. The 11 5-cells are islanded (not bound to each other), and the 11 rhombicosadodecahedra are bound both to 5-cells (at {{radic|5}} 5-cell faces) and to each other (at {{radic|6}} central triangle faces). In the section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#What's in the box|§What's in the box]]'', below, we will look closely at the relation between the {{radic|5}} chord and the {{radic|6}} chord, which is the relation between the pentad and the hexad, and at the general phenomenon of hemi-polytopes which meet each other at a central plane. Then in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Cell rings of the 11-cells|§Cell rings of the 11-cells]]'' we will look at the two-cell cellular structure of the 11-cell in more detail.

Considered as a single complex of overlapping cells rather than as distinct honeycombs of volumetrically disjoint cells, the 120-cell contains 60-point (rhombicosadodecahedral cells), islanded 5-point (5-cells), 20-point (dodecahedral cells), and 10-point ([[w:Pentagonal_prism|pentagonal prism]] cells). The 60-points meet each other in pairs at their pentagon faces, meet regular dodecahedra at those pentagon faces also, meet each other in quads at their 600-cell faces where they share an interior 4-point (600-cell tetrahedral cell) and its four faces (just as they share a larger interior 5-cell tetrahedron and its faces), and meet each other and the 10-point (pentagonal prisms) at their rectangle faces. We might try to imagine such a complex with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with parts truncated from 30 neighboring truncated dodecahedra pentagonal prisms). There are 120 of each of these 3 kinds of dodecahedron. 

Moxness's 60-point (rhombicosadodecahedron cell) is a central object of the 120-cell, and contains many wonders. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra by expansion and contraction operations, as Alicia Boole Stott discovered before 1910,{{Sfn|Polo-Blanco: ''Theory and history of geometric models of Alicia Boole Stott''|2007|loc=§5.3.2 1910 paper on semi-regular polytopes|pp=152-158|ps=; summarizes Boole Stott's method and results from {{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}}, including her tables of expansion-contraction dimensional analogies and a few of her illustrations.}} and those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, rediscovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter &amp; Petrie|1938|p=4|loc=''[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]''}}|name=Concentric polyhedra}} Before we move on from the 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice one more thing about the hidden {{radic|5}} chords lurking below its surface. The 12 little pentagon faces are spanned by sets of 6 {{radic|5}} chords (5-cell edges). The 6 chords of each set are disjoint (they don't touch or form 5-cell faces), but they are symmetrically arranged in 3 parallel pairs, {{radic|3}} apart, which lie in 3 orthogonal central planes. It turns out they are the 6 reflex edges of a 12-point non-convex polyhedron called the [[w:Jessen's_icosahedron|Jessen's icosahedron]], inscribed in the 60-point cell. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again and work our way outwards, before eventually in ''[[#The perfection of Fuller's cyclic design|§The perfection of Fuller's cyclic design]]'' we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cell)s ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks, showing its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Pentahemidemicube.png|120px]]
|[[File:Tetrahemihexahedron_rotation.gif|120px]]
|[[File:Pentahemicosahedron.png|120px]]
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts needed to build the 120-cell (the picture on the cover of the box), and everything else that's invisible in that picture which fits inside the 120-cell: all the astonishing regular 4-polytopes. The pentad building block is half a pair of orthogonal 5-point (regular 5-cells), the hexad building block is half an 8-point (16-cell), and the heptad building block is a quarter of an 11-point (11-cell). Everything can be built from these blocks by embedding them at the center of the 3-sphere, inscribing them as 4-polytopes to make compounds as with the 16-cell building block, rather than by assembling them face-to-face as 3-polytopes to make honeycombs in which they are cells or vertex figures.{{Efn|name=Embedding point}}

The pentad building block is the 5-point '''pentahemidemicube''', an abstract polyhedron with 10 equilateral triangle faces, 10 {{radic|5}} edges, and 5 vertices.{{Sfn|Christie: Pentahemidemicube|2024|loc=File:Pentahemidemicube.png|ps=; &quot;The 5-point ''pentahemidemicube'' is an abstract polyhedron with 10 equilateral triangle faces, 10 edges, and 5 vertices. It has a real presentation as the regular 5-cell.&quot;}} It has a real presentation as the 5-point (regular 5-cell 4-polytope), a regular tetrahedron expanded by addition of a fifth vertex equidistant from all the others in a fourth dimension. The pentad block is a tetrahedral pyramid (a 5-cell) inscribed in a cube: a tetrahedron inscribed in a cube (a demicube) expanded by addition of a fifth vertex, one of the cube's other 4 vertices. Each pentad (5-cell) building block lies in the 120-cell orthogonal to three other pentad (5-cell) building blocks. They are four completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges. Each regular 5-cell in the 120-cell is a pentad (pentahemidemicube) building block, in one of 30 orthogonal sets of 4. There are 120 pentad building blocks in the box.

Every vertex of the 16-cell, and therefore every vertex of the 8-cell, 24-cell, 600-cell and 120-cell, is surrounded by a 6-point (octahedral vertex figure). The hexad building block is that 6-point octahedron embedded at the center of the 4-polytope instead of at a vertex, that is, inscribed in the 3-sphere as a 4-polytope, the 6-point '''[[w:Tetrahemihexahedron|tetrahemihexahedron]]''' or '''hemi-cuboctahedron'''.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; &quot;The tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U&lt;sub&gt;4&lt;/sub&gt;. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices.&quot;}} The hexads have 4 red triangle 5-cell faces, bounded by 12 {{radic|5}} edges. Their 3 yellow square faces lie in central planes, because the building block is an abstract 4-polytope. The hexad is an irregular abstract polyhedron that also has 6 {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

Two orthogonal 5-point (pentad) building blocks intersect in a 4-point (inscribed tetrahedron) to form a 6-point (hexad) building block. Two orthogonal 6-point (hexad) building blocks intersect in a 4-point (central square) to form an 8-point (16-cell) building block, the most usable building block, from which everything larger can be built by compounding.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 120 pentads to pair with a completely orthogonal hexad, and although there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed) they only result in 120 pairs two different ways, so there are 120 11-cells. The 11-cell is 6 pentads' and 5 hexads' smallest joint compound, and the 120-cell is their largest, 120 pentads and 100 hexads.

The heptad{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''|pages=22–31}} building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|4}} edges and 9 {{radic|5}} edges), and 7 vertices.{{Sfn|Christie: Pentahemicosahedron|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;}} It is a compound of a 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]) and a 6-point (hexad [[W:hemi-icosahedron|hemi-icosahedron]]) intersecting in a 4-point (tetrahedron), that is, of three orthogonal 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]&lt;nowiki/&gt;s) intersecting in their common 4-point (tetrahedron). It is also the abstract cube contracted by removal of one vertex, which reduces one of the cube's two inscribed tetrahedra (demicubes) to a single remaining triangular face. As with all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Four &lt;small&gt;&lt;math&gt;(11-7)&lt;/math&gt;&lt;/small&gt; orthogonal 7-point (pentahemicosahedron) blocks can be joined at a common Petrie polygon (a skew hexagon which contains their orthogonal 6-edge paths) to construct a real 11-point ([[W:11-cell|11-cell]] semi-regular 4-polytope), which magically contains six &lt;small&gt;&lt;math&gt;(11-5)&lt;/math&gt;&lt;/small&gt; 5-point (1-pentad regular decahedra) as cells and five &lt;small&gt;&lt;math&gt;(11-6)&lt;/math&gt;&lt;/small&gt; 12-point (2-hexad semi-regular cuboctahedra) as cells. There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like [[W:Euclid's postulates|Euclid's postulates]], and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as honeycombs of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles. It possesses the complete symmetry of the polytope without any redundancy, because it contains one of each of the polytope's characteristic root elements.{{Efn|A [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be partitioned radially by its planes of symmetry (Coxeter's &quot;reflecting circles&quot;) into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated rotationally to generate its regular 4-polytope because it is the complete [[W:gene|gene]] for it, containing all of its elements and capturing all of its symmetry without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring that triggers the reflections.|name=Characteristic orthoscheme}} It is the gene for the polytope, which can be replicated to construct the polytope.

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its &lt;math&gt;A_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the regular compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;small&gt;&lt;math&gt;120\times 120 = 14400&lt;/math&gt;&lt;/small&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[16--cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its &lt;math&gt;B_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 384. The 120-cell is the convex hull of the regular compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 24-point (24-cell) is subdivided into 1152 instances of its [[24-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;F_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 1152. The 120-cell is the convex hull of the regular compound of 25 disjoint 24-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;25\times 1152 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has three equivalent origins in distinct root systems, though most are obviously constructed from one root, and not so obviously from the others. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything larger than the pentad can be constructed by compounding hexads, and we shall see that everything as large as a 24-cell has a construction from heptads which are a product of a pentad and a hexad. Surely every uniform polytope can be constructed by some function of expansions and contractions from any of three root systems, however indirect the recipe. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes, related by an equals sign between them, then give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics by [[W:Noether's theorem|Noether's theorem]].

=== Pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 octahedral cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell contains the 600 octahedral cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It also contains the 600 cubical cells of its 75 8-cells. It is a 600-cell in three ways, although the octahedra and cubes are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell and the 8-cell, respectively.|name=600 cells three ways}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (1-pentad 5-cell), out of the 8-point (2-hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (1-pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope that contains 5-point (1-pentad 5-cell) building blocks beside the largest one, it just isn't a ''regular'' 4-polytope. It is semi-regular with two kinds of cells, the 4-point (regular tetrahedron) and the 12-point (truncated cuboctahedron), which have just one kind of regular face between them, a 5-cell face. That semi-regular 4-polytope is the 11-cell.

We said earlier that the 11-cell had 11 disjoint 5-cells and 11 hemi-icosahedral cells in it, and that its construction from them might be hard to see, but now we are going to account numerically, at least, for how it can have 6 pentad cells plus 5 hexad cells magically adding to 11 of the same kind of elevenad cell. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, 11. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (5-cell regular decahedron) cells, and the 12 6-point (hexad) building blocks into 100 12-point (regular icosahedron) cells, and then magically adding one whole 5-point (5-cell regular decahedron) cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 12-point (regular icosahedron) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such (cell bundles of 6 + 5 = 11 cells), with each 11-cell bundle containing 11 of both kinds of cell. Very, very strange!{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden'': §II To Add is to Multiply|p=38|ps=; the sequence of regular polygons with harmonious proportions beginning {5}, {7}, {9}, {11} as a family of natural numbers; &quot;The numbers in this optimal family have remarkable properties [that allow] similar figures to be arranged easily or allow a figure to be dissected into a set of similar figures. This is the origin of the Greek idea of geometric progression — that ''multiplication can be accomplished instead by addition''.... For instance, it is well known that the square of the golden ratio is one more than itself (written above as 𝜙&lt;sup&gt;2&lt;/sup&gt; = 𝜙 + 1) and one less than 𝜙 is its reciprocal (1/𝜙 = 𝜙 - 1). The first relation equates multiplication with addition, while the second accomplishes division through subtraction. [These are pentagon {5} bisectional golden ratios, but] the heptagonal [{7} tri-sectional] ratios also behave in surprising ways — adding to multiply, subtracting to divide.... There are analogous constructions for [nonagon {9}] quadrisectional and [hendecagon {11}] pentasectional systems.&quot;}}

The 120-cell has three interchangeable cyclic expansion-contraction cycles, the pentad, hexad and heptad cyclic designs, rooted in the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems, respectively. Each of them constructs the whole sequence of regular 4-polytopes from a single building block. The key to all three cycles is that 5 + 6 = 11, and there is an 11-point quasi-regular 4-polytope in the sequence of convex 4-polytopes. It lies between the 8-point (16-cell) and the 16-point (8-cell), and is itself a &lt;math&gt;B_4&lt;/math&gt; polytope. Its role is to be a unified single building block common to all three cyclic expansion-contraction cycles. It is involved with the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems and expresses in one building block their relations to each other.

We have already seen how the hexad building cycle works. It is plain compounding of 4-orthoplexes at the beginning, increasing to densely intricate non-disjoint symmetry toward the end, making it difficult to see why it works once past the 24-cell. This is how the build works, in terms of sixes. The 16-cell is a 2-hexad (3 pairs of completely orthogonal great square planes). The 24-cell is 3 16-cells and a 4-hexad, 4 different ways: 16 non-disjoint hexads (great hexagon planes). The 120-point (600-cell) is 25 24-cells and a 20-hexad, 10 different ways: 200 non-disjoint hexads (great hexagon planes). The 600-point (120-cell) is 225 (9 times 25) 24-cells. ....

The pentad building cycle doesn't do plain compounding to regular 4-polytopes until the last one, the 120-cell. Fives don't even appear as pentagons until after the 24-cell. We have seen how there are 11 5-cells in the quasi-regular 11-cell, which occurs before the 8-cell, and also how everything can be compounded from irregular 5-cells (4-orthoschemes specialized for each 4-polytope). It is still not evident how a pentad cycle can work to build all the regular 4-polytopes from the regular 4-simplex. This is how is how the build works, in terms of fives. ....

This is how the build works, in terms of elevens. The 11-cell is 4 orthogonal heptads (intersecting in their skew hexagon Petrie polygon). The 24-cell is also 4 heptads (intersecting in 4 orthogonal digon axes, so each heptad contributes 6 unique points to a single fibration of 24 points). Heptads are quarter elevenads, so the 11-cell and 24-cell are both 1 elevenad. ....

See the tabular sequence of convex 4-polytopes in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Build with the blocks|§Build with the blocks]]'', below: table rows ''Inscribed, Pentad, Hexad, Heptad''.

=== 11 of them ===
There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of eleven 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum eleven of them. There is only a single Hopf fibration of the 11 great circles of an 11-cell. 

The 5-cell and the 11-cell are limit cases. The 5-cells are completely disjoint, but the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 10 other instances of itself.

== Cell rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer &amp; Segerman|2013|loc=''Puzzling the 120-cell''}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his section of the Wikipedia article on the 120-cell entitled ''[[120-cell#Visualization|§Visualization]]''.{{Sfn|Ruen &amp; Goucher et al. eds. 120-cell|2024|loc=Goucher's ''[[120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious place to look for the 11-cell's Hopf map is in Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other place it could be hiding. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The quasi-regular convex 11-cell|§The quasi-regular convex 11-cell]]'', and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen: triacontagon|2011|loc=''Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them except his illustrations. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Four orthogonal projections of the 120-cell to a central plane, selected from the table of 3 regular and 11 compound 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|100 icosahedra&lt;br&gt;120 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|5 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 icosahedra&lt;br&gt;6 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|120
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|4
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|50
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|7.2°
|align=center|36°
|align=center|6°
|align=center|7.2°
|}

....

Here we summarize our findings on the 11-cells so far, including in advance some results to be obtained in the next section two sections.

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section....

== The 15 major chords ring everything ==
The natural numbers are each a distinct flavor.

Each integer has its own distinct taste. The first six integers {0..5} are the six strongest, most familiar tastes in nature, instantly distinguished and utterly unique.

The seventh flavor {6} is the first ratio of two of the others. Thereafter follows in strange sequence all the ratios of all the flavors, each marriage of every two flavors, most of which are fractions &lt;math&gt;\{k/d\}&lt;/math&gt;, but some of which are integers &lt;math&gt;\{k\}&lt;/math&gt;.

Some of these ratios are successful flavors in their own right, as distinct and inimitable as the marriage of chocolate and oranges, but the marriage of many flavor pairs is mud. It is a rule of thumb among flavor specialists that nearly any taste can be successfully alloyed with either onions or chocolate, but the direct marriage of these two flavors is notably unsuccessful. Some flavors make famous pairings or infamous ones, and subtle new pairings are always to be discovered that work surprisingly well for some purpose, but the majority of flavor marriages are undistinguished compounds.

In polytope geometry, each chord of a polytope is both is a distinct 1-dimensional object, a chord of the sphere of a distinct length &lt;math&gt;l&lt;/math&gt;, and a distinct natural number &lt;math&gt;h&lt;/math&gt;, a unique flavor. If the polytope is regular, it is a noteworthy and successful flavor. The chord's length &lt;math&gt;l&lt;/math&gt; is a square root, related to the natural number &lt;math&gt;h = k/d&lt;/math&gt; and to a set of polytopes they both represent, by a formula discovered by Steinbach.{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden''; Figure 5. Optimal sections and proportions|p=37|ps=; the regular polygons {5}, {7}, {9} and {11} with their diagonals define respectively: {5} the golden bisection proportional to 𝜙; {7} an analogous trisection; {9} an analogous quadrasection; {11} an analogous pentasection.}} The chord length &lt;math&gt;l&lt;/math&gt; is related to the number of sides of the regular polygon formed by the chord (its numerator &lt;math&gt;k&lt;/math&gt;), and to the winding number or density of its regular skew Petrie polygon (its denominator &lt;math&gt;d&lt;/math&gt;).

The regular 4-polytopes do not contain an infinite variety of these chord flavors, but only 30 of them: only 15 unique pairs of 180° degree complementary chords, which combine to make every regular ''n''-dimensional thing found in nature.{{Efn|The 15 ''major chords'' suffice to construct all the regular ''n''-polytopes. Their 180° complements include 15 ''minor chords'' which occur only in the 120-cell, not in any smaller regular polytope.{{Efn|name=Major and minor chords}} Including the complement of the 180° {{radic|4}} digon {2} chord (the {0} chord {{radic|0}}), and the skew variant of the 90° {{radic|2}} square {4} chord (the {30/7} chord), there are 32 distinct natural numbers which are all the chords of all the regular polytopes in all dimensions.|name=32 chords}} All the regular polytopes (including the largest one on the cover of the box) can be built not only from a box containing a number of blocks of 3 shapes glued together by gravity, they can be also be built from a box containing a number of sticks of 15 lengths glued together by rubber joints.

[[File:15 major chords.png|500px|thumb|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the others occur only in the interior of 120-cell, not as edges of any of the regular 4-polytopes. There are 30 distinct 4-space chordal distances between vertices within the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to identify any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=Major and minor chords}} chords are properly labelled #1 — #15 in the order they fan out by size.{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''; Figure 3|ps=; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with a &quot;fan of chords&quot; diagram like the one here.|p=23}} They join vertex pairs which are 1 — 15 edges apart on a &lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon. A #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; chord forms &lt;small&gt;&lt;math&gt;f+1&lt;/math&gt;&lt;/small&gt; regular geodesic polygons which include a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; disjoint regular {{nowrap|&lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt;-gons}} (e.g. &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; a compound of 5 disjoint hexagons) plus a single [[W:Triacontagon#Triacontagram|regular skew {{nowrap|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram}}]] that wraps around the sphere &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times before closing its circle. Notice the maverick #11 chord, with its non-conforming origin point for its {{nowrap|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;}} polygram.]]

These 15 chords (natural numbers) include the integers 2, 3, 4, 5, 6, 10, 11, 15 and 30. The others are the fractions 5/2, 10/3, 13/7, 15/2, 15/4, and 15/7. Represented among the chord lengths (the square roots) measured in ''unit radii'' (not in {{radic|2}} radii, as metrics are often given elsewhere in this article) are the {{background color|paleturquoise|roots of the first 4 positive integers {{radic|1}}, {{radic|2}}, {{radic|3}} and {{radic|4}}}}, the {{background color|#FFCCCC|root of the very distinguished rational fraction {{radic|5/2}}}}, and the roots of various distinguished irrational fractions as Steinbach discovered: for example, 4 of them are {{background color|yellow|roots of golden numbers (a small integer ± 𝜙)}}. The 15 chords each form their regular polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, which is just &lt;small&gt;&lt;math&gt;\{k\}&lt;/math&gt;&lt;/small&gt; in 8 cases. Arranged in order #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; from shortest edge-length square root to longest, the #1 — #15 chords bridge vertex pairs which are &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; vertices apart on a [[W:Triacontagon#Petrie polygons|&lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon]]. Each chord makes a [[W:Triacontagon#Triacontagram|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; polygram]] of its edge length, a compound of the smaller &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. The 15 distinct polygons form a sequence you would not anticipate exactly unless you have Steinbach's formula:

{| style=&quot;background-color:transparent; white-space:nowrap;&quot;
|-
| {30/1} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|.073}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{30} 
| 120-cell edges
|-
| {30/2}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.191}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15}
|- style=&quot;background: yellow;&quot;|
| {30/3}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.382}} 
| &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi}&lt;/math&gt;&lt;/small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{10}''' 
| 600-cell edges
|-
| {30/4} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.573}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/2}
|- style=&quot;background: paleturquoise;&quot;|
| {30/5}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|1}}'''
| &lt;math&gt;1&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{6}''' 
| 24-cell, 8-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/6}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|1.382}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{5}'''
|- style=&quot;background: paleturquoise;&quot;|
| {30/7}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|2}}'''
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{4}''' 
| 16-cell edges
|- style=&quot;background: #FFCCCC;&quot;|
| {30/8}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.5}} 
| &lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;&lt;small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/4} 
| 5-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/9}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.618}} 
| &lt;math&gt;\phi&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{10/3}
|- style=&quot;background: paleturquoise;&quot;|
| {30/10}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|3}}''' 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{3}'''
|-
| {30/11}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.426}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{11/11}
|- style=&quot;background: yellow;&quot;|
| {30/12}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.618}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{5/2}
|-
| {30/13}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.810}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{13/7}
|-
| {30/14}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.928}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/7}
|- style=&quot;background: paleturquoise;&quot;|
| {30/15}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|4}}''' 
| &lt;math&gt;2&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{2}'''
|}

{| class=wikitable style=&quot;white-space:nowrap; float:right;&quot;
!colspan=6|Natural number &lt;small&gt;&lt;math&gt;h=\{k/d\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Fibers&lt;br&gt;&lt;small&gt;&lt;math&gt;f\{h\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Cells&lt;br&gt;&lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|4-tope&lt;br&gt;&lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Verts&lt;br&gt;&lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|[[W:Triacontagon#Triacontagram|30-gram]]&lt;br&gt;&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;
|-
!Chord
!Arc
!colspan=2|&lt;small&gt;&lt;math&gt;l/\sqrt{1}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;l/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;h&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#1
|15.5~°
|{{radic|0.073}}
|0.270
|{{radic|0.146}}
|&lt;small&gt;&lt;math&gt;30&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1\{30\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;5\{3,3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/1\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#2
|25.2~°
|{{radic|0.191}}
|0.437
|{{radic|0.382}}
|&lt;small&gt;&lt;math&gt;15&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;..\{3,3\}..&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/2\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#3
|36°
|{{radic|0.382}}
|0.618
|{{radic|0.764}}
|&lt;small&gt;&lt;math&gt;10&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;30\{3,3\}20&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/3\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#4
|44.5~°
|{{radic|0.573}}
|0.757
|{{radic|1.146}}
|&lt;small&gt;&lt;math&gt;15/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/4\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5
|60°
|{{radic|1}}
|1
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;6&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/5\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#6
|72°
|{{radic|1.382}}
|1.175
|{{radic|2.624}}
|&lt;small&gt;&lt;math&gt;5&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}} 
|&lt;small&gt;&lt;math&gt;10\{5,3\}12&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/6\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#7
|90°
|{{radic|2}}
|1.414
|{{radic|4}}
|&lt;small&gt;&lt;math&gt;4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7\{4\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;4\{4,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;9\{3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/7\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: #FFCCCC;&quot;|
|#8
|104.5~°
|{{radic|2.5}}
|1.581
|{{radic|5}}
|&lt;small&gt;&lt;math&gt;15/4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/4\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}} 
|&lt;small&gt;&lt;math&gt;15\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/8\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#9
|108°
|{{radic|2.618}}
|1.618
|{{radic|5.236}}
|&lt;small&gt;&lt;math&gt;10/3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10/3\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/9\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#10
|120°
|{{radic|3}}
|1.732
|{{radic|6}}
|&lt;small&gt;&lt;math&gt;3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;10\{3\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/10\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#11
|135.5~°
|{{radic|3.426}}
|1.851~
|{{radic|6.852}}
|&lt;small&gt;&lt;math&gt;11/11&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;11\{11/11\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;11\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#12
|144°
|{{radic|3.618}}
|1.902
|{{radic|7.235}}
|&lt;small&gt;&lt;math&gt;5/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2},3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/12\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#13
|154.8~°
|{{radic|3.810}}
|1.952
|{{radic|7.621}}
|&lt;small&gt;&lt;math&gt;13/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;13\{13/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;7\{5,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/13\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#14
|164.5~°
|{{radic|3.928}}
|1.982
|{{radic|7.857}}
|&lt;small&gt;&lt;math&gt;15/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;137\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/14\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{radic|8}}
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15\{2\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐𝜙&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;8\{3,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/15\}&lt;/math&gt;&lt;/small&gt;
|}

The polygons '''{10}''', '''{6}''', '''{5}''', '''{4}''', '''{3}''', and '''{2}''' are flat polygons that each lie on a great circle in a central plane of the 3-sphere. The polygons {30}, {15}, {15/2}, {15/4}, {10/3}, {11/11}, {5/2}, {13/7}, {15/7} and all the {30/''n''} polygons are skew polygrams spiraling through curved 3-space, each successive edge in a different central plane of the 3-sphere. Projected to the 1-sphere (a circle in the plane), they are [[W:Star polygon#Regular star polygon|regular star polygrams]], the simplest being the pentagram {5/2}. In 4-space they wind around the 3-sphere more than once ''without self-intersecting'', before they close their circuit. They lie on helical geodesic circles or ''isoclines'' of circumference greater than 2𝝅''r''.

Even the 6 integer chords which form flat polygons also form skew polygrams. Each of the 15 chords occurs as a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, a fiber bundle of Clifford parallel polygons that totals 30 vertices per bundle. Each chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; also forms a single skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram that winds through another, shorter chord's bundle of parallel polygons, intersecting them in succession multiple times each, and weaving that whole bundle of parallel fibers together as the cross-thread does on a loom. The polygram cross-thread has to wind multiple times through the whole bundle (&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times around the [[W:Triacontagon#Triacontagram|30-gram]] in which each edge connects every &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt;th vertex) to complete an enumeration of all 30 vertices in the bundle. For example, the #5 chord forms hexagon &lt;small&gt;&lt;math&gt;\{6\}&lt;/math&gt;&lt;/small&gt; flat planes in parallel bundles of 5, written &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt;, and also a skew &lt;small&gt;&lt;math&gt;\{15/2\}&lt;/math&gt;&lt;/small&gt;-gram of #5 edges which winds 2 times through the #3 chord's bundle of 3 decagon &lt;small&gt;&lt;math&gt;\{10\}&lt;/math&gt;&lt;/small&gt; flat planes. Every chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; forms both its &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram and its fiber bundle of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, and the edges of one chord's parallel polygons are also the edges of its skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram weaving through some shorter chord's bundle of parallel polygons. Each numbered chord is a natural number, a ratio of two integers and their polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, either flat or skew. Each chord number defines a &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; skew polygon that is a compound bundle of the &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. All these flat or skew polygons are regular polygons made of their numerator instances of a single chord #, whose length is that square root, Steinbach's function of the natural number.

The relationship between each bundle of flat Clifford parallel great circle polygons of a distinct length, and its corresponding bundle of skew polygrams of a longer length, is precisely that of a discrete simple rotation and its corresponding discrete isoclinic rotation. For example, the flat great circle hexagons of edge length {{radic|1}} lie in the invariant rotation planes of the 24-cell's characteristic 60° isoclinic rotation, in which the vertices move on skew hexagram rotation paths of edge length {{radic|3}}. The 24-cell contains 4 bundles of 4 Clifford parallel (disjoint) hexagon central planes each. The 24 vertices of the 24-cell in each bundle are linked by rotation in the 4 hexagon invariant planes in two different ways: simply and isoclinically. In a simple rotation the vertices of a single invariant hexagon exchange places on that great circle hexagon of edge {{radic|1}}. In an isoclinic rotation in which any one of these 4 disjoint hexagon central planes is invariant, all 4 of them are necessarily invariant planes in the rotation, because they rotate in parallel. In that isoclinic rotation the vertices of all 4 hexagon invariant planes (all 24 vertices of the 24-cell) exchange places concurrently, moving along 4 disjoint helical hexagrams of edge {{radic|3}} in parallel. It is a characteristic only of this distinct rotation in hexagon invariant planes that the helical vertex path or ''isocline'' is a hexagram of period 6, edge length {{radic|3}}, and circumference 720°. Rotations in other kinds of polygon invariant planes have different characteristic isoclines. The period of their isoclinic rotation must be ''the same or greater'' than the period of their simple rotation; in most cases (unlike this one) it is greater.

The 15 chords form not only distinct fiber bundles of polygons, but also distinct bundles of polyhedra: cell rings &lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt; and compound vertex figures &lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;. These combine to form their characteristic regular 4-polytope &lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Coxeter|1973|p=269|loc=Compounds|ps=; &quot;It is remarkable that the vertices of {5, 3, 3} include the vertices of all the other fifteen regular polytopes in four dimensions.&quot;}} All these elements of a regular 4-polytope constructed from one chord are themselves regular polytopes, and each of the 16 regular 4-polytopes is a function of one characteristic chord. If the regular 4-polytope is convex, its characteristic chord is its unit-radius edge. For a non-convex regular 4-polytope, the characteristic chord is its unit-radius ''mid-edge diameter'', the distance from the midpoint of each edge to the midpoint of the completely orthogonal edge. Each of the characteristic chords (each distinct natural number) has its expression as one of the 16 regular 4-polytopes and all the products of the distinct regular 1-polytopes, 2-polytopes and 3-polytopes that construct it, and the whole distinct construction from the natural number to the 4-polytope is an [[W:SO(4)|SO(4) symmetry group]] expression of that 4-polytope's characteristic discrete [[W:SO(4)#Isoclinic rotations|isoclinic rotation]], in the invariant central planes of its characteristic chord, by the characteristic chord arc.

...demonstrate this theory by proper enumeration of the chord lengths in the table...

...color-code the △ ☐ 𝜙 ✩ symbols ... chord colors in diagram should match them ... some of the diagram colors are wrong and must be changed, but otherwise diagram is pretty good ... i think in a colored diagram all chords of the same length should have the same size line dash, the polygons are not obvious enough and the 15 fanning-out chords are readily identified by their labels...

... add back the counts of chords in the box (very small font number under each table entry), and present these two images as the front and back a box of building sticks and rubber joints...What's in the other box...

...say the rational-root major chords {{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|5/2}} are the ''n''-edges: their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge...

...abbreviated text of the [[120-cell#Relationships among interior polytopes|''§Relationships among interior polytopes'']] section from the 120-cell article...to explain the color-coding

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius=1}}

See also the elements, properties and metrics of the sequence of regular 4-polytopes summarized in tabular form in the section ''[[#Build with the blocks|§Build with the blocks]]'', below.

== Alicia Boole Stott's original formulation of dimensional analogy ==
&lt;blockquote&gt;&quot;...a method by which bodies having a certain kind of semi-regularity may be derived from regular bodies in an Euclidean space of any number of dimensions...&quot;{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''; presents cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues|pp=12-45}}&lt;/blockquote&gt;

Alica Boole Stott's 1910 paper reads like an exceptionally well-written patent application. It is a comprehensive stepwise description of her original invention, a general method of dimensional analogy between regular and semiregular polytopes of the same edge length.

[[File:The Roots of Three.png|thumb|An illustration of Boole Stott's expansion of unit-edge triangle to hexagon captures the geometric basis in 3 of the &lt;math&gt;A_n&lt;/math&gt;root system.]]

The engineering-like articulation of her new principles is entirely discrete and group-theoretic. Boole Stott repeatedly refers to sets of polytope elements as ''groups''. All her transformations are discrete instances of one or another special case of her &quot;two inverse operations which may be called ''expansion'' and ''contraction''.&quot; Hers is an exploration of ''n''-dimensional space at its most basic: space defined by the discrete objects which fill it, themselves defined directly as groups of elements with a few operations on them, which transform them into each other. This is geometry which seems to precede Euclid's postulates, and does not depend on them. Instead it corresponds directly to the deeper group theory mathematics which underlies geometry, even though Boole Stott, like the ancients, did not have that mathematics.

Boole Stott enumerated her expansion and contraction operations as transformations between regular and semi-regular polytopes of a single edge length. In her 1910 paper she did not consider any polytopes of multiple edge lengths, or expansions of the subject polytope by a distance other than that exactly required to produce new edges of the original edge length, so of course she did not enumerate Moxness's non-uniform rhombicosidodecahedron (Hull #8). This may be why section 8&lt;sub&gt;3&lt;/sub&gt; is missing from her earlier drawings of the sections of {5,3,3}; she may not have had a construction for it.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The Boole Stott expansion (or contraction) operation is as precisely defined a transformation as Coxeter's reflection, rotation and translation operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions). Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;
where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;
where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;
For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} An instance of the expansion (or contraction) operation is an application of the general operation to a particular subject polytope, operating on all of its elements symmetrically at once. The general operation is restricted, such that one may only expand (or contract) the subject polytope by pushing all its edges, faces or cells directly away from (or toward) its center, without changing the size of the moved elements. This splits each vertex into several (or coalesces several vertices into one), and has corresponding effects on the other elements, producing new edges ''of the same size'' between pairs of separated vertices (or reducing some edges to single vertices). The distance all the elements are moved away from (or toward) the center must be such that the vertices separate producing new edges of only the original edge length (or come together reducing edges to single vertices). It is permitted to combine these three operations, e.g. by pushing the edges ''and'' the faces, or by pushing all three element sets, provided the combination results in a semi-regular polytope of only one edge length. The three operations are all commutative with one another, so combined operations can be considered either as a sequence of expansions (or contractions) linking several related polytopes, or as their concurrent movements, a single transformation of the subject polytope. Since there are 8 combinations of one to three operations there are 8 polytope variants (including the original regular polytope), and all are adjacent to one another by a single transformation. The variants of the original regular polytope are named for which element set(s) are moved. By enumerating all 8 variants of expansion (and contraction) for each of the six regular convex 4-polytopes, Boole Stott produced a table of regular and semi-regular 4-polytopes divided into 12 sets of 8: six expansion sets and six contraction sets.{{Sfn|Boole Stott|1910|loc=''Table of Polytopes in S&lt;sub&gt;4&lt;/sub&gt;''; note that Boole Stott's usage of ''S&lt;sub&gt;4&lt;/sub&gt;'' denotes a Euclidean 4-space ''E&lt;sub&gt;4&lt;/sub&gt;'', or a 3-sphere space embedded in it, ''not'' a 4-sphere space embedded in a Euclidean 5-space|p=30}} Each set includes just one of the six regular convex 4-polytopes, with its distinct unit-radius edge length, and the 7 possible semi-regular expansions (or contractions) of it, which all preserve that characteristic edge length. The polytopes in each set of 8 all have the same total number of elements as their original regular polytope, but each polytope in the set has a distinct combination of cell, face, and vertex figures. The 12 sets are not completely disjoint, and even within one set the 8 operations do not necessarily produce 8 unique polytopes. For example, the regular 5-cell (the 4-simplex polytope) has 5 cells, 10 faces, 10 edges, and 5 vertices (30 elements). Expanding the 5-cell produces 7 related semi-regular 4-polytopes of 30 elements each (two of which happen to be the same polytope). In each expansion, the 5 cells have expanded into semi-regular or regular polyhedra, the 10 faces and 10 edges ''may'' have expanded into prisms, and the 5 original vertices have expanded into regular or semi-regular cells.

== The regular 11-cell 4-polytope ==

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosadodecahedron). Each non-convex Jessen's icosahedron is a set of 6 reflex edges that are {{radic|5}} 5-cell edges, each set symmetrically arranged in 3 parallel pairs, not as far apart ({{radic|3}}) as they are long, in 3 orthogonal central planes. The 12-point (Jessen's sets of 6 reflex 5-cell edges) are inscribed in the 60-point with their endpoints as a vertex of a pentagon face. The 12 pentagon faces are joined to 5 other pentagon faces by the 5-cell edges; not to their nearest neighbor pentagons but to their next-to-nearest neighbor pentagons; and since there are 5 vertices to each pentagon face, there are 10 12-point (Jessen's sets of 6 reflex 5-cell edges). Each pentagon face is joined to another pentagon face by 5 reflex edges of five distinct Jessen's. The 5 reflex edges are Clifford parallel, the 5 edges of a pentagonal prism column that has a twist to it. The contraction of those 5 edges into 1, and those 10 vertices (two pentagons) into 2, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing all their edges as a single Jessen's. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (octahedron) by conflating its parallel pairs of edges, leaving the 11-cell's abstract 6-point (hexad cell). The hexad's situation is that it occurs in bundles of 5, disjoint but close together in the sense of adjacent, like the great circle rings in a Hopf fiber bundle. 

To summarize, the 12-point (2-hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell). We shall now be able to see how Fuller's perfect hexads combine with Todd's perfect pentads to form the perfect heptads of the 11-cell and 137-cell.  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=alias the hemi-cuboctahedron}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces. (The ''heptad'' arrives at the party, anonymously; they are our distinguished anti-celebrity, who most of the other guests have never even heard of.) The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. This 12-point (truncated tetrahedron) is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 100 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp: Truncated tetrahedron|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp: Truncated tetrahedron|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}} Of course Moxness has been there, seen that, already with his graphics software;{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} see his parameterized illustration of it exactly as it occurs in the 11-cell, below in his bestiary of A3, B3, and H3 Archimedian and Catalan polyhedra in the 120-cell.]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:Archimedean_and_Catalan_solid_hulls_with_their_Weyl_orbit_definitions.svg|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of the 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. In this catalog of Archimedean and Catalan polyhedra sighted in the 120-cell by Moxness's software,{{Sfn|Moxness: Archimedean and Catalan hulls|2023|loc=Hull #1 Archimedean Name A3 110 Truncated Tetrahedron A (upper left)}} it is the first entry (upper left), Moxness's Hull # = 1 with 12 vertices of 3D norm {{radic|11}}/2. On a {{radic|2}}-radius 3-sphere, 5 of these with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell. 20 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges are cells in the [[#The rings of the 11-cells|cell rings of the quasi-regular 11-cell]], which also has 12 regular 5-cells as cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

...

Both kinds of cells in the 11-cell have expressions in two different dimensions (on the 2-sphere and 3-sphere), because they are all centered on the center of the 3-sphere, unlike the cells of most 4-polytopes, which are centered on various off-center points within the 4-polytope's interior. 

One kind of 11-cell cell appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (semi-regular truncated cuboctahedron), a.k.a. the truncated tetrahedron. The other kind of cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now, except in the 120-cell. But this is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron, and a regular tetrahedron behind each face. The 11-cell is thus a honeycomb of mixed regular tetrahedra and truncated cuboctahedra (a.k.a. truncated tetrahedra) that meet at their shared 5-cell faces. 

Both the 5-cells (decahedra) and the truncated cuboctahedra (truncated tetrahedra) are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks in their 4-polytopes (off-center).{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} Therefore all the 11-cell's cells are concentric, which indeed makes a different kind of honeycomb, an inside-out cellular honeycomb with all the cells piled on top of each other overlapping in 4-space. They do not overlap in 3-sphere space however; their 3-volumes (but not their 4-contents) are disjoint. They make a proper 4D honeycomb like the cells of any 4-polytope, with volumetrically disjoint 3D cells that meet each other in the proper way at their shared 2D faces.

An 11-cell contains 6 completely disjoint 5-point (1-pentad 5-cell decahedron cells), and 5 volumetrically disjoint 12-point (2-hexad truncated cuboctahedron cells), 11 volumetrically disjoint cells altogether. Of course it is supposed to have 11 identical 11-point (elevenad hemi-icosahedral cells), and that is also true in its abstract sense, which is not the same sense as disjoint cellular containment: each pentad or hexad cell is inscribed in one of 11 larger 60-point (Moxness's rhombicosadodecahedron cells). The abstract pentads and hexads are only hemi-polyhedra, so the 11-cell's pentad cells are 12 (out of 120) disjoint instances of a pentad 5-point, and its hexad cells are 10 (out of 100) disjoint instances of a hexad 6-point, 1/10th of the 120-cell's cellular volume in both cases. The 11-cell's hemi-icosahedral cells are 11 (out of 120) non-disjoint instances of an elevenad 60-point. Each 11-cell shares each of its hexad cells with 9 other 11-cells, and each of its 11 vertices (out of 600) occurs in 9 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.|name=Tetrahedral vertex figures}}

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

...insert image thumb of 11-cell...

...Construct the 11-cell from the heptad....

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...Four concentric 4-polytopes may be mutually orthogonal. In any set of 4 mutually orthogonal regular 5-cells, one pair of faces in each orthogonal pair are completely orthogonal to each other. Thus three edges in one 5-cell in an orthogonal pair are opposite (completely orthogonal) to a face of the orthogonal 5-cell. .... A 5-cell has 5 vertices in its abstract hemi-icosahedral cells, but only 4 vertices in one concrete icosahedral cell; the 5th lies in an icosahedral cell of another 11-cell (not in another icosahedral cell of the same 11-cell).

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

....

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 11-cell ===

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon.... (wrong, 11-cell's Hopf map is the truncated icosahedron)

....

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk: Truncated icosahedron|2018}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

...

Since the rhombicosadodecahedron has 120 5-cell edges, the 120-cell has 120 completely disjoint 5-cells, and the rhombicosadodecahedron shares each vertex with 11 other rhombicosadodecahedra, how a rhombicosadodecahedron is bound to its neighbor 5-cells is constrained. A further constraint is that the abstract hemi-icosahedron has only 15 5-cell edges, so the 120 actual 5-cell edges of the rhombicosadodecahedron are conflated 8-to-1 in the hemi-icosahedron. To be conflated by contraction into a single edge, those 8 edges must lie parallel to each other in the 120-cell; either actually parallel in the same plane, or Clifford parallel in separate planes, as two edges in completely orthogonal planes may be parallel and orthogonal at the same time. Each 5-cell inscribed in the 120-cell contains 5 such tetrahedra occur in parallel pairs in 4 orthogonal planes (none of them completely orthogonal planes). 

.... 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
Each conflated vertex is the 10 vertices of two orthogonal regular 5-cells...
The 60-point contains 6 pairs of these orthogonal regular 5-cell cells...
...its 20 hexad cells are ..., a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere.

== The regular 137-cell 4-polytope ==

(description originally written for the 11-cell, which applies only to a fibration of 11 11-cells)

It is quasi-regular with two kinds of cells, the 12-point (regular icosahedron) and the 5-point (regular 5-cell), which have just one kind of regular face between them. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of regular facets (it is semi-regular with regular facets), it also means that its facets have expressions of two different dimensionalities because they are concentric in the 3-sphere. The 11-cell has two kinds of regular cells. One appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (regular icosahedron), but since it is embedded at the center of the 3-sphere (unlike the cells of most 4-polytopes) it also acts as an inscribed 4-polytope.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} The other cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now except in the 120-cell. This is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron. The 11-cell is a honeycomb of mixed icosahedron and half-icosahedron cells. Like the icosahedra, the decahedra are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks, off-center in their 4-polytopes. All the 11-cell's cells are concentric, which indeed makes an unusual honeycomb of cells, an inside-out honeycomb. It is a proper 4D honeycomb however, with volumetrically disjoint 3D cells that meet each other in the proper way at shared 2D faces. The embedded 12-point (regular icosahedron) cell is an icosahedral pyramid (as a 4-polytope), which is filled by 20 4-point (irregular tetrahedra) that meet at its center; the center point of the 11-cell acts as a 12th vertex, just as the center point of the 24-cell acts as a 25th vertex in some honeycombs. The 5-point (regular 5-cell) cell contains five 4-point (regular tetrahedra) which meet the 12-point (regular icosahedra)'s irregular tetrahedral cells face-to-face, so the 11-cell is quasi-regular in the usual sense of having two varieties of regular 3-polytope cells, but additionally both varieties of of its cells are bi-dimensional: another clue that the 11-cell has something to do with dimensional analogy

....

The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

....

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur to make an 11-cell. 11 of them occur in a 137-cell, all 120 of them occur in the 120-cell, and they occur nowhere else.

How can this regular 4-polytope possibly exist? By Schläfli's criterion {3, 3, ..} exists, but not as a finite polytope.{{Sfn|Coxeter|1973|loc=§7.7 Schläfli's criterion}} Coxeter found {3, 5, 3} as a finite hyperbolic honeycomb, but not in Euclidean 4-space.{{Sfn|Coxeter|1968|title=The Beauty of Geometry: Twelve Geometric Essays|loc=''Regular Honeycombs in Hyperbolic Space''|ps=; &quot;The honeycombs considered by the above authors [Schlegel, Schläfli, Klein, Sommerville] have finite cells and finite vertex figures. ... A further extension allows a cell or vertex figure to be a ''star''-polytope, so that the honeycomb covers the space several times. .... We shall find that there are four regular star-honeycombs in hyperbolic 4-space, as well as two infinite families of them in the hyperbolic plane.&quot;}} We find that {3, 3, ..} is a central expression of the &lt;math&gt;H_4&lt;/math&gt; root system, being the regular convex hull in 4 dimensions of 137 points, 96 pentads and 600 hexads. The minimal expression of &lt;math&gt;H_4&lt;/math&gt; is the regular convex hull {3, 3, 5} of 120 points, 600 tetrads and 75 hexads, and its maximal expression is the regular convex hull {5, 3, 3} of 600 points, 120 pentads and 675 hexads. {3, 3, ..} is a third regular &lt;math&gt;H_4&lt;/math&gt; polytope which fits between them. The properties and metrics of this regular sequence are presented in tabular form below in ''[[#Build with the blocks|§Build with the blocks]]''.

{|class=&quot;wikitable&quot;
!colspan=3|Regular &lt;math&gt;H_4&lt;/math&gt; polytopes
|-
|[[File:600-cell.gif|200px]]
|[[File:5-cell.gif|200px]]
|[[File:120-cell.gif|200px]]
|-
!120-point 600-cell
!137-point 137-cell
!600-point 120-cell
|}

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 137-cell ===
....

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=Apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|2022|loc=''[[Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]''}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie: On Fuller's use of language|2024|loc=''[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry'']]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center, 1948-1949|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with their mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the gender neutral shad can open their six beaks all the way, until they become flat squares and they becomes a cuboctahedron, or they can shut them all tight like a turtle retracting into their octahedron shell. The six mouths always move in unison. This is [[Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do their ''really'' odd trick -- where they flip their 6 jaws 90 degrees in their 6 faces and shut their 6 beaks on the opposite axis of their squares than the one they opened them on -- or whether they will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than their normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make their edge length exactly the same size as their radius, when they open their mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing their mouths in spherical synchrony, their 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron (the hexad), then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=''[[W:SO(4)|SO(4)]]''}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=Isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its orthogonal hexad's planes (wz, wx, wy).{{Efn|name=Six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of (expanded octahederon 16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh &amp; Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== Legendre's 12-point hexad binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a 12-point double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's perfectly realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, this 2-hexad expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre hexad) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction.{{Efn|name=Concentric polyhedra}} In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== ''n''-simplexes ===
....

The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== ''n''-orthoplexes ===
....
[[File:Great 5-cell √5 digons rectangle.png|thumb|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} ''regular'' great hexagons. These are the same 200 dodecagon central planes illustrated above, which also contain 6 120-cell edges and two opposing ''irregular'' great hexagons (truncated {{radic|6}} triangles). The 1200 {{radic|5}} 5-cell face triangles occupy 1200 other, non-central planes.]]

...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's Hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[16--cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[16--cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=Concentric polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[16--cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== ''n''-taliesins ===
The 4-simplex pentad building block combines with the 4-orthoplex hexad building block to make the 4-taliesin heptad building block. The heptad is the essential building block of the ''taliesin'' family of polytopes. The 11-cell and 57-cell hemi-polytopes and the 137-cell and 120-cell regular 4-polytopes are nuclear members of this trans-dimensional family. The 24-cell and 600-cell are extended members of it, because they also contain instances of the heptad building block.

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its spherical symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* [[W:Taliesin Myrddin Namkai-Meche|Taliesin Myrddin Namkai-Meche]], a young Reed College scholar who was stabbed to death by a crazy man in 2017 on a Portland, Oregon light rail train, when he rose to the nonviolent defense of two black teenagers who were being violently assaulted in racist and anti-Muslim language by a right-wing extremist.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in orthogonal [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].
* '''Taliesan polytope''', an (''n+1)''-polytope whose ''n''-polytope dimensional analogue inscribed concentrically in the ''n''-sphere forms a convex honeycomb of itself which is the (''n+1)''-polytope.{{Efn|name=Embedding point}}
* The taliesan polytope's honeycomb, a Hopf fibration of its great circles.
* The ''n''-dimensional family of dimensionally analogous polytopes related to a taliesan polytope by expansion and contraction operations.
* The characteristic [[w:Orthoscheme|orthoscheme]] of a teliesan polytope, an irregular simplex that is the complete gene for the entire expression of trans-dimensional symmetry relationships among the polytopes of the ''n''-dimensional family, as well as the complete gene for the teliesan polytope's rotational symmetry.{{Efn|name=Characteristic orthoscheme}}

=== ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== ''n''-equilaterals ===
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed at four different [[#n-leviathon|chord]] lengths from each other. In the {{radic|2}}-radius polytopes we have using as examples in this paper they are {{radic|2}}, {{radic|4}}, {{radic|6}} and {{radic|8}}, but in a ''unit-radius'' polytope they are: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}.

Each vertex is joined to 8 others by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}. The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== ''n''-pentagonals ===

[[File:600-cell.gif|thumb|The 120-point (600-cell), &lt;math&gt;H^4&lt;/math&gt;.]]
...

[[File:600-cell vertex geometry.png|thumb|Vertex geometry of the 600-cell, showing the 5 regular great circle polygons and the 8 vertex-to-vertex chord lengths with angles of arc. The golden ratio{{Efn|name=golden chords|group=}} governs the fractional roots of every other chord,{{Efn|name=fractional root chords}} and the radial golden triangles which meet at the center.|alt=|400x400px]]
The 120 vertices of the 600-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏&lt;sup&gt;−1&lt;/sup&gt;) beginning with a vertex; see column ''a''}} at eight different [[#n-leviathon|chord]] lengths from each other. These edges and chords of the 600-cell are simply the edges and chords of its five great circle polygons. In ascending order of length in a ''unit-radius'' polytope (not in a {{radic|2}} radius polytopes like our usual examples), they are {{radic|0.𝚫}}, {{radic|1}}, {{radic|1.𝚫}}, {{radic|2}}, {{radic|2.𝚽}}, {{radic|3}}, {{radic|3.𝚽}}, and {{radic|4}}.{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=Decimal fractions of square roots|group=}}

Notice that the four [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#''n-''equilaterals|hypercubic chords]] of the 24-cell ({{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|4}}) alternate with the four new chords of the 600-cell's additional great circles, the decagons and pentagons. The new ''golden chord'' lengths are necessarily square roots of fractions, but very special fractions related to the golden ratio{{Efn|1=The fractional-root ''golden chords'' are irrational fractions that are functions of {{radic|5}}. They exemplify that the [[W:golden ratio|golden ratio]] &lt;big&gt;𝜙&lt;/big&gt; {{=}} {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618 is a circle ratio related to &lt;big&gt;𝜋&lt;/big&gt;:&lt;br&gt;
: {{sfrac|𝜋|5}} = arccos (𝜙/2)
is one decagon edge, the 𝚽 = {{radic|0.𝚫}} = {{radic|0.382~}} ≈ 0.618 chord.
Reciprocally, in this function discovered by Robert Everest expressing &lt;big&gt;𝜙&lt;/big&gt; as a function of &lt;big&gt;𝜋&lt;/big&gt; and the numbers 1, 2, 3 and 5 of the Fibonacci series:&lt;br&gt;
: &lt;big&gt;𝜙&lt;/big&gt; = 1 – 2 cos ({{sfrac|3𝜋|5}})
{{sfrac|3𝜋|5}} is the arc length of the &lt;big&gt;𝜙&lt;/big&gt; = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618 chord.|name=golden chords|group=}} including the two [[W:golden section|golden section]]s of {{radic|5}}, as shown in the diagram.{{Efn|The 600-cell edges are decagon edges of length {{radic|0.𝚫}}, which is 𝚽, the ''smaller'' golden section of {{radic|5}}; the edges are in the inverse [[W:golden ratio|golden ratio]] {{sfrac|1|φ}} to the {{radic|1}} hexagon chords (the 24-cell edges).
The other fractional-root chords exhibit golden relationships as well. The chord of length {{radic|1.𝚫}} is a pentagon edge.
The next fractional-root chord is a decagon diagonal of length {{radic|2.𝚽}} which is &lt;big&gt;𝜙&lt;/big&gt;, the ''larger'' golden section of {{radic|5}}; it is in the golden ratio{{Efn|name=golden chords|group=}} to the {{radic|1}} chord (and the radius).{{Efn|Notice in the diagram how the &lt;big&gt;𝜙&lt;/big&gt; chord (the ''larger'' golden section) sums with the adjacent 𝚽 edge (the ''smaller'' golden section) to {{radic|5}}, as if together they were a {{radic|5}} chord bent to fit inside the {{radic|4}} diameter.}}
The last fractional-root chord is the pentagon diagonal of length {{radic|3.𝚽}}.
The [[W:Pentagon#Side length is given|diagonal of a regular pentagon]] is always in the golden ratio to its edge, and indeed &lt;big&gt;𝜙&lt;/big&gt;{{radic|1.𝚫}} is {{radic|3.𝚽}}.|name=fractional root chords|group=}}

....


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== ''n''-leviathon ===
...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of its &lt;math&gt;10^2&lt;/math&gt; disjoint instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction, before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]].{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}} .... It is only because she did not have the 11-cell to start with that she did not discover the regular 137-cell {3, 3, ..} by her method.

Between Alicia Boole Stott and her identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and her identification of symmetry groups with conservation laws (in her 1st theorem), we owe the deepest mathematics of geometry and physics originally to two pioneering women, colleagues of their contemporaries the greatest men of the academy, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just an [[W:abstract polytope|abstract 4-polytope]], not just a singleton convex 4-polytope, and not just a honeycomb.{{Sfn|Coxeter|1970|loc=''Twisted Honeycombs''}} The 11-point (11-cell) has a concrete ...regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}&lt;nowiki/&gt;} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, 3, 3} as the 137-point (137-cell).

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 11-cell, 137-cell and 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=''Clifford Spinors and Root System Induction: H4 and the Grand Antiprism''}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=Apology}}

== Build with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Duveneck|1978|loc=Carl Jung, quoted in ''Life on Two Levels''|p=ii|ps=.{{Sfn|Jung|1961|ps=: &quot;The best of truths is of no use unless it has become one's most personal inner experience. It is the duty of everyone who takes a solitary path to share with society what he finds on his journey of discovery.&quot;}}}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1954|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius={{radic|2}}}}

== Acknowledgements ==
...

== Notes ==
{{Regular convex 4-polytopes Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Citation|last=Boole Stott|first=Alicia|author-link=W:Alicia Boole Stott|title=Geometrical deduction of semiregular from regular polytopes and space fillings|date=1910|pages=12-45|publisher=Johannes Muller|place=Amsterdam|url=https://dwc.knaw.nl/DL/publications/PU00011492.pdf}}
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      <text bytes="208605" sha1="gsh28aiyzqvuui8e4p3qm0f44r93mzp" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|June 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a regular 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the &lt;s&gt;120-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 5, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 11 [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] cells. Eleven 11-cells (plural) are the 137-cell, the &lt;s&gt;600-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 3, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 137 [[W:Rhombic triacontahedron|triacontahedron]] cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=''Regularity of Graphs, Complexes and Designs''}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=''A Symmetrical Arrangement of Eleven Hemi-Icosahedra''}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]]
The 4-dimensional regular polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:Simplex|simplex]] (called the [[5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:Orthoplex|orthoplex]] (called the [[16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular convex 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the [[w:Tesseract|16-point (8-cell) tesseract]], the 4-cubic, which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found within any of the other regular convex 4-polytopes, except in the largest and most complex one, the [[120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). However, we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the 16-cell is their very starting point, and the most frequently used tool in the box.

== 5-cells and hemi-icosahedra in the 11-cell ==
[[File:Symmetrical_5-set_Venn_diagram.svg|thumb|The 5-point (10-face) regular 5-cell (the regular 4-simplex). Grünbaum's rotationally symmetrical 5-set Venn diagram{{Sfn|Grünbaum|1975|loc=''Rotationally symmetrical 5-set Venn diagram'', Fig 1 (e)|ps=; partitions the individual elements of the 5-cell.}} is an illustration of the 5-cell labeling each of its &lt;math&gt;2^5&lt;/math&gt; elements.{{Sfn|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled; image includes the Python code to render it, optimising for maximum area of the smallest regions.}}]]
[[File:Hemi-icosahedron.png|thumb|The 6-point (10 face) [[W:hemi-icosahedron|hemi-icosahedron]], an abstraction of the regular icosahedron, has half as many faces, edges and vertices. Each element of the abstract polyhedron represents two real elements found in different places in the 11-cell's realization.{{Sfn|Ruen: hemi-icosahedron|2007}}]]
The most apparent relationship between the pentad 5-cell and the hexad hemi-icosahedron is that they both have 10 triangular faces. When we see a facet congruence between a 4-polytope and a 3-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 6-point 5-simplex. It is also the same size as the 6-point 3-orthoplex; thus it is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no chord congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals, pentad and hexad, who are famously stand-offish with each other. Interesting!

The cell of the 11-cell is an abstract hexad hemi-icosahedron with 5 central planes, most handsomely illustrated by Séquin.{{Sfn|Séquin &amp; Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing rings of polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The 11 hemi-icosahedral cells have 10 triangle faces each; the 5-cell's 5 tetrahedral cells have 10 faces and 10 edges all together. Cells meet face-to-face, so the only way 11-cells and 5-cells can share the same faces is if two hemi-icosahedral cells and two tetrahedral cells all meet at each face. Then 3 of each 5-cell's 5 vertices are a hemi-icosahedron face, and its other 2 vertices must be some 11-cell edge lying opposite the face. Coxeter determined that the 11-cell does indeed have an edge opposite each face, but it does not belong to the same hemi-icosahedral cell as its opposing face. He found that the 10 edges opposite each hemi-icosahedron's faces are the 10 edges of a single 5-cell, which does not share any vertices, edges or faces with the hemi-icosahedron. Of course, there is just one such 5-point 5-cell that is completely disjoint from each 6-point hemi-icosahedron in the 11-point 11-cell.{{Sfn|Coxeter|1984|p=110|loc=§6. The Petrie polygon [of the 11-cell]|ps=; &quot;We may reasonably call this edge and face ''opposites''. It is easy to find the face opposite to a given edge by looking at the faces to which a given edge belongs. ... Conversely, given a face, we can find the opposite edge by seeing which vertices belong to neither of the hemi-icosahedra which share that face. The ten edges opposite to the ten faces of one hemi-icosahedron are the edges of the complementary &lt;math&gt;a_4&lt;/math&gt; [4-simplex], that is, the joins of all pairs of the five vertices [of the 11-cell] not belonging to the given hemi-icosahedron.&quot;}}

There are 11 disjoint 5-cells in each 11-cell, which is comprised of 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. The real 11-cell is harder to see than the abstract 11-cell that represents it, because the real hemi-icosahedron that the abstract hemi-icosahedron represents is hard to see, but seeing these objects will be easier once we have identified the real hemi-icosahedron and exactly where the 11-cell's real elements reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each hemi-icosahedron's 10 faces come from 10 disjoint 5-cells, and there is just one 5-cell with which it does not share a face.{{Efn|As Coxeter observes (in the previous citation), that unrepresented 5-point (5-cell) is the other 5 vertices of the 11-point (11-cell) that are not vertices of this 6-point (hemi-icosahedron), the hemi-icosahedron's disjoint complement.}} Each 5-cell has 10 faces from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share a face.

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in two different planes which are completely orthogonal to each other. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must be sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness: 8 concentric hulls|2022|loc=Hull #8 (lower right)|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell of the 11-cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Compounds in the 120-cell|§Compounds in the 120-cell]]'' that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software which uses Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s to render the polyhedra which are found in the interior of ''n''-dimensional polytopes.{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the tetrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is a real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (Hull #8) is a concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this {12} irregular great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing {6} irregular great hexagons (truncated triangles) of alternating 120-cell edges and 5-cell edges. The great dodecagon is a great circle of the hemi-icosahedral cell, and also a great circle of the 120-cell, because unlike the cells of most 4-polytopes, the 11-cell's cells are embedded in 4-space at the exact center of the 4-polytope, making them inscribed 4-polytopes themselves.{{Efn|name=Embedding point}} There are 25 great dodecagons and 120 5-cell edges in Moxness's 60-point (Hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (Hull #8) polyhedron altogether.

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point (Hull #8) is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] similar to the one from the catalog shown here,{{Sfn|Piesk: Rhombicosidodecahedron|2018}} but a slightly shallower truncation with smaller red pentagons (120-cell faces) and narrower rhombs. Rhombicosidodecahedra are made by truncating the [[W:Rhombic triacontahedron|rhombic triacontahedron]], which is the 30-sided polyhedron with only one kind of face, the dual of the 30-point icosidodecahedron. The 120-cell contains 120 of these 11-cell cells.]]
Moxness's 60-point (Hull #8) is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (Hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|The 5-point (regular 5-cell) can be another abstraction of Moxness's 60-point (Hull #8), 12-vertices-into-1, but it is not strictly the result of a contraction operation as described by Boole Stott, since she did not apply her expansion and contraction operations to uniform polytopes with more than one edge length.}} In fact Moxness was not the first to find a rhombicosidodecahedron in the 120-cell; Alicia Boole Stott was. It is the 6th section of {5,3,3} beginning with a cell, which Boole Stott identified as the ''e&lt;sub&gt;2&lt;/sub&gt; expansion'' of the icosahedron (or equivalently of its dual polyhedron the dodecahedron).{{Sfn|Boole Stott|1910|loc=§Examples of the e&lt;sub&gt;2&lt;/sub&gt; expansion|p=7}} But that rhombicosidodecahedron identified by Boole Stott is not Moxness's Hull #8: rather, it is the semi-regular Archimedean solid, with a single edge length and square faces. Moxness's Hull #8 with its two distinct edge lengths and rectangular faces is Coxeter's 8th section of {5,3,3} beginning with a cell, which is missing from the sections illustrated by Boole Stott.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than tetrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. The 120-cell contains 120 of Moxness's 60-point (11-cell cells). They are non-disjoint, with 12 60-points sharing each vertex.

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (Hull #8) is the real hemi-icosahedron cell, with 120 5-cell edges in it as chords lurking under its surface. The rhombicosadodecahedron has enough of those 5-cell edges in its interior to make 20 disjoint tetrahedral cells which belong to 20 distinct 5-cells.

The 600-point (120-cell) may be constructed as a semi-regular honeycomb of two kinds of cells, the 120 non-disjoint 60-point (Moxness's rhombicosadodecahedra) and their neighbors the 120 disjoint 5-point (5-cells), cell-bonded together by shared tetrahedral cells of edge-length {{radic|5}}. Each rhombicosadodecahedron is bonded to 20 islanded 5-cells, sharing with each a 5-cell tetrahedron (4 of its interior 5-cell faces, and 6 of its interior 5-cell edges). The rhombicosadodecahedron is also bonded directly to 10 other rhombicosadodecahedra (since each 11-cell has 11 rhombicosadodecahedron cells), again not by sharing one of its exterior faces, but this time by sharing an interior face, a {{radic|6}} triangle which lies in a central plane.{{Efn|The {{radic|6}} chord is another hidden chord lurking beneath the 60-point rhombicosadodecahedron's surface, like the {{radic|5}} 5-cell edge. Both these hidden chords lie on the same great circle with the 120-cell edges (the pentagon face edges). Three {{radic|6}} chords make a triangle which lies in a central plane. This {{radic|6}} great triangle must not be confused with the {{radic|5}} face triangle of the 5-cell, which does not lie in a central plane.}} The 120 rhombicosadodecahedra are also bonded directly together at ''all'' their exterior faces: at their pentagon faces (which are 120-cell faces of dodecahedral cells); at their small triangular faces (which are 600-cell faces of small tetrahedral cells, 10 of which are inscribed in each dodecahedral cell); and at their rectangular faces (which are faces of 10-point [[w:Pentagonal_prism|pentagonal prism]] cells, where the pentagonal prism is a column inscribed in a dodecahedral cell, joining its opposite pentagon faces). The 120 rhombicosadodecahedra are bonded directly to each other at their own exterior faces; no dodecahedra, small tetrahedra, or pentagonal prisms lie between them, and those objects are not cells of this honeycomb; we mention them here only to help locate the rhombicosadodecahedra within the 120-cell. The only objects which fill holes between the rhombicosadodecahedra are the 120 islanded 5-cells.{{Efn|This honeycomb of the 120-cell resembles the [[600-cell#Icosahedra|honeycomb of the 600-cell consisting of 24 face-bonded regular icosahedra]] with 24 clusters of 5 tetrahedra filling the holes between them. In the 120-cell there are 5 times as many of everything as in the 600-cell, so there are 120 (rhombicosadodecahedron) icosahedral cells with 600 tetrahedra filling the holes between them.}}

It is possible for rhombicosadodecahedra to meet each other at more than one kind of exterior face because they are not meeting there as cells of the same 11-cell; they belong to different 11-cells. The 11 rhombicosadodecahedra in the same 11-cell do not meet at their own exterior faces, they meet at their interior {{radic|6}} triangle faces. As a cellular 4-polytope, the 11-cell is inside-out. Its cells meet each other at interior planes, sharing interior {{radic|6}} triangle faces, instead of in the usual way that 4-polytopes' cells share an exterior face. It is a deeper kind of intertwining to be sure, but the 11 rhombicosadodecahedra in the same 11-cell are still volumetrically disjoint cells, and the 11-cell is a perfectly real 4-polytope with real cells. It is a semi-regular 4-polytope, because it has two different kinds of cells: 11 rhombicosadodecahedra and 11 5-cells. The 11 5-cells are islanded (not bound to each other), and the 11 rhombicosadodecahedra are bound both to 5-cells (at {{radic|5}} 5-cell faces) and to each other (at {{radic|6}} central triangle faces). In the section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#What's in the box|§What's in the box]]'', below, we will look closely at the relation between the {{radic|5}} chord and the {{radic|6}} chord, which is the relation between the pentad and the hexad, and at the general phenomenon of hemi-polytopes which meet each other at a central plane. Then in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Cell rings of the 11-cells|§Cell rings of the 11-cells]]'' we will look at the two-cell cellular structure of the 11-cell in more detail.

Considered as a single complex of overlapping cells rather than as distinct honeycombs of volumetrically disjoint cells, the 120-cell contains 60-point (rhombicosadodecahedral cells), islanded 5-point (5-cells), 20-point (dodecahedral cells), and 10-point ([[w:Pentagonal_prism|pentagonal prism]] cells). The 60-points meet each other in pairs at their pentagon faces, meet regular dodecahedra at those pentagon faces also, meet each other in quads at their 600-cell faces where they share an interior 4-point (600-cell tetrahedral cell) and its four faces (just as they share a larger interior 5-cell tetrahedron and its faces), and meet each other and the 10-point (pentagonal prisms) at their rectangle faces. We might try to imagine such a complex with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with parts truncated from 30 neighboring truncated dodecahedra pentagonal prisms). There are 120 of each of these 3 kinds of dodecahedron. 

Moxness's 60-point (rhombicosadodecahedron cell) is a central object of the 120-cell, and contains many wonders. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra by expansion and contraction operations, as Alicia Boole Stott discovered before 1910,{{Sfn|Polo-Blanco: ''Theory and history of geometric models of Alicia Boole Stott''|2007|loc=§5.3.2 1910 paper on semi-regular polytopes|pp=152-158|ps=; summarizes Boole Stott's method and results from {{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}}, including her tables of expansion-contraction dimensional analogies and a few of her illustrations.}} and those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, rediscovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter &amp; Petrie|1938|p=4|loc=''[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]''}}|name=Concentric polyhedra}} Before we move on from the 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice one more thing about the hidden {{radic|5}} chords lurking below its surface. The 12 little pentagon faces are spanned by sets of 6 {{radic|5}} chords (5-cell edges). The 6 chords of each set are disjoint (they don't touch or form 5-cell faces), but they are symmetrically arranged in 3 parallel pairs, {{radic|3}} apart, which lie in 3 orthogonal central planes. It turns out they are the 6 reflex edges of a 12-point non-convex polyhedron called the [[w:Jessen's_icosahedron|Jessen's icosahedron]], inscribed in the 60-point cell. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again and work our way outwards, before eventually in ''[[#The perfection of Fuller's cyclic design|§The perfection of Fuller's cyclic design]]'' we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cell)s ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks, showing its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Pentahemidemicube.png|120px]]
|[[File:Tetrahemihexahedron_rotation.gif|120px]]
|[[File:Pentahemicosahedron.png|120px]]
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts needed to build the 120-cell (the picture on the cover of the box), and everything else that's invisible in that picture which fits inside the 120-cell: all the astonishing regular 4-polytopes. The pentad building block is half a pair of orthogonal 5-point (regular 5-cells), the hexad building block is half an 8-point (16-cell), and the heptad building block is a quarter of an 11-point (11-cell). Everything can be built from these blocks by embedding them at the center of the 3-sphere, inscribing them as 4-polytopes to make compounds as with the 16-cell building block, rather than by assembling them face-to-face as 3-polytopes to make honeycombs in which they are cells or vertex figures.{{Efn|name=Embedding point}}

The pentad building block is the 5-point '''pentahemidemicube''', an abstract polyhedron with 10 equilateral triangle faces, 10 {{radic|5}} edges, and 5 vertices.{{Sfn|Christie: Pentahemidemicube|2024|loc=File:Pentahemidemicube.png|ps=; &quot;The 5-point ''pentahemidemicube'' is an abstract polyhedron with 10 equilateral triangle faces, 10 edges, and 5 vertices. It has a real presentation as the regular 5-cell.&quot;}} It has a real presentation as the 5-point (regular 5-cell 4-polytope), a regular tetrahedron expanded by addition of a fifth vertex equidistant from all the others in a fourth dimension. The pentad block is a tetrahedral pyramid (a 5-cell) inscribed in a cube: a tetrahedron inscribed in a cube (a demicube) expanded by addition of a fifth vertex, one of the cube's other 4 vertices. Each pentad (5-cell) building block lies in the 120-cell orthogonal to three other pentad (5-cell) building blocks. They are four completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges. Each regular 5-cell in the 120-cell is a pentad (pentahemidemicube) building block, in one of 30 orthogonal sets of 4. There are 120 pentad building blocks in the box.

Every vertex of the 16-cell, and therefore every vertex of the 8-cell, 24-cell, 600-cell and 120-cell, is surrounded by a 6-point (octahedral vertex figure). The hexad building block is that 6-point octahedron embedded at the center of the 4-polytope instead of at a vertex, that is, inscribed in the 3-sphere as a 4-polytope, the 6-point '''[[w:Tetrahemihexahedron|tetrahemihexahedron]]''' or '''hemi-cuboctahedron'''.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; &quot;The tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U&lt;sub&gt;4&lt;/sub&gt;. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices.&quot;}} The hexads have 4 red triangle 5-cell faces, bounded by 12 {{radic|5}} edges. Their 3 yellow square faces lie in central planes, because the building block is an abstract 4-polytope. The hexad is an irregular abstract polyhedron that also has 6 {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

Two orthogonal 5-point (pentad) building blocks intersect in a 4-point (inscribed tetrahedron) to form a 6-point (hexad) building block. Two orthogonal 6-point (hexad) building blocks intersect in a 4-point (central square) to form an 8-point (16-cell) building block, the most usable building block, from which everything larger can be built by compounding.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 120 pentads to pair with a completely orthogonal hexad, and although there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed) they only result in 120 pairs two different ways, so there are 120 11-cells. The 11-cell is 6 pentads' and 5 hexads' smallest joint compound, and the 120-cell is their largest, 120 pentads and 100 hexads.

The heptad{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''|pages=22–31}} building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|4}} edges and 9 {{radic|5}} edges), and 7 vertices.{{Sfn|Christie: Pentahemicosahedron|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;}} It is a compound of a 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]) and a 6-point (hexad [[W:hemi-icosahedron|hemi-icosahedron]]) intersecting in a 4-point (tetrahedron), that is, of three orthogonal 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]&lt;nowiki/&gt;s) intersecting in their common 4-point (tetrahedron). It is also the abstract cube contracted by removal of one vertex, which reduces one of the cube's two inscribed tetrahedra (demicubes) to a single remaining triangular face. As with all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Four &lt;small&gt;&lt;math&gt;(11-7)&lt;/math&gt;&lt;/small&gt; orthogonal 7-point (pentahemicosahedron) blocks can be joined at a common Petrie polygon (a skew hexagon which contains their orthogonal 6-edge paths) to construct a real 11-point ([[W:11-cell|11-cell]] semi-regular 4-polytope), which magically contains six &lt;small&gt;&lt;math&gt;(11-5)&lt;/math&gt;&lt;/small&gt; 5-point (1-pentad regular decahedra) as cells and five &lt;small&gt;&lt;math&gt;(11-6)&lt;/math&gt;&lt;/small&gt; 12-point (2-hexad semi-regular cuboctahedra) as cells. There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like [[W:Euclid's postulates|Euclid's postulates]], and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as honeycombs of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles. It possesses the complete symmetry of the polytope without any redundancy, because it contains one of each of the polytope's characteristic root elements.{{Efn|A [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be partitioned radially by its planes of symmetry (Coxeter's &quot;reflecting circles&quot;) into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated rotationally to generate its regular 4-polytope because it is the complete [[W:gene|gene]] for it, containing all of its elements and capturing all of its symmetry without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring that triggers the reflections.|name=Characteristic orthoscheme}} It is the gene for the polytope, which can be replicated to construct the polytope.

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its &lt;math&gt;A_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the regular compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;small&gt;&lt;math&gt;120\times 120 = 14400&lt;/math&gt;&lt;/small&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[16--cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its &lt;math&gt;B_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 384. The 120-cell is the convex hull of the regular compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 24-point (24-cell) is subdivided into 1152 instances of its [[24-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;F_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 1152. The 120-cell is the convex hull of the regular compound of 25 disjoint 24-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;25\times 1152 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has three equivalent origins in distinct root systems, though most are obviously constructed from one root, and not so obviously from the others. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything larger than the pentad can be constructed by compounding hexads, and we shall see that everything as large as a 24-cell has a construction from heptads which are a product of a pentad and a hexad. Surely every uniform polytope can be constructed by some function of expansions and contractions from any of three root systems, however indirect the recipe. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes, related by an equals sign between them, then give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics by [[W:Noether's theorem|Noether's theorem]].

=== Pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 octahedral cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell contains the 600 octahedral cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It also contains the 600 cubical cells of its 75 8-cells. It is a 600-cell in three ways, although the octahedra and cubes are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell and the 8-cell, respectively.|name=600 cells three ways}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (1-pentad 5-cell), out of the 8-point (2-hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (1-pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope that contains 5-point (1-pentad 5-cell) building blocks beside the largest one, it just isn't a ''regular'' 4-polytope. It is semi-regular with two kinds of cells, the 4-point (regular tetrahedron) and the 12-point (truncated cuboctahedron), which have just one kind of regular face between them, a 5-cell face. That semi-regular 4-polytope is the 11-cell.

We said earlier that the 11-cell had 11 disjoint 5-cells and 11 hemi-icosahedral cells in it, and that its construction from them might be hard to see, but now we are going to account numerically, at least, for how it can have 6 pentad cells plus 5 hexad cells magically adding to 11 of the same kind of elevenad cell. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, 11. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (5-cell regular decahedron) cells, and the 12 6-point (hexad) building blocks into 100 12-point (regular icosahedron) cells, and then magically adding one whole 5-point (5-cell regular decahedron) cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 12-point (regular icosahedron) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such (cell bundles of 6 + 5 = 11 cells), with each 11-cell bundle containing 11 of both kinds of cell. Very, very strange!{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden'': §II To Add is to Multiply|p=38|ps=; the sequence of regular polygons with harmonious proportions beginning {5}, {7}, {9}, {11} as a family of natural numbers; &quot;The numbers in this optimal family have remarkable properties [that allow] similar figures to be arranged easily or allow a figure to be dissected into a set of similar figures. This is the origin of the Greek idea of geometric progression — that ''multiplication can be accomplished instead by addition''.... For instance, it is well known that the square of the golden ratio is one more than itself (written above as 𝜙&lt;sup&gt;2&lt;/sup&gt; = 𝜙 + 1) and one less than 𝜙 is its reciprocal (1/𝜙 = 𝜙 - 1). The first relation equates multiplication with addition, while the second accomplishes division through subtraction. [These are pentagon {5} bisectional golden ratios, but] the heptagonal [{7} tri-sectional] ratios also behave in surprising ways — adding to multiply, subtracting to divide.... There are analogous constructions for [nonagon {9}] quadrisectional and [hendecagon {11}] pentasectional systems.&quot;}}

The 120-cell has three interchangeable cyclic expansion-contraction cycles, the pentad, hexad and heptad cyclic designs, rooted in the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems, respectively. Each of them constructs the whole sequence of regular 4-polytopes from a single building block. The key to all three cycles is that 5 + 6 = 11, and there is an 11-point quasi-regular 4-polytope in the sequence of convex 4-polytopes. It lies between the 8-point (16-cell) and the 16-point (8-cell), and is itself a &lt;math&gt;B_4&lt;/math&gt; polytope. Its role is to be a unified single building block common to all three cyclic expansion-contraction cycles. It is involved with the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems and expresses in one building block their relations to each other.

We have already seen how the hexad building cycle works. It is plain compounding of 4-orthoplexes at the beginning, increasing to densely intricate non-disjoint symmetry toward the end, making it difficult to see why it works once past the 24-cell. This is how the build works, in terms of sixes. The 16-cell is a 2-hexad (3 pairs of completely orthogonal great square planes). The 24-cell is 3 16-cells and a 4-hexad, 4 different ways: 16 non-disjoint hexads (great hexagon planes). The 120-point (600-cell) is 25 24-cells and a 20-hexad, 10 different ways: 200 non-disjoint hexads (great hexagon planes). The 600-point (120-cell) is 225 (9 times 25) 24-cells. ....

The pentad building cycle doesn't do plain compounding to regular 4-polytopes until the last one, the 120-cell. Fives don't even appear as pentagons until after the 24-cell. We have seen how there are 11 5-cells in the quasi-regular 11-cell, which occurs before the 8-cell, and also how everything can be compounded from irregular 5-cells (4-orthoschemes specialized for each 4-polytope). It is still not evident how a pentad cycle can work to build all the regular 4-polytopes from the regular 4-simplex. This is how is how the build works, in terms of fives. ....

This is how the build works, in terms of elevens. The 11-cell is 4 orthogonal heptads (intersecting in their skew hexagon Petrie polygon). The 24-cell is also 4 heptads (intersecting in 4 orthogonal digon axes, so each heptad contributes 6 unique points to a single fibration of 24 points). Heptads are quarter elevenads, so the 11-cell and 24-cell are both 1 elevenad. ....

See the tabular sequence of convex 4-polytopes in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Build with the blocks|§Build with the blocks]]'', below: table rows ''Inscribed, Pentad, Hexad, Heptad''.

=== 11 of them ===
There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of eleven 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum eleven of them. There is only a single Hopf fibration of the 11 great circles of an 11-cell. 

The 5-cell and the 11-cell are limit cases. The 5-cells are completely disjoint, but the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 10 other instances of itself.

== Cell rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer &amp; Segerman|2013|loc=''Puzzling the 120-cell''}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his section of the Wikipedia article on the 120-cell entitled ''[[120-cell#Visualization|§Visualization]]''.{{Sfn|Ruen &amp; Goucher et al. eds. 120-cell|2024|loc=Goucher's ''[[120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious place to look for the 11-cell's Hopf map is in Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other place it could be hiding. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The quasi-regular convex 11-cell|§The quasi-regular convex 11-cell]]'', and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen: triacontagon|2011|loc=''Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them except his illustrations. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Four orthogonal projections of the 120-cell to a central plane, selected from the table of 3 regular and 11 compound 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|100 icosahedra&lt;br&gt;120 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|5 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 icosahedra&lt;br&gt;6 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|120
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|4
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|50
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|7.2°
|align=center|36°
|align=center|6°
|align=center|7.2°
|}

....

Here we summarize our findings on the 11-cells so far, including in advance some results to be obtained in the next section two sections.

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section....

== The 15 major chords ring everything ==
The natural numbers are each a distinct flavor.

Each integer has its own distinct taste. The first six integers {0..5} are the six strongest, most familiar tastes in nature, instantly distinguished and utterly unique.

The seventh flavor {6} is the first ratio of two of the others. Thereafter follows in strange sequence all the ratios of all the flavors, each marriage of every two flavors, most of which are fractions &lt;math&gt;\{k/d\}&lt;/math&gt;, but some of which are integers &lt;math&gt;\{k\}&lt;/math&gt;.

Some of these ratios are successful flavors in their own right, as distinct and inimitable as the marriage of chocolate and oranges, but the marriage of many flavor pairs is mud. It is a rule of thumb among flavor specialists that nearly any taste can be successfully alloyed with either onions or chocolate, but the direct marriage of these two flavors is notably unsuccessful. Some flavors make famous pairings or infamous ones, and subtle new pairings are always to be discovered that work surprisingly well for some purpose, but the majority of flavor marriages are undistinguished compounds.

In polytope geometry, each chord of a polytope is both is a distinct 1-dimensional object, a chord of the sphere of a distinct length &lt;math&gt;l&lt;/math&gt;, and a distinct natural number &lt;math&gt;h&lt;/math&gt;, a unique flavor. If the polytope is regular, it is a noteworthy and successful flavor. The chord's length &lt;math&gt;l&lt;/math&gt; is a square root, related to the natural number &lt;math&gt;h = k/d&lt;/math&gt; and to a set of polytopes they both represent, by a formula discovered by Steinbach.{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden''; Figure 5. Optimal sections and proportions|p=37|ps=; the regular polygons {5}, {7}, {9} and {11} with their diagonals define respectively: {5} the golden bisection proportional to 𝜙; {7} an analogous trisection; {9} an analogous quadrasection; {11} an analogous pentasection.}} The chord length &lt;math&gt;l&lt;/math&gt; is related to the number of sides of the regular polygon formed by the chord (its numerator &lt;math&gt;k&lt;/math&gt;), and to the winding number or density of its regular skew Petrie polygon (its denominator &lt;math&gt;d&lt;/math&gt;).

The regular 4-polytopes do not contain an infinite variety of these chord flavors, but only 30 of them: only 15 unique pairs of 180° degree complementary chords, which combine to make every regular ''n''-dimensional thing found in nature.{{Efn|The 15 ''major chords'' suffice to construct all the regular ''n''-polytopes. Their 180° complements include 15 ''minor chords'' which occur only in the 120-cell, not in any smaller regular polytope.{{Efn|name=Major and minor chords}} Including the complement of the 180° {{radic|4}} digon {2} chord (the {0} chord {{radic|0}}), and the skew variant of the 90° {{radic|2}} square {4} chord (the {30/7} chord), there are 32 distinct natural numbers which are all the chords of all the regular polytopes in all dimensions.|name=32 chords}} All the regular polytopes (including the largest one on the cover of the box) can be built not only from a box containing a number of blocks of 3 shapes glued together by gravity, they can be also be built from a box containing a number of sticks of 15 lengths glued together by rubber joints.

[[File:15 major chords.png|500px|thumb|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the others occur only in the interior of 120-cell, not as edges of any of the regular 4-polytopes. There are 30 distinct 4-space chordal distances between vertices within the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to identify any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=Major and minor chords}} chords are properly labelled #1 — #15 in the order they fan out by size.{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''; Figure 3|ps=; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with a &quot;fan of chords&quot; diagram like the one here.|p=23}} They join vertex pairs which are 1 — 15 edges apart on a &lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon. A #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; chord forms &lt;small&gt;&lt;math&gt;f+1&lt;/math&gt;&lt;/small&gt; regular geodesic polygons which include a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; disjoint regular {{nowrap|&lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt;-gons}} (e.g. &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; a compound of 5 disjoint hexagons) plus a single [[W:Triacontagon#Triacontagram|regular skew {{nowrap|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram}}]] that wraps around the sphere &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times before closing its circle. Notice the maverick #11 chord, with its non-conforming origin point for its {{nowrap|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;}} polygram.]]

These 15 chords (natural numbers) include the integers 2, 3, 4, 5, 6, 10, 11, 15 and 30. The others are the fractions 5/2, 10/3, 13/7, 15/2, 15/4, and 15/7. Represented among the chord lengths (the square roots) measured in ''unit radii'' (not in {{radic|2}} radii, as metrics are often given elsewhere in this article) are the {{background color|paleturquoise|roots of the first 4 positive integers {{radic|1}}, {{radic|2}}, {{radic|3}} and {{radic|4}}}}, the {{background color|#FFCCCC|root of the very distinguished rational fraction {{radic|5/2}}}}, and the roots of various distinguished irrational fractions as Steinbach discovered: for example, 4 of them are {{background color|yellow|roots of golden numbers (a small integer ± 𝜙)}}. The 15 chords each form their regular polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, which is just &lt;small&gt;&lt;math&gt;\{k\}&lt;/math&gt;&lt;/small&gt; in 8 cases. Arranged in order #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; from shortest edge-length square root to longest, the #1 — #15 chords bridge vertex pairs which are &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; vertices apart on a [[W:Triacontagon#Petrie polygons|&lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon]]. Each chord makes a [[W:Triacontagon#Triacontagram|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; polygram]] of its edge length, a compound of the smaller &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. The 15 distinct polygons form a sequence you would not anticipate exactly unless you have Steinbach's formula:

{| style=&quot;background-color:transparent; white-space:nowrap;&quot;
|-
| {30/1} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|.073}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{30} 
| 120-cell edges
|-
| {30/2}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.191}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15}
|- style=&quot;background: yellow;&quot;|
| {30/3}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.382}} 
| &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi}&lt;/math&gt;&lt;/small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{10}''' 
| 600-cell edges
|-
| {30/4} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.573}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/2}
|- style=&quot;background: paleturquoise;&quot;|
| {30/5}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|1}}'''
| &lt;math&gt;1&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{6}''' 
| 24-cell, 8-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/6}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|1.382}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{5}'''
|- style=&quot;background: paleturquoise;&quot;|
| {30/7}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|2}}'''
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{4}''' 
| 16-cell edges
|- style=&quot;background: #FFCCCC;&quot;|
| {30/8}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.5}} 
| &lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;&lt;small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/4} 
| 5-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/9}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.618}} 
| &lt;math&gt;\phi&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{10/3}
|- style=&quot;background: paleturquoise;&quot;|
| {30/10}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|3}}''' 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{3}'''
|-
| {30/11}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.426}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{11/11}
|- style=&quot;background: yellow;&quot;|
| {30/12}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.618}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{5/2}
|-
| {30/13}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.810}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{13/7}
|-
| {30/14}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.928}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/7}
|- style=&quot;background: paleturquoise;&quot;|
| {30/15}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|4}}''' 
| &lt;math&gt;2&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{2}'''
|}

{| class=wikitable style=&quot;white-space:nowrap; float:right;&quot;
!colspan=6|Natural number &lt;small&gt;&lt;math&gt;h=\{k/d\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Fibers&lt;br&gt;&lt;small&gt;&lt;math&gt;f\{h\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Cells&lt;br&gt;&lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|4-tope&lt;br&gt;&lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Verts&lt;br&gt;&lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|[[W:Triacontagon#Triacontagram|30-gram]]&lt;br&gt;&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;
|-
!Chord
!Arc
!colspan=2|&lt;small&gt;&lt;math&gt;l/\sqrt{1}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;l/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;h&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#1
|15.5~°
|{{radic|0.073}}
|0.270
|{{radic|0.146}}
|&lt;small&gt;&lt;math&gt;30&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1\{30\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;5\{3,3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/1\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#2
|25.2~°
|{{radic|0.191}}
|0.437
|{{radic|0.382}}
|&lt;small&gt;&lt;math&gt;15&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;..\{3,3\}..&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/2\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#3
|36°
|{{radic|0.382}}
|0.618
|{{radic|0.764}}
|&lt;small&gt;&lt;math&gt;10&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;30\{3,3\}20&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/3\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#4
|44.5~°
|{{radic|0.573}}
|0.757
|{{radic|1.146}}
|&lt;small&gt;&lt;math&gt;15/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/4\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5
|60°
|{{radic|1}}
|1
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;6&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/5\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#6
|72°
|{{radic|1.382}}
|1.175
|{{radic|2.624}}
|&lt;small&gt;&lt;math&gt;5&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}} 
|&lt;small&gt;&lt;math&gt;10\{5,3\}12&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/6\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#7
|90°
|{{radic|2}}
|1.414
|{{radic|4}}
|&lt;small&gt;&lt;math&gt;4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7\{4\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;4\{4,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;9\{3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/7\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: #FFCCCC;&quot;|
|#8
|104.5~°
|{{radic|2.5}}
|1.581
|{{radic|5}}
|&lt;small&gt;&lt;math&gt;15/4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/4\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}} 
|&lt;small&gt;&lt;math&gt;15\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/8\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#9
|108°
|{{radic|2.618}}
|1.618
|{{radic|5.236}}
|&lt;small&gt;&lt;math&gt;10/3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10/3\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/9\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#10
|120°
|{{radic|3}}
|1.732
|{{radic|6}}
|&lt;small&gt;&lt;math&gt;3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;10\{3\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/10\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#11
|135.5~°
|{{radic|3.426}}
|1.851~
|{{radic|6.852}}
|&lt;small&gt;&lt;math&gt;11/11&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;11\{11/11\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;11\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#12
|144°
|{{radic|3.618}}
|1.902
|{{radic|7.235}}
|&lt;small&gt;&lt;math&gt;5/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2},3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/12\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#13
|154.8~°
|{{radic|3.810}}
|1.952
|{{radic|7.621}}
|&lt;small&gt;&lt;math&gt;13/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;13\{13/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;7\{5,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/13\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#14
|164.5~°
|{{radic|3.928}}
|1.982
|{{radic|7.857}}
|&lt;small&gt;&lt;math&gt;15/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;137\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/14\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{radic|8}}
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15\{2\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐𝜙&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;8\{3,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/15\}&lt;/math&gt;&lt;/small&gt;
|}

The polygons '''{10}''', '''{6}''', '''{5}''', '''{4}''', '''{3}''', and '''{2}''' are flat polygons that each lie on a great circle in a central plane of the 3-sphere. The polygons {30}, {15}, {15/2}, {15/4}, {10/3}, {11/11}, {5/2}, {13/7}, {15/7} and all the {30/''n''} polygons are skew polygrams spiraling through curved 3-space, each successive edge in a different central plane of the 3-sphere. Projected to the 1-sphere (a circle in the plane), they are [[W:Star polygon#Regular star polygon|regular star polygrams]], the simplest being the pentagram {5/2}. In 4-space they wind around the 3-sphere more than once ''without self-intersecting'', before they close their circuit. They lie on helical geodesic circles or ''isoclines'' of circumference greater than 2𝝅''r''.

Even the 6 integer chords which form flat polygons also form skew polygrams. Each of the 15 chords occurs as a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, a fiber bundle of Clifford parallel polygons that totals 30 vertices per bundle. Each chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; also forms a single skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram that winds through another, shorter chord's bundle of parallel polygons, intersecting them in succession multiple times each, and weaving that whole bundle of parallel fibers together as the cross-thread does on a loom. The polygram cross-thread has to wind multiple times through the whole bundle (&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times around the [[W:Triacontagon#Triacontagram|30-gram]] in which each edge connects every &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt;th vertex) to complete an enumeration of all 30 vertices in the bundle. For example, the #5 chord forms hexagon &lt;small&gt;&lt;math&gt;\{6\}&lt;/math&gt;&lt;/small&gt; flat planes in parallel bundles of 5, written &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt;, and also a skew &lt;small&gt;&lt;math&gt;\{15/2\}&lt;/math&gt;&lt;/small&gt;-gram of #5 edges which winds 2 times through the #3 chord's bundle of 3 decagon &lt;small&gt;&lt;math&gt;\{10\}&lt;/math&gt;&lt;/small&gt; flat planes. Every chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; forms both its &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram and its fiber bundle of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, and the edges of one chord's parallel polygons are also the edges of its skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram weaving through some shorter chord's bundle of parallel polygons. Each numbered chord is a natural number, a ratio of two integers and their polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, either flat or skew. Each chord number defines a &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; skew polygon that is a compound bundle of the &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. All these flat or skew polygons are regular polygons made of their numerator instances of a single chord #, whose length is that square root, Steinbach's function of the natural number.

The relationship between each bundle of flat Clifford parallel great circle polygons of a distinct length, and its corresponding bundle of skew polygrams of a longer length, is precisely that of a discrete simple rotation and its corresponding discrete isoclinic rotation. For example, the flat great circle hexagons of edge length {{radic|1}} lie in the invariant rotation planes of the 24-cell's characteristic 60° isoclinic rotation, in which the vertices move on skew hexagram rotation paths of edge length {{radic|3}}. The 24-cell contains 4 bundles of 4 Clifford parallel (disjoint) hexagon central planes each. The 24 vertices of the 24-cell in each bundle are linked by rotation in the 4 hexagon invariant planes in two different ways: simply and isoclinically. In a simple rotation the vertices of a single invariant hexagon exchange places on that great circle hexagon of edge {{radic|1}}. In an isoclinic rotation in which any one of these 4 disjoint hexagon central planes is invariant, all 4 of them are necessarily invariant planes in the rotation, because they rotate in parallel. In that isoclinic rotation the vertices of all 4 hexagon invariant planes (all 24 vertices of the 24-cell) exchange places concurrently, moving along 4 disjoint helical hexagrams of edge {{radic|3}} in parallel. It is a characteristic only of this distinct rotation in hexagon invariant planes that the helical vertex path or ''isocline'' is a hexagram of period 6, edge length {{radic|3}}, and circumference 720°. Rotations in other kinds of polygon invariant planes have different characteristic isoclines. The period of their isoclinic rotation must be ''the same or greater'' than the period of their simple rotation; in most cases (unlike this one) it is greater.

The 15 chords form not only distinct fiber bundles of polygons, but also distinct bundles of polyhedra: cell rings &lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt; and compound vertex figures &lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;. These combine to form their characteristic regular 4-polytope &lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Coxeter|1973|p=269|loc=Compounds|ps=; &quot;It is remarkable that the vertices of {5, 3, 3} include the vertices of all the other fifteen regular polytopes in four dimensions.&quot;}} All these elements of a regular 4-polytope constructed from one chord are themselves regular polytopes, and each of the 16 regular 4-polytopes is a function of one characteristic chord. If the regular 4-polytope is convex, its characteristic chord is its unit-radius edge. For a non-convex regular 4-polytope, the characteristic chord is its unit-radius ''mid-edge diameter'', the distance from the midpoint of each edge to the midpoint of the completely orthogonal edge. Each of the characteristic chords (each distinct natural number) has its expression as one of the 16 regular 4-polytopes and all the products of the distinct regular 1-polytopes, 2-polytopes and 3-polytopes that construct it, and the whole distinct construction from the natural number to the 4-polytope is an [[W:SO(4)|SO(4) symmetry group]] expression of that 4-polytope's characteristic discrete [[W:SO(4)#Isoclinic rotations|isoclinic rotation]], in the invariant central planes of its characteristic chord, by the characteristic chord arc.

...demonstrate this theory by proper enumeration of the chord lengths in the table...

...color-code the △ ☐ 𝜙 ✩ symbols ... chord colors in diagram should match them ... some of the diagram colors are wrong and must be changed, but otherwise diagram is pretty good ... i think in a colored diagram all chords of the same length should have the same size line dash, the polygons are not obvious enough and the 15 fanning-out chords are readily identified by their labels...

... add back the counts of chords in the box (very small font number under each table entry), and present these two images as the front and back a box of building sticks and rubber joints...What's in the other box...

...say the rational-root major chords {{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|5/2}} are the ''n''-edges: their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge...

...abbreviated text of the [[120-cell#Relationships among interior polytopes|''§Relationships among interior polytopes'']] section from the 120-cell article...to explain the color-coding

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius=1}}

See also the elements, properties and metrics of the sequence of regular 4-polytopes summarized in tabular form in the section ''[[#Build with the blocks|§Build with the blocks]]'', below.

== Alicia Boole Stott's original formulation of dimensional analogy ==
&lt;blockquote&gt;&quot;...a method by which bodies having a certain kind of semi-regularity may be derived from regular bodies in an Euclidean space of any number of dimensions...&quot;{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''; presents cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues|pp=12-45}}&lt;/blockquote&gt;

Alica Boole Stott's 1910 paper reads like an exceptionally well-written patent application. It is a comprehensive stepwise description of her original invention, a general method of dimensional analogy between regular and semiregular polytopes of the same edge length.

[[File:The Roots of Three.png|thumb|An illustration of Boole Stott's expansion of unit-edge triangle to hexagon captures the geometric basis in 3 of the &lt;math&gt;A_n&lt;/math&gt;root system.]]

The engineering-like articulation of her new principles is entirely discrete and group-theoretic. Boole Stott repeatedly refers to sets of polytope elements as ''groups''. All her transformations are discrete instances of one or another special case of her &quot;two inverse operations which may be called ''expansion'' and ''contraction''.&quot; Hers is an exploration of ''n''-dimensional space at its most basic: space defined by the discrete objects which fill it, themselves defined directly as groups of elements with a few operations on them, which transform them into each other. This is geometry which seems to precede Euclid's postulates, and does not depend on them. Instead it corresponds directly to the deeper group theory mathematics which underlies geometry, even though Boole Stott, like the ancients, did not have that mathematics.

Boole Stott enumerated her expansion and contraction operations as transformations between regular and semi-regular polytopes of a single edge length. In her 1910 paper she did not consider any polytopes of multiple edge lengths, or expansions of the subject polytope by a distance other than that exactly required to produce new edges of the original edge length, so of course she did not enumerate Moxness's non-uniform rhombicosidodecahedron (Hull #8). This may be why section 8&lt;sub&gt;3&lt;/sub&gt; is missing from her earlier drawings of the sections of {5,3,3}; she may not have had a construction for it.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The Boole Stott expansion (or contraction) operation is as precisely defined a transformation as Coxeter's reflection, rotation and translation operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions). Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;
where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;
where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;
For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} An instance of the expansion (or contraction) operation is an application of the general operation to a particular subject polytope, operating on all of its elements symmetrically at once. The general operation is restricted, such that one may only expand (or contract) the subject polytope by pushing all its edges, faces or cells directly away from (or toward) its center, without changing the size of the moved elements. This splits each vertex into several (or coalesces several vertices into one), and has corresponding effects on the other elements, producing new edges ''of the same size'' between pairs of separated vertices (or reducing some edges to single vertices). The distance all the elements are moved away from (or toward) the center must be such that the vertices separate producing new edges of only the original edge length (or come together reducing edges to single vertices). It is permitted to combine these three operations, e.g. by pushing the edges ''and'' the faces, or by pushing all three element sets, provided the combination results in a semi-regular polytope of only one edge length. The three operations are all commutative with one another, so combined operations can be considered either as a sequence of expansions (or contractions) linking several related polytopes, or as their concurrent movements, a single transformation of the subject polytope. Since there are 8 combinations of one to three operations there are 8 polytope variants (including the original regular polytope), and all are adjacent to one another by a single transformation. The variants of the original regular polytope are named for which element set(s) are moved. By enumerating all 8 variants of expansion (and contraction) for each of the six regular convex 4-polytopes, Boole Stott produced a table of regular and semi-regular 4-polytopes divided into 12 sets of 8: six expansion sets and six contraction sets.{{Sfn|Boole Stott|1910|loc=''Table of Polytopes in S&lt;sub&gt;4&lt;/sub&gt;''; note that Boole Stott's usage of ''S&lt;sub&gt;4&lt;/sub&gt;'' denotes a Euclidean 4-space ''E&lt;sub&gt;4&lt;/sub&gt;'', or a 3-sphere space embedded in it, ''not'' a 4-sphere space embedded in a Euclidean 5-space|p=30}} Each set includes just one of the six regular convex 4-polytopes, with its distinct unit-radius edge length, and the 7 possible semi-regular expansions (or contractions) of it, which all preserve that characteristic edge length. The polytopes in each set of 8 all have the same total number of elements as their original regular polytope, but each polytope in the set has a distinct combination of cell, face, and vertex figures. The 12 sets are not completely disjoint, and even within one set the 8 operations do not necessarily produce 8 unique polytopes. For example, the regular 5-cell (the 4-simplex polytope) has 5 cells, 10 faces, 10 edges, and 5 vertices (30 elements). Expanding the 5-cell produces 7 related semi-regular 4-polytopes of 30 elements each (two of which happen to be the same polytope). In each expansion, the 5 cells have expanded into semi-regular or regular polyhedra, the 10 faces and 10 edges ''may'' have expanded into prisms, and the 5 original vertices have expanded into regular or semi-regular cells.

== The regular 11-cell 4-polytope ==

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosadodecahedron). Each non-convex Jessen's icosahedron is a set of 6 reflex edges that are {{radic|5}} 5-cell edges, each set symmetrically arranged in 3 parallel pairs, not as far apart ({{radic|3}}) as they are long, in 3 orthogonal central planes. The 12-point (Jessen's sets of 6 reflex 5-cell edges) are inscribed in the 60-point with their endpoints as a vertex of a pentagon face. The 12 pentagon faces are joined to 5 other pentagon faces by the 5-cell edges; not to their nearest neighbor pentagons but to their next-to-nearest neighbor pentagons; and since there are 5 vertices to each pentagon face, there are 10 12-point (Jessen's sets of 6 reflex 5-cell edges). Each pentagon face is joined to another pentagon face by 5 reflex edges of five distinct Jessen's. The 5 reflex edges are Clifford parallel, the 5 edges of a pentagonal prism column that has a twist to it. The contraction of those 5 edges into 1, and those 10 vertices (two pentagons) into 2, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing all their edges as a single Jessen's. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (octahedron) by conflating its parallel pairs of edges, leaving the 11-cell's abstract 6-point (hexad cell). The hexad's situation is that it occurs in bundles of 5, disjoint but close together in the sense of adjacent, like the great circle rings in a Hopf fiber bundle. 

To summarize, the 12-point (2-hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell). We shall now be able to see how Fuller's perfect hexads combine with Todd's perfect pentads to form the perfect heptads of the 11-cell and 137-cell.  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=alias the hemi-cuboctahedron}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces. (The ''heptad'' arrives at the party, anonymously; they are our distinguished anti-celebrity, who most of the other guests have never even heard of.) The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. This 12-point (truncated tetrahedron) is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 100 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp: Truncated tetrahedron|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp: Truncated tetrahedron|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}} Of course Moxness has been there, seen that, already with his graphics software;{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} see his parameterized illustration of it exactly as it occurs in the 11-cell, below in his bestiary of A3, B3, and H3 Archimedian and Catalan polyhedra in the 120-cell.]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:Archimedean_and_Catalan_solid_hulls_with_their_Weyl_orbit_definitions.svg|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of the 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. In this catalog of Archimedean and Catalan polyhedra sighted in the 120-cell by Moxness's software,{{Sfn|Moxness: Archimedean and Catalan hulls|2023|loc=Hull #1 Archimedean Name A3 110 Truncated Tetrahedron A (upper left)}} it is the first entry (upper left), Moxness's Hull # = 1 with 12 vertices of 3D norm {{radic|11}}/2. On a {{radic|2}}-radius 3-sphere, 5 of these with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell. 20 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges are cells in the [[#The rings of the 11-cells|cell rings of the quasi-regular 11-cell]], which also has 12 regular 5-cells as cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

...

Both kinds of cells in the 11-cell have expressions in two different dimensions (on the 2-sphere and 3-sphere), because they are all centered on the center of the 3-sphere, unlike the cells of most 4-polytopes, which are centered on various off-center points within the 4-polytope's interior. 

One kind of 11-cell cell appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (semi-regular truncated cuboctahedron), a.k.a. the truncated tetrahedron. The other kind of cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now, except in the 120-cell. But this is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron, and a regular tetrahedron behind each face. The 11-cell is thus a honeycomb of mixed regular tetrahedra and truncated cuboctahedra (a.k.a. truncated tetrahedra) that meet at their shared 5-cell faces. 

Both the 5-cells (decahedra) and the truncated cuboctahedra (truncated tetrahedra) are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks in their 4-polytopes (off-center).{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} Therefore all the 11-cell's cells are concentric, which indeed makes a different kind of honeycomb, an inside-out cellular honeycomb with all the cells piled on top of each other overlapping in 4-space. They do not overlap in 3-sphere space however; their 3-volumes (but not their 4-contents) are disjoint. They make a proper 4D honeycomb like the cells of any 4-polytope, with volumetrically disjoint 3D cells that meet each other in the proper way at their shared 2D faces.

An 11-cell contains 6 completely disjoint 5-point (1-pentad 5-cell decahedron cells), and 5 volumetrically disjoint 12-point (2-hexad truncated cuboctahedron cells), 11 volumetrically disjoint cells altogether. Of course it is supposed to have 11 identical 11-point (elevenad hemi-icosahedral cells), and that is also true in its abstract sense, which is not the same sense as disjoint cellular containment: each pentad or hexad cell is inscribed in one of 11 larger 60-point (Moxness's rhombicosadodecahedron cells). The abstract pentads and hexads are only hemi-polyhedra, so the 11-cell's pentad cells are 12 (out of 120) disjoint instances of a pentad 5-point, and its hexad cells are 10 (out of 100) disjoint instances of a hexad 6-point, 1/10th of the 120-cell's cellular volume in both cases. The 11-cell's hemi-icosahedral cells are 11 (out of 120) non-disjoint instances of an elevenad 60-point. Each 11-cell shares each of its hexad cells with 9 other 11-cells, and each of its 11 vertices (out of 600) occurs in 9 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.|name=Tetrahedral vertex figures}}

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

...insert image thumb of 11-cell...

...Construct the 11-cell from the heptad....

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...Four concentric 4-polytopes may be mutually orthogonal. In any set of 4 mutually orthogonal regular 5-cells, one pair of faces in each orthogonal pair are completely orthogonal to each other. Thus three edges in one 5-cell in an orthogonal pair are opposite (completely orthogonal) to a face of the orthogonal 5-cell. .... A 5-cell has 5 vertices in its abstract hemi-icosahedral cells, but only 4 vertices in one concrete icosahedral cell; the 5th lies in an icosahedral cell of another 11-cell (not in another icosahedral cell of the same 11-cell).

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

....

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 11-cell ===

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon.... (wrong, 11-cell's Hopf map is the truncated icosahedron)

....

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk: Truncated icosahedron|2018}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

...

Since the rhombicosadodecahedron has 120 5-cell edges, the 120-cell has 120 completely disjoint 5-cells, and the rhombicosadodecahedron shares each vertex with 11 other rhombicosadodecahedra, how a rhombicosadodecahedron is bound to its neighbor 5-cells is constrained. A further constraint is that the abstract hemi-icosahedron has only 15 5-cell edges, so the 120 actual 5-cell edges of the rhombicosadodecahedron are conflated 8-to-1 in the hemi-icosahedron. To be conflated by contraction into a single edge, those 8 edges must lie parallel to each other in the 120-cell; either actually parallel in the same plane, or Clifford parallel in separate planes, as two edges in completely orthogonal planes may be parallel and orthogonal at the same time. Each 5-cell inscribed in the 120-cell contains 5 such tetrahedra occur in parallel pairs in 4 orthogonal planes (none of them completely orthogonal planes). 

.... 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
Each conflated vertex is the 10 vertices of two orthogonal regular 5-cells...
The 60-point contains 6 pairs of these orthogonal regular 5-cell cells...
...its 20 hexad cells are ..., a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere.

== The regular 137-cell 4-polytope ==

(description originally written for the 11-cell, which applies only to a fibration of 11 11-cells)

It is quasi-regular with two kinds of cells, the 12-point (regular icosahedron) and the 5-point (regular 5-cell), which have just one kind of regular face between them. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of regular facets (it is semi-regular with regular facets), it also means that its facets have expressions of two different dimensionalities because they are concentric in the 3-sphere. The 11-cell has two kinds of regular cells. One appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (regular icosahedron), but since it is embedded at the center of the 3-sphere (unlike the cells of most 4-polytopes) it also acts as an inscribed 4-polytope.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} The other cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now except in the 120-cell. This is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron. The 11-cell is a honeycomb of mixed icosahedron and half-icosahedron cells. Like the icosahedra, the decahedra are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks, off-center in their 4-polytopes. All the 11-cell's cells are concentric, which indeed makes an unusual honeycomb of cells, an inside-out honeycomb. It is a proper 4D honeycomb however, with volumetrically disjoint 3D cells that meet each other in the proper way at shared 2D faces. The embedded 12-point (regular icosahedron) cell is an icosahedral pyramid (as a 4-polytope), which is filled by 20 4-point (irregular tetrahedra) that meet at its center; the center point of the 11-cell acts as a 12th vertex, just as the center point of the 24-cell acts as a 25th vertex in some honeycombs. The 5-point (regular 5-cell) cell contains five 4-point (regular tetrahedra) which meet the 12-point (regular icosahedra)'s irregular tetrahedral cells face-to-face, so the 11-cell is quasi-regular in the usual sense of having two varieties of regular 3-polytope cells, but additionally both varieties of of its cells are bi-dimensional: another clue that the 11-cell has something to do with dimensional analogy

....

The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

....

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur to make an 11-cell. 11 of them occur in a 137-cell, all 120 of them occur in the 120-cell, and they occur nowhere else.

How can this regular 4-polytope possibly exist? By Schläfli's criterion {3, 3, ..} exists, but not as a finite polytope.{{Sfn|Coxeter|1973|loc=§7.7 Schläfli's criterion}} Coxeter found {3, 5, 3} as a finite hyperbolic honeycomb, but not in Euclidean 4-space.{{Sfn|Coxeter|1968|title=The Beauty of Geometry: Twelve Geometric Essays|loc=''Regular Honeycombs in Hyperbolic Space''|ps=; &quot;The honeycombs considered by the above authors [Schlegel, Schläfli, Klein, Sommerville] have finite cells and finite vertex figures. ... A further extension allows a cell or vertex figure to be a ''star''-polytope, so that the honeycomb covers the space several times. .... We shall find that there are four regular star-honeycombs in hyperbolic 4-space, as well as two infinite families of them in the hyperbolic plane.&quot;}} We find that {3, 3, ..} is a central expression of the &lt;math&gt;H_4&lt;/math&gt; root system, being the regular convex hull in 4 dimensions of 137 points, 96 pentads and 600 hexads. The minimal expression of &lt;math&gt;H_4&lt;/math&gt; is the regular convex hull {3, 3, 5} of 120 points, 600 tetrads and 75 hexads, and its maximal expression is the regular convex hull {5, 3, 3} of 600 points, 120 pentads and 675 hexads. {3, 3, ..} is a third regular &lt;math&gt;H_4&lt;/math&gt; polytope which fits between them. The properties and metrics of this regular sequence are presented in tabular form below in ''[[#Build with the blocks|§Build with the blocks]]''.

{|class=&quot;wikitable&quot;
!colspan=3|Regular &lt;math&gt;H_4&lt;/math&gt; polytopes
|-
|[[File:600-cell.gif|200px]]
|[[File:5-cell.gif|200px]]
|[[File:120-cell.gif|200px]]
|-
!120-point 600-cell
!137-point 137-cell
!600-point 120-cell
|}

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 137-cell ===
....

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=Apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|2022|loc=''[[Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]''}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie: On Fuller's use of language|2024|loc=''[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry'']]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center, 1948-1949|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with their mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the gender neutral shad can open their six beaks all the way, until they become flat squares and they becomes a cuboctahedron, or they can shut them all tight like a turtle retracting into their octahedron shell. The six mouths always move in unison. This is [[Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do their ''really'' odd trick -- where they flip their 6 jaws 90 degrees in their 6 faces and shut their 6 beaks on the opposite axis of their squares than the one they opened them on -- or whether they will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than their normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make their edge length exactly the same size as their radius, when they open their mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing their mouths in spherical synchrony, their 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron (the hexad), then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=''[[W:SO(4)|SO(4)]]''}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=Isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its orthogonal hexad's planes (wz, wx, wy).{{Efn|name=Six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of (expanded octahederon 16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh &amp; Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== Legendre's 12-point hexad binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a 12-point double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's perfectly realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, this 2-hexad expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre hexad) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction.{{Efn|name=Concentric polyhedra}} In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== ''n''-simplexes ===
....

The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== ''n''-orthoplexes ===
....
[[File:Great 5-cell √5 digons rectangle.png|thumb|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} ''regular'' great hexagons. These are the same 200 dodecagon central planes illustrated above, which also contain 6 120-cell edges and two opposing ''irregular'' great hexagons (truncated {{radic|6}} triangles). The 1200 {{radic|5}} 5-cell face triangles occupy 1200 other, non-central planes.]]

...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's Hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[16--cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[16--cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=Concentric polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[16--cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== ''n''-taliesins ===
The 4-simplex pentad building block combines with the 4-orthoplex hexad building block to make the 4-taliesin heptad building block. The heptad is the essential building block of the ''taliesin'' family of polytopes. The 11-cell and 57-cell hemi-polytopes and the 137-cell and 120-cell regular 4-polytopes are nuclear members of this trans-dimensional family. The 24-cell and 600-cell are extended members of it, because they also contain instances of the heptad building block.

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its spherical symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* [[W:Taliesin Myrddin Namkai-Meche|Taliesin Myrddin Namkai-Meche]], a young Reed College scholar who was stabbed to death by a crazy man in 2017 on a Portland, Oregon light rail train, when he rose to the nonviolent defense of two black teenagers who were being violently assaulted in racist and anti-Muslim language by a right-wing extremist.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in orthogonal [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].
* '''Taliesan polytope''', an (''n+1)''-polytope whose ''n''-polytope dimensional analogue inscribed concentrically in the ''n''-sphere forms a convex honeycomb of itself which is the (''n+1)''-polytope.{{Efn|name=Embedding point}}
* The taliesan polytope's honeycomb, a Hopf fibration of its great circles.
* The ''n''-dimensional family of dimensionally analogous polytopes related to a taliesan polytope by expansion and contraction operations.
* The characteristic [[w:Orthoscheme|orthoscheme]] of a teliesan polytope, an irregular simplex that is the complete gene for the entire expression of trans-dimensional symmetry relationships among the polytopes of the ''n''-dimensional family, as well as the complete gene for the teliesan polytope's rotational symmetry.{{Efn|name=Characteristic orthoscheme}}

=== ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== ''n''-equilaterals ===
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed at four different [[#n-leviathon|chord]] lengths from each other. In the {{radic|2}}-radius polytopes we have using as examples in this paper they are {{radic|2}}, {{radic|4}}, {{radic|6}} and {{radic|8}}, but in a ''unit-radius'' polytope they are: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}.

Each vertex is joined to 8 others by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}. The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== ''n''-pentagonals ===

[[File:600-cell.gif|thumb|The 120-point (600-cell), &lt;math&gt;H^4&lt;/math&gt;.]]
...

[[File:600-cell vertex geometry.png|thumb|Vertex geometry of the 600-cell, showing the 5 regular great circle polygons and the 8 vertex-to-vertex chord lengths with angles of arc. The golden ratio{{Efn|name=golden chords|group=}} governs the fractional roots of every other chord,{{Efn|name=fractional root chords}} and the radial golden triangles which meet at the center.|alt=|400x400px]]
The 120 vertices of the 600-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏&lt;sup&gt;−1&lt;/sup&gt;) beginning with a vertex; see column ''a''}} at eight different [[#n-leviathon|chord]] lengths from each other. These edges and chords of the 600-cell are simply the edges and chords of its five great circle polygons. In ascending order of length in a ''unit-radius'' polytope (not in a {{radic|2}} radius polytopes like our usual examples), they are {{radic|0.𝚫}}, {{radic|1}}, {{radic|1.𝚫}}, {{radic|2}}, {{radic|2.𝚽}}, {{radic|3}}, {{radic|3.𝚽}}, and {{radic|4}}.{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=Decimal fractions of square roots|group=}}

Notice that the four [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#''n-''equilaterals|hypercubic chords]] of the 24-cell ({{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|4}}) alternate with the four new chords of the 600-cell's additional great circles, the decagons and pentagons. The new ''golden chord'' lengths are necessarily square roots of fractions, but very special fractions related to the golden ratio{{Efn|1=The fractional-root ''golden chords'' are irrational fractions that are functions of {{radic|5}}. They exemplify that the [[W:golden ratio|golden ratio]] &lt;big&gt;𝜙&lt;/big&gt; {{=}} {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618 is a circle ratio related to &lt;big&gt;𝜋&lt;/big&gt;:&lt;br&gt;
: {{sfrac|𝜋|5}} = arccos (𝜙/2)
is one decagon edge, the 𝚽 = {{radic|0.𝚫}} = {{radic|0.382~}} ≈ 0.618 chord.
Reciprocally, in this function discovered by Robert Everest expressing &lt;big&gt;𝜙&lt;/big&gt; as a function of &lt;big&gt;𝜋&lt;/big&gt; and the numbers 1, 2, 3 and 5 of the Fibonacci series:&lt;br&gt;
: &lt;big&gt;𝜙&lt;/big&gt; = 1 – 2 cos ({{sfrac|3𝜋|5}})
{{sfrac|3𝜋|5}} is the arc length of the &lt;big&gt;𝜙&lt;/big&gt; = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618 chord.|name=golden chords|group=}} including the two [[W:golden section|golden section]]s of {{radic|5}}, as shown in the diagram.{{Efn|The 600-cell edges are decagon edges of length {{radic|0.𝚫}}, which is 𝚽, the ''smaller'' golden section of {{radic|5}}; the edges are in the inverse [[W:golden ratio|golden ratio]] {{sfrac|1|φ}} to the {{radic|1}} hexagon chords (the 24-cell edges).
The other fractional-root chords exhibit golden relationships as well. The chord of length {{radic|1.𝚫}} is a pentagon edge.
The next fractional-root chord is a decagon diagonal of length {{radic|2.𝚽}} which is &lt;big&gt;𝜙&lt;/big&gt;, the ''larger'' golden section of {{radic|5}}; it is in the golden ratio{{Efn|name=golden chords|group=}} to the {{radic|1}} chord (and the radius).{{Efn|Notice in the diagram how the &lt;big&gt;𝜙&lt;/big&gt; chord (the ''larger'' golden section) sums with the adjacent 𝚽 edge (the ''smaller'' golden section) to {{radic|5}}, as if together they were a {{radic|5}} chord bent to fit inside the {{radic|4}} diameter.}}
The last fractional-root chord is the pentagon diagonal of length {{radic|3.𝚽}}.
The [[W:Pentagon#Side length is given|diagonal of a regular pentagon]] is always in the golden ratio to its edge, and indeed &lt;big&gt;𝜙&lt;/big&gt;{{radic|1.𝚫}} is {{radic|3.𝚽}}.|name=fractional root chords|group=}}

....


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== ''n''-leviathon ===
...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of its &lt;math&gt;10^2&lt;/math&gt; disjoint instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction, before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]].{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}} .... It is only because she did not have the 11-cell to start with that she did not discover the regular 137-cell {3, 3, ..} by her method.

Between Alicia Boole Stott and her identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and her identification of symmetry groups with conservation laws (in her 1st theorem), we owe the deepest mathematics of geometry and physics originally to two pioneering women, colleagues of their contemporaries the greatest men of the academy, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just an [[W:abstract polytope|abstract 4-polytope]], not just a singleton convex 4-polytope, and not just a honeycomb.{{Sfn|Coxeter|1970|loc=''Twisted Honeycombs''}} The 11-point (11-cell) has a concrete ...regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}&lt;nowiki/&gt;} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, 3, 3} as the 137-point (137-cell).

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 11-cell, 137-cell and 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=''Clifford Spinors and Root System Induction: H4 and the Grand Antiprism''}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=Apology}}

== Build with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Duveneck|1978|loc=Carl Jung, quoted in ''Life on Two Levels''|p=ii|ps=.{{Sfn|Jung|1961|ps=: &quot;The best of truths is of no use unless it has become one's most personal inner experience. It is the duty of everyone who takes a solitary path to share with society what he finds on his journey of discovery.&quot;}}}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1954|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius={{radic|2}}}}

== Acknowledgements ==
...

== Notes ==
{{Regular convex 4-polytopes Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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{{Refend}}</text>
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      <text bytes="208614" sha1="spb3e91601ij8gnmr2gkq2kw0mdqx0d" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|June 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a regular 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the &lt;s&gt;120-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 5, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 11 [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] cells. Eleven 11-cells (plural) are the 137-cell, the &lt;s&gt;600-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 3, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 137 [[W:Rhombic triacontahedron|triacontahedron]] cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=''Regularity of Graphs, Complexes and Designs''}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=''A Symmetrical Arrangement of Eleven Hemi-Icosahedra''}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]]
The 4-dimensional regular polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:Simplex|simplex]] (called the [[5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:Orthoplex|orthoplex]] (called the [[16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular convex 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the [[w:Tesseract|16-point (8-cell) tesseract]], the 4-cubic, which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found within any of the other regular convex 4-polytopes, except in the largest and most complex one, the [[120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). However, we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the 16-cell is their very starting point, and the most frequently used tool in the box.

== 5-cells and hemi-icosahedra in the 11-cell ==
[[File:Symmetrical_5-set_Venn_diagram.svg|thumb|The 5-point (10-face) regular 5-cell (the regular 4-simplex). Grünbaum's rotationally symmetrical 5-set Venn diagram{{Sfn|Grünbaum|1975|loc=''Rotationally symmetrical 5-set Venn diagram'', Fig 1 (e)|ps=; partitions the individual elements of the 5-cell.}} is an illustration of the 5-cell labeling each of its &lt;math&gt;2^5&lt;/math&gt; elements.{{Sfn|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled; image includes the Python code to render it, optimising for maximum area of the smallest regions.}}]]
[[File:Hemi-icosahedron.png|thumb|The 6-point (10 face) [[W:hemi-icosahedron|hemi-icosahedron]], an abstraction of the regular icosahedron, has half as many faces, edges and vertices. Each element of the abstract polyhedron represents two real elements found in different places in the 11-cell's realization.{{Sfn|Ruen: hemi-icosahedron|2007}}]]
The most apparent relationship between the pentad 5-cell and the hexad hemi-icosahedron is that they both have 10 triangular faces. When we see a facet congruence between a 4-polytope and a 3-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 6-point 5-simplex. Another hexad is the 6-point 3-orthoplex; thus the hemi-icosahedron is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no chord congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals, 5-cell and 16-cell, who are famously stand-offish with each other. Interesting!

The cell of the 11-cell is an abstract hexad hemi-icosahedron with 5 central planes, most handsomely illustrated by Séquin.{{Sfn|Séquin &amp; Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing rings of polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The 11 hemi-icosahedral cells have 10 triangle faces each; the 5-cell's 5 tetrahedral cells have 10 faces and 10 edges all together. Cells meet face-to-face, so the only way 11-cells and 5-cells can share the same faces is if two hemi-icosahedral cells and two tetrahedral cells all meet at each face. Then 3 of each 5-cell's 5 vertices are a hemi-icosahedron face, and its other 2 vertices must be some 11-cell edge lying opposite the face. Coxeter determined that the 11-cell does indeed have an edge opposite each face, but it does not belong to the same hemi-icosahedral cell as its opposing face. He found that the 10 edges opposite each hemi-icosahedron's faces are the 10 edges of a single 5-cell, which does not share any vertices, edges or faces with the hemi-icosahedron. Of course, in the 11-point 11-cell there is just one such 5-point 5-cell that is completely disjoint from each 6-point hemi-icosahedron.{{Sfn|Coxeter|1984|p=110|loc=§6. The Petrie polygon [of the 11-cell]|ps=; &quot;We may reasonably call this edge and face ''opposites''. It is easy to find the face opposite to a given edge by looking at the faces to which a given edge belongs. ... Conversely, given a face, we can find the opposite edge by seeing which vertices belong to neither of the hemi-icosahedra which share that face. The ten edges opposite to the ten faces of one hemi-icosahedron are the edges of the complementary &lt;math&gt;a_4&lt;/math&gt; [4-simplex], that is, the joins of all pairs of the five vertices [of the 11-cell] not belonging to the given hemi-icosahedron.&quot;}}

There are 11 disjoint 5-cells in each 11-cell, which is comprised of 11 hemi-icosahedra, 55 faces, 55 edges and 11 vertices. The real 11-cell is harder to see than the abstract 11-cell that represents it, because the real hemi-icosahedron that the abstract hemi-icosahedron represents is hard to see, but seeing these objects will be easier once we have identified the real hemi-icosahedron and exactly where the 11-cell's real elements reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each hemi-icosahedron's 10 faces come from 10 disjoint 5-cells, and there is just one 5-cell with which it does not share a face.{{Efn|As Coxeter observes (in the previous citation), that unrepresented 5-point (5-cell) is the other 5 vertices of the 11-point (11-cell) that are not vertices of this 6-point (hemi-icosahedron), the hemi-icosahedron's disjoint complement.}} Each 5-cell has 10 faces from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share a face.

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in two different planes which are completely orthogonal to each other. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must be sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness: 8 concentric hulls|2022|loc=Hull #8 (lower right)|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell of the 11-cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Compounds in the 120-cell|§Compounds in the 120-cell]]'' that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software which uses Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s to render the polyhedra which are found in the interior of ''n''-dimensional polytopes.{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the tetrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is a real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (Hull #8) is a concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this {12} irregular great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing {6} irregular great hexagons (truncated triangles) of alternating 120-cell edges and 5-cell edges. The great dodecagon is a great circle of the hemi-icosahedral cell, and also a great circle of the 120-cell, because unlike the cells of most 4-polytopes, the 11-cell's cells are embedded in 4-space at the exact center of the 4-polytope, making them inscribed 4-polytopes themselves.{{Efn|name=Embedding point}} There are 25 great dodecagons and 120 5-cell edges in Moxness's 60-point (Hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (Hull #8) polyhedron altogether.

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point (Hull #8) is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] similar to the one from the catalog shown here,{{Sfn|Piesk: Rhombicosidodecahedron|2018}} but a slightly shallower truncation with smaller red pentagons (120-cell faces) and narrower rhombs. Rhombicosidodecahedra are made by truncating the [[W:Rhombic triacontahedron|rhombic triacontahedron]], which is the 30-sided polyhedron with only one kind of face, the dual of the 30-point icosidodecahedron. The 120-cell contains 120 of these 11-cell cells.]]
Moxness's 60-point (Hull #8) is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (Hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|The 5-point (regular 5-cell) can be another abstraction of Moxness's 60-point (Hull #8), 12-vertices-into-1, but it is not strictly the result of a contraction operation as described by Boole Stott, since she did not apply her expansion and contraction operations to uniform polytopes with more than one edge length.}} In fact Moxness was not the first to find a rhombicosidodecahedron in the 120-cell; Alicia Boole Stott was. It is the 6th section of {5,3,3} beginning with a cell, which Boole Stott identified as the ''e&lt;sub&gt;2&lt;/sub&gt; expansion'' of the icosahedron (or equivalently of its dual polyhedron the dodecahedron).{{Sfn|Boole Stott|1910|loc=§Examples of the e&lt;sub&gt;2&lt;/sub&gt; expansion|p=7}} But that rhombicosidodecahedron identified by Boole Stott is not Moxness's Hull #8: rather, it is the semi-regular Archimedean solid, with a single edge length and square faces. Moxness's Hull #8 with its two distinct edge lengths and rectangular faces is Coxeter's 8th section of {5,3,3} beginning with a cell, which is missing from the sections illustrated by Boole Stott.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than tetrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. The 120-cell contains 120 of Moxness's 60-point (11-cell cells). They are non-disjoint, with 12 60-points sharing each vertex.

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (Hull #8) is the real hemi-icosahedron cell, with 120 5-cell edges in it as chords lurking under its surface. The rhombicosadodecahedron has enough of those 5-cell edges in its interior to make 20 disjoint tetrahedral cells which belong to 20 distinct 5-cells.

The 600-point (120-cell) may be constructed as a semi-regular honeycomb of two kinds of cells, the 120 non-disjoint 60-point (Moxness's rhombicosadodecahedra) and their neighbors the 120 disjoint 5-point (5-cells), cell-bonded together by shared tetrahedral cells of edge-length {{radic|5}}. Each rhombicosadodecahedron is bonded to 20 islanded 5-cells, sharing with each a 5-cell tetrahedron (4 of its interior 5-cell faces, and 6 of its interior 5-cell edges). The rhombicosadodecahedron is also bonded directly to 10 other rhombicosadodecahedra (since each 11-cell has 11 rhombicosadodecahedron cells), again not by sharing one of its exterior faces, but this time by sharing an interior face, a {{radic|6}} triangle which lies in a central plane.{{Efn|The {{radic|6}} chord is another hidden chord lurking beneath the 60-point rhombicosadodecahedron's surface, like the {{radic|5}} 5-cell edge. Both these hidden chords lie on the same great circle with the 120-cell edges (the pentagon face edges). Three {{radic|6}} chords make a triangle which lies in a central plane. This {{radic|6}} great triangle must not be confused with the {{radic|5}} face triangle of the 5-cell, which does not lie in a central plane.}} The 120 rhombicosadodecahedra are also bonded directly together at ''all'' their exterior faces: at their pentagon faces (which are 120-cell faces of dodecahedral cells); at their small triangular faces (which are 600-cell faces of small tetrahedral cells, 10 of which are inscribed in each dodecahedral cell); and at their rectangular faces (which are faces of 10-point [[w:Pentagonal_prism|pentagonal prism]] cells, where the pentagonal prism is a column inscribed in a dodecahedral cell, joining its opposite pentagon faces). The 120 rhombicosadodecahedra are bonded directly to each other at their own exterior faces; no dodecahedra, small tetrahedra, or pentagonal prisms lie between them, and those objects are not cells of this honeycomb; we mention them here only to help locate the rhombicosadodecahedra within the 120-cell. The only objects which fill holes between the rhombicosadodecahedra are the 120 islanded 5-cells.{{Efn|This honeycomb of the 120-cell resembles the [[600-cell#Icosahedra|honeycomb of the 600-cell consisting of 24 face-bonded regular icosahedra]] with 24 clusters of 5 tetrahedra filling the holes between them. In the 120-cell there are 5 times as many of everything as in the 600-cell, so there are 120 (rhombicosadodecahedron) icosahedral cells with 600 tetrahedra filling the holes between them.}}

It is possible for rhombicosadodecahedra to meet each other at more than one kind of exterior face because they are not meeting there as cells of the same 11-cell; they belong to different 11-cells. The 11 rhombicosadodecahedra in the same 11-cell do not meet at their own exterior faces, they meet at their interior {{radic|6}} triangle faces. As a cellular 4-polytope, the 11-cell is inside-out. Its cells meet each other at interior planes, sharing interior {{radic|6}} triangle faces, instead of in the usual way that 4-polytopes' cells share an exterior face. It is a deeper kind of intertwining to be sure, but the 11 rhombicosadodecahedra in the same 11-cell are still volumetrically disjoint cells, and the 11-cell is a perfectly real 4-polytope with real cells. It is a semi-regular 4-polytope, because it has two different kinds of cells: 11 rhombicosadodecahedra and 11 5-cells. The 11 5-cells are islanded (not bound to each other), and the 11 rhombicosadodecahedra are bound both to 5-cells (at {{radic|5}} 5-cell faces) and to each other (at {{radic|6}} central triangle faces). In the section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#What's in the box|§What's in the box]]'', below, we will look closely at the relation between the {{radic|5}} chord and the {{radic|6}} chord, which is the relation between the pentad and the hexad, and at the general phenomenon of hemi-polytopes which meet each other at a central plane. Then in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Cell rings of the 11-cells|§Cell rings of the 11-cells]]'' we will look at the two-cell cellular structure of the 11-cell in more detail.

Considered as a single complex of overlapping cells rather than as distinct honeycombs of volumetrically disjoint cells, the 120-cell contains 60-point (rhombicosadodecahedral cells), islanded 5-point (5-cells), 20-point (dodecahedral cells), and 10-point ([[w:Pentagonal_prism|pentagonal prism]] cells). The 60-points meet each other in pairs at their pentagon faces, meet regular dodecahedra at those pentagon faces also, meet each other in quads at their 600-cell faces where they share an interior 4-point (600-cell tetrahedral cell) and its four faces (just as they share a larger interior 5-cell tetrahedron and its faces), and meet each other and the 10-point (pentagonal prisms) at their rectangle faces. We might try to imagine such a complex with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with parts truncated from 30 neighboring truncated dodecahedra pentagonal prisms). There are 120 of each of these 3 kinds of dodecahedron. 

Moxness's 60-point (rhombicosadodecahedron cell) is a central object of the 120-cell, and contains many wonders. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra by expansion and contraction operations, as Alicia Boole Stott discovered before 1910,{{Sfn|Polo-Blanco: ''Theory and history of geometric models of Alicia Boole Stott''|2007|loc=§5.3.2 1910 paper on semi-regular polytopes|pp=152-158|ps=; summarizes Boole Stott's method and results from {{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}}, including her tables of expansion-contraction dimensional analogies and a few of her illustrations.}} and those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, rediscovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter &amp; Petrie|1938|p=4|loc=''[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]''}}|name=Concentric polyhedra}} Before we move on from the 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice one more thing about the hidden {{radic|5}} chords lurking below its surface. The 12 little pentagon faces are spanned by sets of 6 {{radic|5}} chords (5-cell edges). The 6 chords of each set are disjoint (they don't touch or form 5-cell faces), but they are symmetrically arranged in 3 parallel pairs, {{radic|3}} apart, which lie in 3 orthogonal central planes. It turns out they are the 6 reflex edges of a 12-point non-convex polyhedron called the [[w:Jessen's_icosahedron|Jessen's icosahedron]], inscribed in the 60-point cell. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again and work our way outwards, before eventually in ''[[#The perfection of Fuller's cyclic design|§The perfection of Fuller's cyclic design]]'' we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cell)s ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks, showing its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Pentahemidemicube.png|120px]]
|[[File:Tetrahemihexahedron_rotation.gif|120px]]
|[[File:Pentahemicosahedron.png|120px]]
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts needed to build the 120-cell (the picture on the cover of the box), and everything else that's invisible in that picture which fits inside the 120-cell: all the astonishing regular 4-polytopes. The pentad building block is half a pair of orthogonal 5-point (regular 5-cells), the hexad building block is half an 8-point (16-cell), and the heptad building block is a quarter of an 11-point (11-cell). Everything can be built from these blocks by embedding them at the center of the 3-sphere, inscribing them as 4-polytopes to make compounds as with the 16-cell building block, rather than by assembling them face-to-face as 3-polytopes to make honeycombs in which they are cells or vertex figures.{{Efn|name=Embedding point}}

The pentad building block is the 5-point '''pentahemidemicube''', an abstract polyhedron with 10 equilateral triangle faces, 10 {{radic|5}} edges, and 5 vertices.{{Sfn|Christie: Pentahemidemicube|2024|loc=File:Pentahemidemicube.png|ps=; &quot;The 5-point ''pentahemidemicube'' is an abstract polyhedron with 10 equilateral triangle faces, 10 edges, and 5 vertices. It has a real presentation as the regular 5-cell.&quot;}} It has a real presentation as the 5-point (regular 5-cell 4-polytope), a regular tetrahedron expanded by addition of a fifth vertex equidistant from all the others in a fourth dimension. The pentad block is a tetrahedral pyramid (a 5-cell) inscribed in a cube: a tetrahedron inscribed in a cube (a demicube) expanded by addition of a fifth vertex, one of the cube's other 4 vertices. Each pentad (5-cell) building block lies in the 120-cell orthogonal to three other pentad (5-cell) building blocks. They are four completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges. Each regular 5-cell in the 120-cell is a pentad (pentahemidemicube) building block, in one of 30 orthogonal sets of 4. There are 120 pentad building blocks in the box.

Every vertex of the 16-cell, and therefore every vertex of the 8-cell, 24-cell, 600-cell and 120-cell, is surrounded by a 6-point (octahedral vertex figure). The hexad building block is that 6-point octahedron embedded at the center of the 4-polytope instead of at a vertex, that is, inscribed in the 3-sphere as a 4-polytope, the 6-point '''[[w:Tetrahemihexahedron|tetrahemihexahedron]]''' or '''hemi-cuboctahedron'''.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; &quot;The tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U&lt;sub&gt;4&lt;/sub&gt;. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices.&quot;}} The hexads have 4 red triangle 5-cell faces, bounded by 12 {{radic|5}} edges. Their 3 yellow square faces lie in central planes, because the building block is an abstract 4-polytope. The hexad is an irregular abstract polyhedron that also has 6 {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

Two orthogonal 5-point (pentad) building blocks intersect in a 4-point (inscribed tetrahedron) to form a 6-point (hexad) building block. Two orthogonal 6-point (hexad) building blocks intersect in a 4-point (central square) to form an 8-point (16-cell) building block, the most usable building block, from which everything larger can be built by compounding.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 120 pentads to pair with a completely orthogonal hexad, and although there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed) they only result in 120 pairs two different ways, so there are 120 11-cells. The 11-cell is 6 pentads' and 5 hexads' smallest joint compound, and the 120-cell is their largest, 120 pentads and 100 hexads.

The heptad{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''|pages=22–31}} building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|4}} edges and 9 {{radic|5}} edges), and 7 vertices.{{Sfn|Christie: Pentahemicosahedron|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;}} It is a compound of a 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]) and a 6-point (hexad [[W:hemi-icosahedron|hemi-icosahedron]]) intersecting in a 4-point (tetrahedron), that is, of three orthogonal 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]&lt;nowiki/&gt;s) intersecting in their common 4-point (tetrahedron). It is also the abstract cube contracted by removal of one vertex, which reduces one of the cube's two inscribed tetrahedra (demicubes) to a single remaining triangular face. As with all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Four &lt;small&gt;&lt;math&gt;(11-7)&lt;/math&gt;&lt;/small&gt; orthogonal 7-point (pentahemicosahedron) blocks can be joined at a common Petrie polygon (a skew hexagon which contains their orthogonal 6-edge paths) to construct a real 11-point ([[W:11-cell|11-cell]] semi-regular 4-polytope), which magically contains six &lt;small&gt;&lt;math&gt;(11-5)&lt;/math&gt;&lt;/small&gt; 5-point (1-pentad regular decahedra) as cells and five &lt;small&gt;&lt;math&gt;(11-6)&lt;/math&gt;&lt;/small&gt; 12-point (2-hexad semi-regular cuboctahedra) as cells. There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like [[W:Euclid's postulates|Euclid's postulates]], and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as honeycombs of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles. It possesses the complete symmetry of the polytope without any redundancy, because it contains one of each of the polytope's characteristic root elements.{{Efn|A [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be partitioned radially by its planes of symmetry (Coxeter's &quot;reflecting circles&quot;) into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated rotationally to generate its regular 4-polytope because it is the complete [[W:gene|gene]] for it, containing all of its elements and capturing all of its symmetry without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring that triggers the reflections.|name=Characteristic orthoscheme}} It is the gene for the polytope, which can be replicated to construct the polytope.

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its &lt;math&gt;A_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the regular compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;small&gt;&lt;math&gt;120\times 120 = 14400&lt;/math&gt;&lt;/small&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[16--cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its &lt;math&gt;B_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 384. The 120-cell is the convex hull of the regular compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 24-point (24-cell) is subdivided into 1152 instances of its [[24-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;F_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 1152. The 120-cell is the convex hull of the regular compound of 25 disjoint 24-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;25\times 1152 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has three equivalent origins in distinct root systems, though most are obviously constructed from one root, and not so obviously from the others. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything larger than the pentad can be constructed by compounding hexads, and we shall see that everything as large as a 24-cell has a construction from heptads which are a product of a pentad and a hexad. Surely every uniform polytope can be constructed by some function of expansions and contractions from any of three root systems, however indirect the recipe. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes, related by an equals sign between them, then give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics by [[W:Noether's theorem|Noether's theorem]].

=== Pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 octahedral cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell contains the 600 octahedral cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It also contains the 600 cubical cells of its 75 8-cells. It is a 600-cell in three ways, although the octahedra and cubes are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell and the 8-cell, respectively.|name=600 cells three ways}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (1-pentad 5-cell), out of the 8-point (2-hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (1-pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope that contains 5-point (1-pentad 5-cell) building blocks beside the largest one, it just isn't a ''regular'' 4-polytope. It is semi-regular with two kinds of cells, the 4-point (regular tetrahedron) and the 12-point (truncated cuboctahedron), which have just one kind of regular face between them, a 5-cell face. That semi-regular 4-polytope is the 11-cell.

We said earlier that the 11-cell had 11 disjoint 5-cells and 11 hemi-icosahedral cells in it, and that its construction from them might be hard to see, but now we are going to account numerically, at least, for how it can have 6 pentad cells plus 5 hexad cells magically adding to 11 of the same kind of elevenad cell. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, 11. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (5-cell regular decahedron) cells, and the 12 6-point (hexad) building blocks into 100 12-point (regular icosahedron) cells, and then magically adding one whole 5-point (5-cell regular decahedron) cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 12-point (regular icosahedron) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such (cell bundles of 6 + 5 = 11 cells), with each 11-cell bundle containing 11 of both kinds of cell. Very, very strange!{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden'': §II To Add is to Multiply|p=38|ps=; the sequence of regular polygons with harmonious proportions beginning {5}, {7}, {9}, {11} as a family of natural numbers; &quot;The numbers in this optimal family have remarkable properties [that allow] similar figures to be arranged easily or allow a figure to be dissected into a set of similar figures. This is the origin of the Greek idea of geometric progression — that ''multiplication can be accomplished instead by addition''.... For instance, it is well known that the square of the golden ratio is one more than itself (written above as 𝜙&lt;sup&gt;2&lt;/sup&gt; = 𝜙 + 1) and one less than 𝜙 is its reciprocal (1/𝜙 = 𝜙 - 1). The first relation equates multiplication with addition, while the second accomplishes division through subtraction. [These are pentagon {5} bisectional golden ratios, but] the heptagonal [{7} tri-sectional] ratios also behave in surprising ways — adding to multiply, subtracting to divide.... There are analogous constructions for [nonagon {9}] quadrisectional and [hendecagon {11}] pentasectional systems.&quot;}}

The 120-cell has three interchangeable cyclic expansion-contraction cycles, the pentad, hexad and heptad cyclic designs, rooted in the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems, respectively. Each of them constructs the whole sequence of regular 4-polytopes from a single building block. The key to all three cycles is that 5 + 6 = 11, and there is an 11-point quasi-regular 4-polytope in the sequence of convex 4-polytopes. It lies between the 8-point (16-cell) and the 16-point (8-cell), and is itself a &lt;math&gt;B_4&lt;/math&gt; polytope. Its role is to be a unified single building block common to all three cyclic expansion-contraction cycles. It is involved with the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems and expresses in one building block their relations to each other.

We have already seen how the hexad building cycle works. It is plain compounding of 4-orthoplexes at the beginning, increasing to densely intricate non-disjoint symmetry toward the end, making it difficult to see why it works once past the 24-cell. This is how the build works, in terms of sixes. The 16-cell is a 2-hexad (3 pairs of completely orthogonal great square planes). The 24-cell is 3 16-cells and a 4-hexad, 4 different ways: 16 non-disjoint hexads (great hexagon planes). The 120-point (600-cell) is 25 24-cells and a 20-hexad, 10 different ways: 200 non-disjoint hexads (great hexagon planes). The 600-point (120-cell) is 225 (9 times 25) 24-cells. ....

The pentad building cycle doesn't do plain compounding to regular 4-polytopes until the last one, the 120-cell. Fives don't even appear as pentagons until after the 24-cell. We have seen how there are 11 5-cells in the quasi-regular 11-cell, which occurs before the 8-cell, and also how everything can be compounded from irregular 5-cells (4-orthoschemes specialized for each 4-polytope). It is still not evident how a pentad cycle can work to build all the regular 4-polytopes from the regular 4-simplex. This is how is how the build works, in terms of fives. ....

This is how the build works, in terms of elevens. The 11-cell is 4 orthogonal heptads (intersecting in their skew hexagon Petrie polygon). The 24-cell is also 4 heptads (intersecting in 4 orthogonal digon axes, so each heptad contributes 6 unique points to a single fibration of 24 points). Heptads are quarter elevenads, so the 11-cell and 24-cell are both 1 elevenad. ....

See the tabular sequence of convex 4-polytopes in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Build with the blocks|§Build with the blocks]]'', below: table rows ''Inscribed, Pentad, Hexad, Heptad''.

=== 11 of them ===
There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of eleven 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum eleven of them. There is only a single Hopf fibration of the 11 great circles of an 11-cell. 

The 5-cell and the 11-cell are limit cases. The 5-cells are completely disjoint, but the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 10 other instances of itself.

== Cell rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer &amp; Segerman|2013|loc=''Puzzling the 120-cell''}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his section of the Wikipedia article on the 120-cell entitled ''[[120-cell#Visualization|§Visualization]]''.{{Sfn|Ruen &amp; Goucher et al. eds. 120-cell|2024|loc=Goucher's ''[[120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious place to look for the 11-cell's Hopf map is in Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other place it could be hiding. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The quasi-regular convex 11-cell|§The quasi-regular convex 11-cell]]'', and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen: triacontagon|2011|loc=''Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them except his illustrations. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Four orthogonal projections of the 120-cell to a central plane, selected from the table of 3 regular and 11 compound 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|100 icosahedra&lt;br&gt;120 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|5 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 icosahedra&lt;br&gt;6 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|120
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|4
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|50
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|7.2°
|align=center|36°
|align=center|6°
|align=center|7.2°
|}

....

Here we summarize our findings on the 11-cells so far, including in advance some results to be obtained in the next section two sections.

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section....

== The 15 major chords ring everything ==
The natural numbers are each a distinct flavor.

Each integer has its own distinct taste. The first six integers {0..5} are the six strongest, most familiar tastes in nature, instantly distinguished and utterly unique.

The seventh flavor {6} is the first ratio of two of the others. Thereafter follows in strange sequence all the ratios of all the flavors, each marriage of every two flavors, most of which are fractions &lt;math&gt;\{k/d\}&lt;/math&gt;, but some of which are integers &lt;math&gt;\{k\}&lt;/math&gt;.

Some of these ratios are successful flavors in their own right, as distinct and inimitable as the marriage of chocolate and oranges, but the marriage of many flavor pairs is mud. It is a rule of thumb among flavor specialists that nearly any taste can be successfully alloyed with either onions or chocolate, but the direct marriage of these two flavors is notably unsuccessful. Some flavors make famous pairings or infamous ones, and subtle new pairings are always to be discovered that work surprisingly well for some purpose, but the majority of flavor marriages are undistinguished compounds.

In polytope geometry, each chord of a polytope is both is a distinct 1-dimensional object, a chord of the sphere of a distinct length &lt;math&gt;l&lt;/math&gt;, and a distinct natural number &lt;math&gt;h&lt;/math&gt;, a unique flavor. If the polytope is regular, it is a noteworthy and successful flavor. The chord's length &lt;math&gt;l&lt;/math&gt; is a square root, related to the natural number &lt;math&gt;h = k/d&lt;/math&gt; and to a set of polytopes they both represent, by a formula discovered by Steinbach.{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden''; Figure 5. Optimal sections and proportions|p=37|ps=; the regular polygons {5}, {7}, {9} and {11} with their diagonals define respectively: {5} the golden bisection proportional to 𝜙; {7} an analogous trisection; {9} an analogous quadrasection; {11} an analogous pentasection.}} The chord length &lt;math&gt;l&lt;/math&gt; is related to the number of sides of the regular polygon formed by the chord (its numerator &lt;math&gt;k&lt;/math&gt;), and to the winding number or density of its regular skew Petrie polygon (its denominator &lt;math&gt;d&lt;/math&gt;).

The regular 4-polytopes do not contain an infinite variety of these chord flavors, but only 30 of them: only 15 unique pairs of 180° degree complementary chords, which combine to make every regular ''n''-dimensional thing found in nature.{{Efn|The 15 ''major chords'' suffice to construct all the regular ''n''-polytopes. Their 180° complements include 15 ''minor chords'' which occur only in the 120-cell, not in any smaller regular polytope.{{Efn|name=Major and minor chords}} Including the complement of the 180° {{radic|4}} digon {2} chord (the {0} chord {{radic|0}}), and the skew variant of the 90° {{radic|2}} square {4} chord (the {30/7} chord), there are 32 distinct natural numbers which are all the chords of all the regular polytopes in all dimensions.|name=32 chords}} All the regular polytopes (including the largest one on the cover of the box) can be built not only from a box containing a number of blocks of 3 shapes glued together by gravity, they can be also be built from a box containing a number of sticks of 15 lengths glued together by rubber joints.

[[File:15 major chords.png|500px|thumb|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the others occur only in the interior of 120-cell, not as edges of any of the regular 4-polytopes. There are 30 distinct 4-space chordal distances between vertices within the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to identify any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=Major and minor chords}} chords are properly labelled #1 — #15 in the order they fan out by size.{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''; Figure 3|ps=; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with a &quot;fan of chords&quot; diagram like the one here.|p=23}} They join vertex pairs which are 1 — 15 edges apart on a &lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon. A #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; chord forms &lt;small&gt;&lt;math&gt;f+1&lt;/math&gt;&lt;/small&gt; regular geodesic polygons which include a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; disjoint regular {{nowrap|&lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt;-gons}} (e.g. &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; a compound of 5 disjoint hexagons) plus a single [[W:Triacontagon#Triacontagram|regular skew {{nowrap|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram}}]] that wraps around the sphere &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times before closing its circle. Notice the maverick #11 chord, with its non-conforming origin point for its {{nowrap|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;}} polygram.]]

These 15 chords (natural numbers) include the integers 2, 3, 4, 5, 6, 10, 11, 15 and 30. The others are the fractions 5/2, 10/3, 13/7, 15/2, 15/4, and 15/7. Represented among the chord lengths (the square roots) measured in ''unit radii'' (not in {{radic|2}} radii, as metrics are often given elsewhere in this article) are the {{background color|paleturquoise|roots of the first 4 positive integers {{radic|1}}, {{radic|2}}, {{radic|3}} and {{radic|4}}}}, the {{background color|#FFCCCC|root of the very distinguished rational fraction {{radic|5/2}}}}, and the roots of various distinguished irrational fractions as Steinbach discovered: for example, 4 of them are {{background color|yellow|roots of golden numbers (a small integer ± 𝜙)}}. The 15 chords each form their regular polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, which is just &lt;small&gt;&lt;math&gt;\{k\}&lt;/math&gt;&lt;/small&gt; in 8 cases. Arranged in order #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; from shortest edge-length square root to longest, the #1 — #15 chords bridge vertex pairs which are &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; vertices apart on a [[W:Triacontagon#Petrie polygons|&lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon]]. Each chord makes a [[W:Triacontagon#Triacontagram|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; polygram]] of its edge length, a compound of the smaller &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. The 15 distinct polygons form a sequence you would not anticipate exactly unless you have Steinbach's formula:

{| style=&quot;background-color:transparent; white-space:nowrap;&quot;
|-
| {30/1} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|.073}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{30} 
| 120-cell edges
|-
| {30/2}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.191}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15}
|- style=&quot;background: yellow;&quot;|
| {30/3}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.382}} 
| &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi}&lt;/math&gt;&lt;/small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{10}''' 
| 600-cell edges
|-
| {30/4} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.573}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/2}
|- style=&quot;background: paleturquoise;&quot;|
| {30/5}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|1}}'''
| &lt;math&gt;1&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{6}''' 
| 24-cell, 8-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/6}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|1.382}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{5}'''
|- style=&quot;background: paleturquoise;&quot;|
| {30/7}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|2}}'''
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{4}''' 
| 16-cell edges
|- style=&quot;background: #FFCCCC;&quot;|
| {30/8}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.5}} 
| &lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;&lt;small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/4} 
| 5-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/9}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.618}} 
| &lt;math&gt;\phi&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{10/3}
|- style=&quot;background: paleturquoise;&quot;|
| {30/10}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|3}}''' 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{3}'''
|-
| {30/11}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.426}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{11/11}
|- style=&quot;background: yellow;&quot;|
| {30/12}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.618}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{5/2}
|-
| {30/13}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.810}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{13/7}
|-
| {30/14}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.928}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/7}
|- style=&quot;background: paleturquoise;&quot;|
| {30/15}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|4}}''' 
| &lt;math&gt;2&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{2}'''
|}

{| class=wikitable style=&quot;white-space:nowrap; float:right;&quot;
!colspan=6|Natural number &lt;small&gt;&lt;math&gt;h=\{k/d\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Fibers&lt;br&gt;&lt;small&gt;&lt;math&gt;f\{h\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Cells&lt;br&gt;&lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|4-tope&lt;br&gt;&lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Verts&lt;br&gt;&lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|[[W:Triacontagon#Triacontagram|30-gram]]&lt;br&gt;&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;
|-
!Chord
!Arc
!colspan=2|&lt;small&gt;&lt;math&gt;l/\sqrt{1}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;l/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;h&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#1
|15.5~°
|{{radic|0.073}}
|0.270
|{{radic|0.146}}
|&lt;small&gt;&lt;math&gt;30&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1\{30\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;5\{3,3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/1\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#2
|25.2~°
|{{radic|0.191}}
|0.437
|{{radic|0.382}}
|&lt;small&gt;&lt;math&gt;15&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;..\{3,3\}..&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/2\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#3
|36°
|{{radic|0.382}}
|0.618
|{{radic|0.764}}
|&lt;small&gt;&lt;math&gt;10&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;30\{3,3\}20&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/3\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#4
|44.5~°
|{{radic|0.573}}
|0.757
|{{radic|1.146}}
|&lt;small&gt;&lt;math&gt;15/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/4\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5
|60°
|{{radic|1}}
|1
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;6&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/5\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#6
|72°
|{{radic|1.382}}
|1.175
|{{radic|2.624}}
|&lt;small&gt;&lt;math&gt;5&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}} 
|&lt;small&gt;&lt;math&gt;10\{5,3\}12&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/6\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#7
|90°
|{{radic|2}}
|1.414
|{{radic|4}}
|&lt;small&gt;&lt;math&gt;4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7\{4\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;4\{4,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;9\{3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/7\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: #FFCCCC;&quot;|
|#8
|104.5~°
|{{radic|2.5}}
|1.581
|{{radic|5}}
|&lt;small&gt;&lt;math&gt;15/4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/4\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}} 
|&lt;small&gt;&lt;math&gt;15\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/8\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#9
|108°
|{{radic|2.618}}
|1.618
|{{radic|5.236}}
|&lt;small&gt;&lt;math&gt;10/3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10/3\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/9\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#10
|120°
|{{radic|3}}
|1.732
|{{radic|6}}
|&lt;small&gt;&lt;math&gt;3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;10\{3\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/10\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#11
|135.5~°
|{{radic|3.426}}
|1.851~
|{{radic|6.852}}
|&lt;small&gt;&lt;math&gt;11/11&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;11\{11/11\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;11\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#12
|144°
|{{radic|3.618}}
|1.902
|{{radic|7.235}}
|&lt;small&gt;&lt;math&gt;5/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2},3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/12\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#13
|154.8~°
|{{radic|3.810}}
|1.952
|{{radic|7.621}}
|&lt;small&gt;&lt;math&gt;13/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;13\{13/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;7\{5,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/13\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#14
|164.5~°
|{{radic|3.928}}
|1.982
|{{radic|7.857}}
|&lt;small&gt;&lt;math&gt;15/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;137\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/14\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{radic|8}}
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15\{2\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐𝜙&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;8\{3,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/15\}&lt;/math&gt;&lt;/small&gt;
|}

The polygons '''{10}''', '''{6}''', '''{5}''', '''{4}''', '''{3}''', and '''{2}''' are flat polygons that each lie on a great circle in a central plane of the 3-sphere. The polygons {30}, {15}, {15/2}, {15/4}, {10/3}, {11/11}, {5/2}, {13/7}, {15/7} and all the {30/''n''} polygons are skew polygrams spiraling through curved 3-space, each successive edge in a different central plane of the 3-sphere. Projected to the 1-sphere (a circle in the plane), they are [[W:Star polygon#Regular star polygon|regular star polygrams]], the simplest being the pentagram {5/2}. In 4-space they wind around the 3-sphere more than once ''without self-intersecting'', before they close their circuit. They lie on helical geodesic circles or ''isoclines'' of circumference greater than 2𝝅''r''.

Even the 6 integer chords which form flat polygons also form skew polygrams. Each of the 15 chords occurs as a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, a fiber bundle of Clifford parallel polygons that totals 30 vertices per bundle. Each chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; also forms a single skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram that winds through another, shorter chord's bundle of parallel polygons, intersecting them in succession multiple times each, and weaving that whole bundle of parallel fibers together as the cross-thread does on a loom. The polygram cross-thread has to wind multiple times through the whole bundle (&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times around the [[W:Triacontagon#Triacontagram|30-gram]] in which each edge connects every &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt;th vertex) to complete an enumeration of all 30 vertices in the bundle. For example, the #5 chord forms hexagon &lt;small&gt;&lt;math&gt;\{6\}&lt;/math&gt;&lt;/small&gt; flat planes in parallel bundles of 5, written &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt;, and also a skew &lt;small&gt;&lt;math&gt;\{15/2\}&lt;/math&gt;&lt;/small&gt;-gram of #5 edges which winds 2 times through the #3 chord's bundle of 3 decagon &lt;small&gt;&lt;math&gt;\{10\}&lt;/math&gt;&lt;/small&gt; flat planes. Every chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; forms both its &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram and its fiber bundle of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, and the edges of one chord's parallel polygons are also the edges of its skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram weaving through some shorter chord's bundle of parallel polygons. Each numbered chord is a natural number, a ratio of two integers and their polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, either flat or skew. Each chord number defines a &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; skew polygon that is a compound bundle of the &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. All these flat or skew polygons are regular polygons made of their numerator instances of a single chord #, whose length is that square root, Steinbach's function of the natural number.

The relationship between each bundle of flat Clifford parallel great circle polygons of a distinct length, and its corresponding bundle of skew polygrams of a longer length, is precisely that of a discrete simple rotation and its corresponding discrete isoclinic rotation. For example, the flat great circle hexagons of edge length {{radic|1}} lie in the invariant rotation planes of the 24-cell's characteristic 60° isoclinic rotation, in which the vertices move on skew hexagram rotation paths of edge length {{radic|3}}. The 24-cell contains 4 bundles of 4 Clifford parallel (disjoint) hexagon central planes each. The 24 vertices of the 24-cell in each bundle are linked by rotation in the 4 hexagon invariant planes in two different ways: simply and isoclinically. In a simple rotation the vertices of a single invariant hexagon exchange places on that great circle hexagon of edge {{radic|1}}. In an isoclinic rotation in which any one of these 4 disjoint hexagon central planes is invariant, all 4 of them are necessarily invariant planes in the rotation, because they rotate in parallel. In that isoclinic rotation the vertices of all 4 hexagon invariant planes (all 24 vertices of the 24-cell) exchange places concurrently, moving along 4 disjoint helical hexagrams of edge {{radic|3}} in parallel. It is a characteristic only of this distinct rotation in hexagon invariant planes that the helical vertex path or ''isocline'' is a hexagram of period 6, edge length {{radic|3}}, and circumference 720°. Rotations in other kinds of polygon invariant planes have different characteristic isoclines. The period of their isoclinic rotation must be ''the same or greater'' than the period of their simple rotation; in most cases (unlike this one) it is greater.

The 15 chords form not only distinct fiber bundles of polygons, but also distinct bundles of polyhedra: cell rings &lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt; and compound vertex figures &lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;. These combine to form their characteristic regular 4-polytope &lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Coxeter|1973|p=269|loc=Compounds|ps=; &quot;It is remarkable that the vertices of {5, 3, 3} include the vertices of all the other fifteen regular polytopes in four dimensions.&quot;}} All these elements of a regular 4-polytope constructed from one chord are themselves regular polytopes, and each of the 16 regular 4-polytopes is a function of one characteristic chord. If the regular 4-polytope is convex, its characteristic chord is its unit-radius edge. For a non-convex regular 4-polytope, the characteristic chord is its unit-radius ''mid-edge diameter'', the distance from the midpoint of each edge to the midpoint of the completely orthogonal edge. Each of the characteristic chords (each distinct natural number) has its expression as one of the 16 regular 4-polytopes and all the products of the distinct regular 1-polytopes, 2-polytopes and 3-polytopes that construct it, and the whole distinct construction from the natural number to the 4-polytope is an [[W:SO(4)|SO(4) symmetry group]] expression of that 4-polytope's characteristic discrete [[W:SO(4)#Isoclinic rotations|isoclinic rotation]], in the invariant central planes of its characteristic chord, by the characteristic chord arc.

...demonstrate this theory by proper enumeration of the chord lengths in the table...

...color-code the △ ☐ 𝜙 ✩ symbols ... chord colors in diagram should match them ... some of the diagram colors are wrong and must be changed, but otherwise diagram is pretty good ... i think in a colored diagram all chords of the same length should have the same size line dash, the polygons are not obvious enough and the 15 fanning-out chords are readily identified by their labels...

... add back the counts of chords in the box (very small font number under each table entry), and present these two images as the front and back a box of building sticks and rubber joints...What's in the other box...

...say the rational-root major chords {{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|5/2}} are the ''n''-edges: their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge...

...abbreviated text of the [[120-cell#Relationships among interior polytopes|''§Relationships among interior polytopes'']] section from the 120-cell article...to explain the color-coding

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius=1}}

See also the elements, properties and metrics of the sequence of regular 4-polytopes summarized in tabular form in the section ''[[#Build with the blocks|§Build with the blocks]]'', below.

== Alicia Boole Stott's original formulation of dimensional analogy ==
&lt;blockquote&gt;&quot;...a method by which bodies having a certain kind of semi-regularity may be derived from regular bodies in an Euclidean space of any number of dimensions...&quot;{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''; presents cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues|pp=12-45}}&lt;/blockquote&gt;

Alica Boole Stott's 1910 paper reads like an exceptionally well-written patent application. It is a comprehensive stepwise description of her original invention, a general method of dimensional analogy between regular and semiregular polytopes of the same edge length.

[[File:The Roots of Three.png|thumb|An illustration of Boole Stott's expansion of unit-edge triangle to hexagon captures the geometric basis in 3 of the &lt;math&gt;A_n&lt;/math&gt;root system.]]

The engineering-like articulation of her new principles is entirely discrete and group-theoretic. Boole Stott repeatedly refers to sets of polytope elements as ''groups''. All her transformations are discrete instances of one or another special case of her &quot;two inverse operations which may be called ''expansion'' and ''contraction''.&quot; Hers is an exploration of ''n''-dimensional space at its most basic: space defined by the discrete objects which fill it, themselves defined directly as groups of elements with a few operations on them, which transform them into each other. This is geometry which seems to precede Euclid's postulates, and does not depend on them. Instead it corresponds directly to the deeper group theory mathematics which underlies geometry, even though Boole Stott, like the ancients, did not have that mathematics.

Boole Stott enumerated her expansion and contraction operations as transformations between regular and semi-regular polytopes of a single edge length. In her 1910 paper she did not consider any polytopes of multiple edge lengths, or expansions of the subject polytope by a distance other than that exactly required to produce new edges of the original edge length, so of course she did not enumerate Moxness's non-uniform rhombicosidodecahedron (Hull #8). This may be why section 8&lt;sub&gt;3&lt;/sub&gt; is missing from her earlier drawings of the sections of {5,3,3}; she may not have had a construction for it.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The Boole Stott expansion (or contraction) operation is as precisely defined a transformation as Coxeter's reflection, rotation and translation operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions). Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;
where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;
where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;
For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} An instance of the expansion (or contraction) operation is an application of the general operation to a particular subject polytope, operating on all of its elements symmetrically at once. The general operation is restricted, such that one may only expand (or contract) the subject polytope by pushing all its edges, faces or cells directly away from (or toward) its center, without changing the size of the moved elements. This splits each vertex into several (or coalesces several vertices into one), and has corresponding effects on the other elements, producing new edges ''of the same size'' between pairs of separated vertices (or reducing some edges to single vertices). The distance all the elements are moved away from (or toward) the center must be such that the vertices separate producing new edges of only the original edge length (or come together reducing edges to single vertices). It is permitted to combine these three operations, e.g. by pushing the edges ''and'' the faces, or by pushing all three element sets, provided the combination results in a semi-regular polytope of only one edge length. The three operations are all commutative with one another, so combined operations can be considered either as a sequence of expansions (or contractions) linking several related polytopes, or as their concurrent movements, a single transformation of the subject polytope. Since there are 8 combinations of one to three operations there are 8 polytope variants (including the original regular polytope), and all are adjacent to one another by a single transformation. The variants of the original regular polytope are named for which element set(s) are moved. By enumerating all 8 variants of expansion (and contraction) for each of the six regular convex 4-polytopes, Boole Stott produced a table of regular and semi-regular 4-polytopes divided into 12 sets of 8: six expansion sets and six contraction sets.{{Sfn|Boole Stott|1910|loc=''Table of Polytopes in S&lt;sub&gt;4&lt;/sub&gt;''; note that Boole Stott's usage of ''S&lt;sub&gt;4&lt;/sub&gt;'' denotes a Euclidean 4-space ''E&lt;sub&gt;4&lt;/sub&gt;'', or a 3-sphere space embedded in it, ''not'' a 4-sphere space embedded in a Euclidean 5-space|p=30}} Each set includes just one of the six regular convex 4-polytopes, with its distinct unit-radius edge length, and the 7 possible semi-regular expansions (or contractions) of it, which all preserve that characteristic edge length. The polytopes in each set of 8 all have the same total number of elements as their original regular polytope, but each polytope in the set has a distinct combination of cell, face, and vertex figures. The 12 sets are not completely disjoint, and even within one set the 8 operations do not necessarily produce 8 unique polytopes. For example, the regular 5-cell (the 4-simplex polytope) has 5 cells, 10 faces, 10 edges, and 5 vertices (30 elements). Expanding the 5-cell produces 7 related semi-regular 4-polytopes of 30 elements each (two of which happen to be the same polytope). In each expansion, the 5 cells have expanded into semi-regular or regular polyhedra, the 10 faces and 10 edges ''may'' have expanded into prisms, and the 5 original vertices have expanded into regular or semi-regular cells.

== The regular 11-cell 4-polytope ==

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosadodecahedron). Each non-convex Jessen's icosahedron is a set of 6 reflex edges that are {{radic|5}} 5-cell edges, each set symmetrically arranged in 3 parallel pairs, not as far apart ({{radic|3}}) as they are long, in 3 orthogonal central planes. The 12-point (Jessen's sets of 6 reflex 5-cell edges) are inscribed in the 60-point with their endpoints as a vertex of a pentagon face. The 12 pentagon faces are joined to 5 other pentagon faces by the 5-cell edges; not to their nearest neighbor pentagons but to their next-to-nearest neighbor pentagons; and since there are 5 vertices to each pentagon face, there are 10 12-point (Jessen's sets of 6 reflex 5-cell edges). Each pentagon face is joined to another pentagon face by 5 reflex edges of five distinct Jessen's. The 5 reflex edges are Clifford parallel, the 5 edges of a pentagonal prism column that has a twist to it. The contraction of those 5 edges into 1, and those 10 vertices (two pentagons) into 2, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing all their edges as a single Jessen's. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (octahedron) by conflating its parallel pairs of edges, leaving the 11-cell's abstract 6-point (hexad cell). The hexad's situation is that it occurs in bundles of 5, disjoint but close together in the sense of adjacent, like the great circle rings in a Hopf fiber bundle. 

To summarize, the 12-point (2-hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell). We shall now be able to see how Fuller's perfect hexads combine with Todd's perfect pentads to form the perfect heptads of the 11-cell and 137-cell.  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=alias the hemi-cuboctahedron}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces. (The ''heptad'' arrives at the party, anonymously; they are our distinguished anti-celebrity, who most of the other guests have never even heard of.) The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. This 12-point (truncated tetrahedron) is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 100 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp: Truncated tetrahedron|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp: Truncated tetrahedron|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}} Of course Moxness has been there, seen that, already with his graphics software;{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} see his parameterized illustration of it exactly as it occurs in the 11-cell, below in his bestiary of A3, B3, and H3 Archimedian and Catalan polyhedra in the 120-cell.]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:Archimedean_and_Catalan_solid_hulls_with_their_Weyl_orbit_definitions.svg|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of the 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. In this catalog of Archimedean and Catalan polyhedra sighted in the 120-cell by Moxness's software,{{Sfn|Moxness: Archimedean and Catalan hulls|2023|loc=Hull #1 Archimedean Name A3 110 Truncated Tetrahedron A (upper left)}} it is the first entry (upper left), Moxness's Hull # = 1 with 12 vertices of 3D norm {{radic|11}}/2. On a {{radic|2}}-radius 3-sphere, 5 of these with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell. 20 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges are cells in the [[#The rings of the 11-cells|cell rings of the quasi-regular 11-cell]], which also has 12 regular 5-cells as cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

...

Both kinds of cells in the 11-cell have expressions in two different dimensions (on the 2-sphere and 3-sphere), because they are all centered on the center of the 3-sphere, unlike the cells of most 4-polytopes, which are centered on various off-center points within the 4-polytope's interior. 

One kind of 11-cell cell appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (semi-regular truncated cuboctahedron), a.k.a. the truncated tetrahedron. The other kind of cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now, except in the 120-cell. But this is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron, and a regular tetrahedron behind each face. The 11-cell is thus a honeycomb of mixed regular tetrahedra and truncated cuboctahedra (a.k.a. truncated tetrahedra) that meet at their shared 5-cell faces. 

Both the 5-cells (decahedra) and the truncated cuboctahedra (truncated tetrahedra) are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks in their 4-polytopes (off-center).{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} Therefore all the 11-cell's cells are concentric, which indeed makes a different kind of honeycomb, an inside-out cellular honeycomb with all the cells piled on top of each other overlapping in 4-space. They do not overlap in 3-sphere space however; their 3-volumes (but not their 4-contents) are disjoint. They make a proper 4D honeycomb like the cells of any 4-polytope, with volumetrically disjoint 3D cells that meet each other in the proper way at their shared 2D faces.

An 11-cell contains 6 completely disjoint 5-point (1-pentad 5-cell decahedron cells), and 5 volumetrically disjoint 12-point (2-hexad truncated cuboctahedron cells), 11 volumetrically disjoint cells altogether. Of course it is supposed to have 11 identical 11-point (elevenad hemi-icosahedral cells), and that is also true in its abstract sense, which is not the same sense as disjoint cellular containment: each pentad or hexad cell is inscribed in one of 11 larger 60-point (Moxness's rhombicosadodecahedron cells). The abstract pentads and hexads are only hemi-polyhedra, so the 11-cell's pentad cells are 12 (out of 120) disjoint instances of a pentad 5-point, and its hexad cells are 10 (out of 100) disjoint instances of a hexad 6-point, 1/10th of the 120-cell's cellular volume in both cases. The 11-cell's hemi-icosahedral cells are 11 (out of 120) non-disjoint instances of an elevenad 60-point. Each 11-cell shares each of its hexad cells with 9 other 11-cells, and each of its 11 vertices (out of 600) occurs in 9 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.|name=Tetrahedral vertex figures}}

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

...insert image thumb of 11-cell...

...Construct the 11-cell from the heptad....

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...Four concentric 4-polytopes may be mutually orthogonal. In any set of 4 mutually orthogonal regular 5-cells, one pair of faces in each orthogonal pair are completely orthogonal to each other. Thus three edges in one 5-cell in an orthogonal pair are opposite (completely orthogonal) to a face of the orthogonal 5-cell. .... A 5-cell has 5 vertices in its abstract hemi-icosahedral cells, but only 4 vertices in one concrete icosahedral cell; the 5th lies in an icosahedral cell of another 11-cell (not in another icosahedral cell of the same 11-cell).

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

....

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 11-cell ===

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon.... (wrong, 11-cell's Hopf map is the truncated icosahedron)

....

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk: Truncated icosahedron|2018}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

...

Since the rhombicosadodecahedron has 120 5-cell edges, the 120-cell has 120 completely disjoint 5-cells, and the rhombicosadodecahedron shares each vertex with 11 other rhombicosadodecahedra, how a rhombicosadodecahedron is bound to its neighbor 5-cells is constrained. A further constraint is that the abstract hemi-icosahedron has only 15 5-cell edges, so the 120 actual 5-cell edges of the rhombicosadodecahedron are conflated 8-to-1 in the hemi-icosahedron. To be conflated by contraction into a single edge, those 8 edges must lie parallel to each other in the 120-cell; either actually parallel in the same plane, or Clifford parallel in separate planes, as two edges in completely orthogonal planes may be parallel and orthogonal at the same time. Each 5-cell inscribed in the 120-cell contains 5 such tetrahedra occur in parallel pairs in 4 orthogonal planes (none of them completely orthogonal planes). 

.... 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
Each conflated vertex is the 10 vertices of two orthogonal regular 5-cells...
The 60-point contains 6 pairs of these orthogonal regular 5-cell cells...
...its 20 hexad cells are ..., a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere.

== The regular 137-cell 4-polytope ==

(description originally written for the 11-cell, which applies only to a fibration of 11 11-cells)

It is quasi-regular with two kinds of cells, the 12-point (regular icosahedron) and the 5-point (regular 5-cell), which have just one kind of regular face between them. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of regular facets (it is semi-regular with regular facets), it also means that its facets have expressions of two different dimensionalities because they are concentric in the 3-sphere. The 11-cell has two kinds of regular cells. One appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (regular icosahedron), but since it is embedded at the center of the 3-sphere (unlike the cells of most 4-polytopes) it also acts as an inscribed 4-polytope.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} The other cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now except in the 120-cell. This is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron. The 11-cell is a honeycomb of mixed icosahedron and half-icosahedron cells. Like the icosahedra, the decahedra are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks, off-center in their 4-polytopes. All the 11-cell's cells are concentric, which indeed makes an unusual honeycomb of cells, an inside-out honeycomb. It is a proper 4D honeycomb however, with volumetrically disjoint 3D cells that meet each other in the proper way at shared 2D faces. The embedded 12-point (regular icosahedron) cell is an icosahedral pyramid (as a 4-polytope), which is filled by 20 4-point (irregular tetrahedra) that meet at its center; the center point of the 11-cell acts as a 12th vertex, just as the center point of the 24-cell acts as a 25th vertex in some honeycombs. The 5-point (regular 5-cell) cell contains five 4-point (regular tetrahedra) which meet the 12-point (regular icosahedra)'s irregular tetrahedral cells face-to-face, so the 11-cell is quasi-regular in the usual sense of having two varieties of regular 3-polytope cells, but additionally both varieties of of its cells are bi-dimensional: another clue that the 11-cell has something to do with dimensional analogy

....

The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

....

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur to make an 11-cell. 11 of them occur in a 137-cell, all 120 of them occur in the 120-cell, and they occur nowhere else.

How can this regular 4-polytope possibly exist? By Schläfli's criterion {3, 3, ..} exists, but not as a finite polytope.{{Sfn|Coxeter|1973|loc=§7.7 Schläfli's criterion}} Coxeter found {3, 5, 3} as a finite hyperbolic honeycomb, but not in Euclidean 4-space.{{Sfn|Coxeter|1968|title=The Beauty of Geometry: Twelve Geometric Essays|loc=''Regular Honeycombs in Hyperbolic Space''|ps=; &quot;The honeycombs considered by the above authors [Schlegel, Schläfli, Klein, Sommerville] have finite cells and finite vertex figures. ... A further extension allows a cell or vertex figure to be a ''star''-polytope, so that the honeycomb covers the space several times. .... We shall find that there are four regular star-honeycombs in hyperbolic 4-space, as well as two infinite families of them in the hyperbolic plane.&quot;}} We find that {3, 3, ..} is a central expression of the &lt;math&gt;H_4&lt;/math&gt; root system, being the regular convex hull in 4 dimensions of 137 points, 96 pentads and 600 hexads. The minimal expression of &lt;math&gt;H_4&lt;/math&gt; is the regular convex hull {3, 3, 5} of 120 points, 600 tetrads and 75 hexads, and its maximal expression is the regular convex hull {5, 3, 3} of 600 points, 120 pentads and 675 hexads. {3, 3, ..} is a third regular &lt;math&gt;H_4&lt;/math&gt; polytope which fits between them. The properties and metrics of this regular sequence are presented in tabular form below in ''[[#Build with the blocks|§Build with the blocks]]''.

{|class=&quot;wikitable&quot;
!colspan=3|Regular &lt;math&gt;H_4&lt;/math&gt; polytopes
|-
|[[File:600-cell.gif|200px]]
|[[File:5-cell.gif|200px]]
|[[File:120-cell.gif|200px]]
|-
!120-point 600-cell
!137-point 137-cell
!600-point 120-cell
|}

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 137-cell ===
....

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=Apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|2022|loc=''[[Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]''}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie: On Fuller's use of language|2024|loc=''[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry'']]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center, 1948-1949|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with their mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the gender neutral shad can open their six beaks all the way, until they become flat squares and they becomes a cuboctahedron, or they can shut them all tight like a turtle retracting into their octahedron shell. The six mouths always move in unison. This is [[Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do their ''really'' odd trick -- where they flip their 6 jaws 90 degrees in their 6 faces and shut their 6 beaks on the opposite axis of their squares than the one they opened them on -- or whether they will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than their normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make their edge length exactly the same size as their radius, when they open their mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing their mouths in spherical synchrony, their 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron (the hexad), then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=''[[W:SO(4)|SO(4)]]''}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=Isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its orthogonal hexad's planes (wz, wx, wy).{{Efn|name=Six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of (expanded octahederon 16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh &amp; Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== Legendre's 12-point hexad binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a 12-point double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's perfectly realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, this 2-hexad expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre hexad) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction.{{Efn|name=Concentric polyhedra}} In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== ''n''-simplexes ===
....

The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== ''n''-orthoplexes ===
....
[[File:Great 5-cell √5 digons rectangle.png|thumb|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} ''regular'' great hexagons. These are the same 200 dodecagon central planes illustrated above, which also contain 6 120-cell edges and two opposing ''irregular'' great hexagons (truncated {{radic|6}} triangles). The 1200 {{radic|5}} 5-cell face triangles occupy 1200 other, non-central planes.]]

...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's Hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[16--cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[16--cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=Concentric polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[16--cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== ''n''-taliesins ===
The 4-simplex pentad building block combines with the 4-orthoplex hexad building block to make the 4-taliesin heptad building block. The heptad is the essential building block of the ''taliesin'' family of polytopes. The 11-cell and 57-cell hemi-polytopes and the 137-cell and 120-cell regular 4-polytopes are nuclear members of this trans-dimensional family. The 24-cell and 600-cell are extended members of it, because they also contain instances of the heptad building block.

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its spherical symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* [[W:Taliesin Myrddin Namkai-Meche|Taliesin Myrddin Namkai-Meche]], a young Reed College scholar who was stabbed to death by a crazy man in 2017 on a Portland, Oregon light rail train, when he rose to the nonviolent defense of two black teenagers who were being violently assaulted in racist and anti-Muslim language by a right-wing extremist.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in orthogonal [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].
* '''Taliesan polytope''', an (''n+1)''-polytope whose ''n''-polytope dimensional analogue inscribed concentrically in the ''n''-sphere forms a convex honeycomb of itself which is the (''n+1)''-polytope.{{Efn|name=Embedding point}}
* The taliesan polytope's honeycomb, a Hopf fibration of its great circles.
* The ''n''-dimensional family of dimensionally analogous polytopes related to a taliesan polytope by expansion and contraction operations.
* The characteristic [[w:Orthoscheme|orthoscheme]] of a teliesan polytope, an irregular simplex that is the complete gene for the entire expression of trans-dimensional symmetry relationships among the polytopes of the ''n''-dimensional family, as well as the complete gene for the teliesan polytope's rotational symmetry.{{Efn|name=Characteristic orthoscheme}}

=== ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== ''n''-equilaterals ===
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed at four different [[#n-leviathon|chord]] lengths from each other. In the {{radic|2}}-radius polytopes we have using as examples in this paper they are {{radic|2}}, {{radic|4}}, {{radic|6}} and {{radic|8}}, but in a ''unit-radius'' polytope they are: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}.

Each vertex is joined to 8 others by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}. The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== ''n''-pentagonals ===

[[File:600-cell.gif|thumb|The 120-point (600-cell), &lt;math&gt;H^4&lt;/math&gt;.]]
...

[[File:600-cell vertex geometry.png|thumb|Vertex geometry of the 600-cell, showing the 5 regular great circle polygons and the 8 vertex-to-vertex chord lengths with angles of arc. The golden ratio{{Efn|name=golden chords|group=}} governs the fractional roots of every other chord,{{Efn|name=fractional root chords}} and the radial golden triangles which meet at the center.|alt=|400x400px]]
The 120 vertices of the 600-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏&lt;sup&gt;−1&lt;/sup&gt;) beginning with a vertex; see column ''a''}} at eight different [[#n-leviathon|chord]] lengths from each other. These edges and chords of the 600-cell are simply the edges and chords of its five great circle polygons. In ascending order of length in a ''unit-radius'' polytope (not in a {{radic|2}} radius polytopes like our usual examples), they are {{radic|0.𝚫}}, {{radic|1}}, {{radic|1.𝚫}}, {{radic|2}}, {{radic|2.𝚽}}, {{radic|3}}, {{radic|3.𝚽}}, and {{radic|4}}.{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=Decimal fractions of square roots|group=}}

Notice that the four [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#''n-''equilaterals|hypercubic chords]] of the 24-cell ({{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|4}}) alternate with the four new chords of the 600-cell's additional great circles, the decagons and pentagons. The new ''golden chord'' lengths are necessarily square roots of fractions, but very special fractions related to the golden ratio{{Efn|1=The fractional-root ''golden chords'' are irrational fractions that are functions of {{radic|5}}. They exemplify that the [[W:golden ratio|golden ratio]] &lt;big&gt;𝜙&lt;/big&gt; {{=}} {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618 is a circle ratio related to &lt;big&gt;𝜋&lt;/big&gt;:&lt;br&gt;
: {{sfrac|𝜋|5}} = arccos (𝜙/2)
is one decagon edge, the 𝚽 = {{radic|0.𝚫}} = {{radic|0.382~}} ≈ 0.618 chord.
Reciprocally, in this function discovered by Robert Everest expressing &lt;big&gt;𝜙&lt;/big&gt; as a function of &lt;big&gt;𝜋&lt;/big&gt; and the numbers 1, 2, 3 and 5 of the Fibonacci series:&lt;br&gt;
: &lt;big&gt;𝜙&lt;/big&gt; = 1 – 2 cos ({{sfrac|3𝜋|5}})
{{sfrac|3𝜋|5}} is the arc length of the &lt;big&gt;𝜙&lt;/big&gt; = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618 chord.|name=golden chords|group=}} including the two [[W:golden section|golden section]]s of {{radic|5}}, as shown in the diagram.{{Efn|The 600-cell edges are decagon edges of length {{radic|0.𝚫}}, which is 𝚽, the ''smaller'' golden section of {{radic|5}}; the edges are in the inverse [[W:golden ratio|golden ratio]] {{sfrac|1|φ}} to the {{radic|1}} hexagon chords (the 24-cell edges).
The other fractional-root chords exhibit golden relationships as well. The chord of length {{radic|1.𝚫}} is a pentagon edge.
The next fractional-root chord is a decagon diagonal of length {{radic|2.𝚽}} which is &lt;big&gt;𝜙&lt;/big&gt;, the ''larger'' golden section of {{radic|5}}; it is in the golden ratio{{Efn|name=golden chords|group=}} to the {{radic|1}} chord (and the radius).{{Efn|Notice in the diagram how the &lt;big&gt;𝜙&lt;/big&gt; chord (the ''larger'' golden section) sums with the adjacent 𝚽 edge (the ''smaller'' golden section) to {{radic|5}}, as if together they were a {{radic|5}} chord bent to fit inside the {{radic|4}} diameter.}}
The last fractional-root chord is the pentagon diagonal of length {{radic|3.𝚽}}.
The [[W:Pentagon#Side length is given|diagonal of a regular pentagon]] is always in the golden ratio to its edge, and indeed &lt;big&gt;𝜙&lt;/big&gt;{{radic|1.𝚫}} is {{radic|3.𝚽}}.|name=fractional root chords|group=}}

....


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== ''n''-leviathon ===
...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of its &lt;math&gt;10^2&lt;/math&gt; disjoint instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction, before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]].{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}} .... It is only because she did not have the 11-cell to start with that she did not discover the regular 137-cell {3, 3, ..} by her method.

Between Alicia Boole Stott and her identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and her identification of symmetry groups with conservation laws (in her 1st theorem), we owe the deepest mathematics of geometry and physics originally to two pioneering women, colleagues of their contemporaries the greatest men of the academy, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just an [[W:abstract polytope|abstract 4-polytope]], not just a singleton convex 4-polytope, and not just a honeycomb.{{Sfn|Coxeter|1970|loc=''Twisted Honeycombs''}} The 11-point (11-cell) has a concrete ...regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}&lt;nowiki/&gt;} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, 3, 3} as the 137-point (137-cell).

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 11-cell, 137-cell and 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=''Clifford Spinors and Root System Induction: H4 and the Grand Antiprism''}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=Apology}}

== Build with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Duveneck|1978|loc=Carl Jung, quoted in ''Life on Two Levels''|p=ii|ps=.{{Sfn|Jung|1961|ps=: &quot;The best of truths is of no use unless it has become one's most personal inner experience. It is the duty of everyone who takes a solitary path to share with society what he finds on his journey of discovery.&quot;}}}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1954|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius={{radic|2}}}}

== Acknowledgements ==
...

== Notes ==
{{Regular convex 4-polytopes Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{Refend}}</text>
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    <revision>
      <id>2692241</id>
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      <comment>/* 5-cells and hemi-icosahedra in the 11-cell */</comment>
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      <text bytes="208646" sha1="9g91q7u92nhahen2m75oumeuthj3rk3" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|June 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a regular 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the &lt;s&gt;120-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 5, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 11 [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] cells. Eleven 11-cells (plural) are the 137-cell, the &lt;s&gt;600-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 3, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 137 [[W:Rhombic triacontahedron|triacontahedron]] cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=''Regularity of Graphs, Complexes and Designs''}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=''A Symmetrical Arrangement of Eleven Hemi-Icosahedra''}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]]
The 4-dimensional regular polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:Simplex|simplex]] (called the [[5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:Orthoplex|orthoplex]] (called the [[16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular convex 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the [[w:Tesseract|16-point (8-cell) tesseract]], the 4-cubic, which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found within any of the other regular convex 4-polytopes, except in the largest and most complex one, the [[120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). However, we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the 16-cell is their very starting point, and the most frequently used tool in the box.

== 5-cells and hemi-icosahedra in the 11-cell ==
[[File:Symmetrical_5-set_Venn_diagram.svg|thumb|The 5-point (10-face) regular 5-cell (the regular 4-simplex). Grünbaum's rotationally symmetrical 5-set Venn diagram{{Sfn|Grünbaum|1975|loc=''Rotationally symmetrical 5-set Venn diagram'', Fig 1 (e)|ps=; partitions the individual elements of the 5-cell.}} is an illustration of the 5-cell labeling each of its &lt;math&gt;2^5&lt;/math&gt; elements.{{Sfn|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled; image includes the Python code to render it, optimising for maximum area of the smallest regions.}}]]
[[File:Hemi-icosahedron.png|thumb|The 6-point (10 face) [[W:hemi-icosahedron|hemi-icosahedron]], an abstraction of the regular icosahedron, has half as many faces, edges and vertices. Each element of the abstract polyhedron represents two real elements found in different places in the 11-cell's realization.{{Sfn|Ruen: hemi-icosahedron|2007}}]]
The most apparent relationship between the pentad 5-cell and the hexad hemi-icosahedron is that they both have 10 triangular faces. When we see a facet congruence between a 4-polytope and a 3-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 6-point 5-simplex. Another hexad is the 6-point 3-orthoplex; thus the hemi-icosahedron is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no chord congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals, 5-cell and 16-cell, who are famously stand-offish with each other. Interesting!

The cell of the 11-cell is an abstract hexad hemi-icosahedron with 5 central planes, most handsomely illustrated by Séquin.{{Sfn|Séquin &amp; Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing rings of polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The 11 hemi-icosahedral cells have 10 triangle faces each; the 5-cell's 5 tetrahedral cells have 10 faces and 10 edges all together. Cells meet face-to-face, so the only way 11-cells and 5-cells can share the same faces is if two hemi-icosahedral cells and two tetrahedral cells all meet at each face. Then 3 of each 5-cell's 5 vertices are a hemi-icosahedron face, and its other 2 vertices must be some 11-cell edge lying opposite the face. Coxeter determined that the 11-cell does indeed have an edge opposite each face, but it does not belong to the same hemi-icosahedral cell as its opposing face. He found that the 10 edges opposite each hemi-icosahedron's faces are the 10 edges of a single 5-cell, which does not share any vertices, edges or faces with the hemi-icosahedron. Of course, in the 11-point 11-cell there is just one such 5-point 5-cell that is completely disjoint from each 6-point hemi-icosahedron.{{Sfn|Coxeter|1984|p=110|loc=§6. The Petrie polygon [of the 11-cell]|ps=; &quot;We may reasonably call this edge and face ''opposites''. It is easy to find the face opposite to a given edge by looking at the faces to which a given edge belongs. ... Conversely, given a face, we can find the opposite edge by seeing which vertices belong to neither of the hemi-icosahedra which share that face. The ten edges opposite to the ten faces of one hemi-icosahedron are the edges of the complementary &lt;math&gt;a_4&lt;/math&gt; [4-simplex], that is, the joins of all pairs of the five vertices [of the 11-cell] not belonging to the given hemi-icosahedron.&quot;}}

There are 11 disjoint 5-cell 4-polytopes inscribed in each 11-cell, which also contains 11 hemi-icosahedral cells, 55 faces, 55 edges and 11 vertices. The real 11-cell is harder to see than the abstract 11-cell that represents it, because the real hemi-icosahedron that the abstract hemi-icosahedron represents is hard to see, but seeing these objects will be easier once we have identified the real hemi-icosahedron, and seen exactly where the 11-cell's real elements reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each hemi-icosahedron's 10 faces come from 10 disjoint 5-cells, and there is just one 5-cell with which it does not share a face.{{Efn|As Coxeter observes (in the previous citation), that unrepresented 5-point (5-cell) is the other 5 vertices of the 11-point (11-cell) that are not vertices of this 6-point (hemi-icosahedron), the hemi-icosahedron's disjoint complement.}} Each 5-cell has 10 faces from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share a face.

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent 5-cell faces do not actually touch! The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in two different planes which are completely orthogonal to each other. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must be sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness: 8 concentric hulls|2022|loc=Hull #8 (lower right)|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell of the 11-cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Compounds in the 120-cell|§Compounds in the 120-cell]]'' that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software which uses Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s to render the polyhedra which are found in the interior of ''n''-dimensional polytopes.{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the tetrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is a real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (Hull #8) is a concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this {12} irregular great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing {6} irregular great hexagons (truncated triangles) of alternating 120-cell edges and 5-cell edges. The great dodecagon is a great circle of the hemi-icosahedral cell, and also a great circle of the 120-cell, because unlike the cells of most 4-polytopes, the 11-cell's cells are embedded in 4-space at the exact center of the 4-polytope, making them inscribed 4-polytopes themselves.{{Efn|name=Embedding point}} There are 25 great dodecagons and 120 5-cell edges in Moxness's 60-point (Hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (Hull #8) polyhedron altogether.

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point (Hull #8) is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] similar to the one from the catalog shown here,{{Sfn|Piesk: Rhombicosidodecahedron|2018}} but a slightly shallower truncation with smaller red pentagons (120-cell faces) and narrower rhombs. Rhombicosidodecahedra are made by truncating the [[W:Rhombic triacontahedron|rhombic triacontahedron]], which is the 30-sided polyhedron with only one kind of face, the dual of the 30-point icosidodecahedron. The 120-cell contains 120 of these 11-cell cells.]]
Moxness's 60-point (Hull #8) is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (Hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|The 5-point (regular 5-cell) can be another abstraction of Moxness's 60-point (Hull #8), 12-vertices-into-1, but it is not strictly the result of a contraction operation as described by Boole Stott, since she did not apply her expansion and contraction operations to uniform polytopes with more than one edge length.}} In fact Moxness was not the first to find a rhombicosidodecahedron in the 120-cell; Alicia Boole Stott was. It is the 6th section of {5,3,3} beginning with a cell, which Boole Stott identified as the ''e&lt;sub&gt;2&lt;/sub&gt; expansion'' of the icosahedron (or equivalently of its dual polyhedron the dodecahedron).{{Sfn|Boole Stott|1910|loc=§Examples of the e&lt;sub&gt;2&lt;/sub&gt; expansion|p=7}} But that rhombicosidodecahedron identified by Boole Stott is not Moxness's Hull #8: rather, it is the semi-regular Archimedean solid, with a single edge length and square faces. Moxness's Hull #8 with its two distinct edge lengths and rectangular faces is Coxeter's 8th section of {5,3,3} beginning with a cell, which is missing from the sections illustrated by Boole Stott.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than tetrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. The 120-cell contains 120 of Moxness's 60-point (11-cell cells). They are non-disjoint, with 12 60-points sharing each vertex.

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (Hull #8) is the real hemi-icosahedron cell, with 120 5-cell edges in it as chords lurking under its surface. The rhombicosadodecahedron has enough of those 5-cell edges in its interior to make 20 disjoint tetrahedral cells which belong to 20 distinct 5-cells.

The 600-point (120-cell) may be constructed as a semi-regular honeycomb of two kinds of cells, the 120 non-disjoint 60-point (Moxness's rhombicosadodecahedra) and their neighbors the 120 disjoint 5-point (5-cells), cell-bonded together by shared tetrahedral cells of edge-length {{radic|5}}. Each rhombicosadodecahedron is bonded to 20 islanded 5-cells, sharing with each a 5-cell tetrahedron (4 of its interior 5-cell faces, and 6 of its interior 5-cell edges). The rhombicosadodecahedron is also bonded directly to 10 other rhombicosadodecahedra (since each 11-cell has 11 rhombicosadodecahedron cells), again not by sharing one of its exterior faces, but this time by sharing an interior face, a {{radic|6}} triangle which lies in a central plane.{{Efn|The {{radic|6}} chord is another hidden chord lurking beneath the 60-point rhombicosadodecahedron's surface, like the {{radic|5}} 5-cell edge. Both these hidden chords lie on the same great circle with the 120-cell edges (the pentagon face edges). Three {{radic|6}} chords make a triangle which lies in a central plane. This {{radic|6}} great triangle must not be confused with the {{radic|5}} face triangle of the 5-cell, which does not lie in a central plane.}} The 120 rhombicosadodecahedra are also bonded directly together at ''all'' their exterior faces: at their pentagon faces (which are 120-cell faces of dodecahedral cells); at their small triangular faces (which are 600-cell faces of small tetrahedral cells, 10 of which are inscribed in each dodecahedral cell); and at their rectangular faces (which are faces of 10-point [[w:Pentagonal_prism|pentagonal prism]] cells, where the pentagonal prism is a column inscribed in a dodecahedral cell, joining its opposite pentagon faces). The 120 rhombicosadodecahedra are bonded directly to each other at their own exterior faces; no dodecahedra, small tetrahedra, or pentagonal prisms lie between them, and those objects are not cells of this honeycomb; we mention them here only to help locate the rhombicosadodecahedra within the 120-cell. The only objects which fill holes between the rhombicosadodecahedra are the 120 islanded 5-cells.{{Efn|This honeycomb of the 120-cell resembles the [[600-cell#Icosahedra|honeycomb of the 600-cell consisting of 24 face-bonded regular icosahedra]] with 24 clusters of 5 tetrahedra filling the holes between them. In the 120-cell there are 5 times as many of everything as in the 600-cell, so there are 120 (rhombicosadodecahedron) icosahedral cells with 600 tetrahedra filling the holes between them.}}

It is possible for rhombicosadodecahedra to meet each other at more than one kind of exterior face because they are not meeting there as cells of the same 11-cell; they belong to different 11-cells. The 11 rhombicosadodecahedra in the same 11-cell do not meet at their own exterior faces, they meet at their interior {{radic|6}} triangle faces. As a cellular 4-polytope, the 11-cell is inside-out. Its cells meet each other at interior planes, sharing interior {{radic|6}} triangle faces, instead of in the usual way that 4-polytopes' cells share an exterior face. It is a deeper kind of intertwining to be sure, but the 11 rhombicosadodecahedra in the same 11-cell are still volumetrically disjoint cells, and the 11-cell is a perfectly real 4-polytope with real cells. It is a semi-regular 4-polytope, because it has two different kinds of cells: 11 rhombicosadodecahedra and 11 5-cells. The 11 5-cells are islanded (not bound to each other), and the 11 rhombicosadodecahedra are bound both to 5-cells (at {{radic|5}} 5-cell faces) and to each other (at {{radic|6}} central triangle faces). In the section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#What's in the box|§What's in the box]]'', below, we will look closely at the relation between the {{radic|5}} chord and the {{radic|6}} chord, which is the relation between the pentad and the hexad, and at the general phenomenon of hemi-polytopes which meet each other at a central plane. Then in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Cell rings of the 11-cells|§Cell rings of the 11-cells]]'' we will look at the two-cell cellular structure of the 11-cell in more detail.

Considered as a single complex of overlapping cells rather than as distinct honeycombs of volumetrically disjoint cells, the 120-cell contains 60-point (rhombicosadodecahedral cells), islanded 5-point (5-cells), 20-point (dodecahedral cells), and 10-point ([[w:Pentagonal_prism|pentagonal prism]] cells). The 60-points meet each other in pairs at their pentagon faces, meet regular dodecahedra at those pentagon faces also, meet each other in quads at their 600-cell faces where they share an interior 4-point (600-cell tetrahedral cell) and its four faces (just as they share a larger interior 5-cell tetrahedron and its faces), and meet each other and the 10-point (pentagonal prisms) at their rectangle faces. We might try to imagine such a complex with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with parts truncated from 30 neighboring truncated dodecahedra pentagonal prisms). There are 120 of each of these 3 kinds of dodecahedron. 

Moxness's 60-point (rhombicosadodecahedron cell) is a central object of the 120-cell, and contains many wonders. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra by expansion and contraction operations, as Alicia Boole Stott discovered before 1910,{{Sfn|Polo-Blanco: ''Theory and history of geometric models of Alicia Boole Stott''|2007|loc=§5.3.2 1910 paper on semi-regular polytopes|pp=152-158|ps=; summarizes Boole Stott's method and results from {{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}}, including her tables of expansion-contraction dimensional analogies and a few of her illustrations.}} and those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, rediscovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter &amp; Petrie|1938|p=4|loc=''[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]''}}|name=Concentric polyhedra}} Before we move on from the 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice one more thing about the hidden {{radic|5}} chords lurking below its surface. The 12 little pentagon faces are spanned by sets of 6 {{radic|5}} chords (5-cell edges). The 6 chords of each set are disjoint (they don't touch or form 5-cell faces), but they are symmetrically arranged in 3 parallel pairs, {{radic|3}} apart, which lie in 3 orthogonal central planes. It turns out they are the 6 reflex edges of a 12-point non-convex polyhedron called the [[w:Jessen's_icosahedron|Jessen's icosahedron]], inscribed in the 60-point cell. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again and work our way outwards, before eventually in ''[[#The perfection of Fuller's cyclic design|§The perfection of Fuller's cyclic design]]'' we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cell)s ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks, showing its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Pentahemidemicube.png|120px]]
|[[File:Tetrahemihexahedron_rotation.gif|120px]]
|[[File:Pentahemicosahedron.png|120px]]
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts needed to build the 120-cell (the picture on the cover of the box), and everything else that's invisible in that picture which fits inside the 120-cell: all the astonishing regular 4-polytopes. The pentad building block is half a pair of orthogonal 5-point (regular 5-cells), the hexad building block is half an 8-point (16-cell), and the heptad building block is a quarter of an 11-point (11-cell). Everything can be built from these blocks by embedding them at the center of the 3-sphere, inscribing them as 4-polytopes to make compounds as with the 16-cell building block, rather than by assembling them face-to-face as 3-polytopes to make honeycombs in which they are cells or vertex figures.{{Efn|name=Embedding point}}

The pentad building block is the 5-point '''pentahemidemicube''', an abstract polyhedron with 10 equilateral triangle faces, 10 {{radic|5}} edges, and 5 vertices.{{Sfn|Christie: Pentahemidemicube|2024|loc=File:Pentahemidemicube.png|ps=; &quot;The 5-point ''pentahemidemicube'' is an abstract polyhedron with 10 equilateral triangle faces, 10 edges, and 5 vertices. It has a real presentation as the regular 5-cell.&quot;}} It has a real presentation as the 5-point (regular 5-cell 4-polytope), a regular tetrahedron expanded by addition of a fifth vertex equidistant from all the others in a fourth dimension. The pentad block is a tetrahedral pyramid (a 5-cell) inscribed in a cube: a tetrahedron inscribed in a cube (a demicube) expanded by addition of a fifth vertex, one of the cube's other 4 vertices. Each pentad (5-cell) building block lies in the 120-cell orthogonal to three other pentad (5-cell) building blocks. They are four completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges. Each regular 5-cell in the 120-cell is a pentad (pentahemidemicube) building block, in one of 30 orthogonal sets of 4. There are 120 pentad building blocks in the box.

Every vertex of the 16-cell, and therefore every vertex of the 8-cell, 24-cell, 600-cell and 120-cell, is surrounded by a 6-point (octahedral vertex figure). The hexad building block is that 6-point octahedron embedded at the center of the 4-polytope instead of at a vertex, that is, inscribed in the 3-sphere as a 4-polytope, the 6-point '''[[w:Tetrahemihexahedron|tetrahemihexahedron]]''' or '''hemi-cuboctahedron'''.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; &quot;The tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U&lt;sub&gt;4&lt;/sub&gt;. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices.&quot;}} The hexads have 4 red triangle 5-cell faces, bounded by 12 {{radic|5}} edges. Their 3 yellow square faces lie in central planes, because the building block is an abstract 4-polytope. The hexad is an irregular abstract polyhedron that also has 6 {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

Two orthogonal 5-point (pentad) building blocks intersect in a 4-point (inscribed tetrahedron) to form a 6-point (hexad) building block. Two orthogonal 6-point (hexad) building blocks intersect in a 4-point (central square) to form an 8-point (16-cell) building block, the most usable building block, from which everything larger can be built by compounding.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 120 pentads to pair with a completely orthogonal hexad, and although there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed) they only result in 120 pairs two different ways, so there are 120 11-cells. The 11-cell is 6 pentads' and 5 hexads' smallest joint compound, and the 120-cell is their largest, 120 pentads and 100 hexads.

The heptad{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''|pages=22–31}} building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|4}} edges and 9 {{radic|5}} edges), and 7 vertices.{{Sfn|Christie: Pentahemicosahedron|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;}} It is a compound of a 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]) and a 6-point (hexad [[W:hemi-icosahedron|hemi-icosahedron]]) intersecting in a 4-point (tetrahedron), that is, of three orthogonal 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]&lt;nowiki/&gt;s) intersecting in their common 4-point (tetrahedron). It is also the abstract cube contracted by removal of one vertex, which reduces one of the cube's two inscribed tetrahedra (demicubes) to a single remaining triangular face. As with all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Four &lt;small&gt;&lt;math&gt;(11-7)&lt;/math&gt;&lt;/small&gt; orthogonal 7-point (pentahemicosahedron) blocks can be joined at a common Petrie polygon (a skew hexagon which contains their orthogonal 6-edge paths) to construct a real 11-point ([[W:11-cell|11-cell]] semi-regular 4-polytope), which magically contains six &lt;small&gt;&lt;math&gt;(11-5)&lt;/math&gt;&lt;/small&gt; 5-point (1-pentad regular decahedra) as cells and five &lt;small&gt;&lt;math&gt;(11-6)&lt;/math&gt;&lt;/small&gt; 12-point (2-hexad semi-regular cuboctahedra) as cells. There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like [[W:Euclid's postulates|Euclid's postulates]], and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as honeycombs of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles. It possesses the complete symmetry of the polytope without any redundancy, because it contains one of each of the polytope's characteristic root elements.{{Efn|A [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be partitioned radially by its planes of symmetry (Coxeter's &quot;reflecting circles&quot;) into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated rotationally to generate its regular 4-polytope because it is the complete [[W:gene|gene]] for it, containing all of its elements and capturing all of its symmetry without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring that triggers the reflections.|name=Characteristic orthoscheme}} It is the gene for the polytope, which can be replicated to construct the polytope.

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its &lt;math&gt;A_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the regular compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;small&gt;&lt;math&gt;120\times 120 = 14400&lt;/math&gt;&lt;/small&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[16--cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its &lt;math&gt;B_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 384. The 120-cell is the convex hull of the regular compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 24-point (24-cell) is subdivided into 1152 instances of its [[24-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;F_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 1152. The 120-cell is the convex hull of the regular compound of 25 disjoint 24-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;25\times 1152 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has three equivalent origins in distinct root systems, though most are obviously constructed from one root, and not so obviously from the others. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything larger than the pentad can be constructed by compounding hexads, and we shall see that everything as large as a 24-cell has a construction from heptads which are a product of a pentad and a hexad. Surely every uniform polytope can be constructed by some function of expansions and contractions from any of three root systems, however indirect the recipe. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes, related by an equals sign between them, then give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics by [[W:Noether's theorem|Noether's theorem]].

=== Pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 octahedral cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell contains the 600 octahedral cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It also contains the 600 cubical cells of its 75 8-cells. It is a 600-cell in three ways, although the octahedra and cubes are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell and the 8-cell, respectively.|name=600 cells three ways}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (1-pentad 5-cell), out of the 8-point (2-hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (1-pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope that contains 5-point (1-pentad 5-cell) building blocks beside the largest one, it just isn't a ''regular'' 4-polytope. It is semi-regular with two kinds of cells, the 4-point (regular tetrahedron) and the 12-point (truncated cuboctahedron), which have just one kind of regular face between them, a 5-cell face. That semi-regular 4-polytope is the 11-cell.

We said earlier that the 11-cell had 11 disjoint 5-cells and 11 hemi-icosahedral cells in it, and that its construction from them might be hard to see, but now we are going to account numerically, at least, for how it can have 6 pentad cells plus 5 hexad cells magically adding to 11 of the same kind of elevenad cell. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, 11. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (5-cell regular decahedron) cells, and the 12 6-point (hexad) building blocks into 100 12-point (regular icosahedron) cells, and then magically adding one whole 5-point (5-cell regular decahedron) cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 12-point (regular icosahedron) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such (cell bundles of 6 + 5 = 11 cells), with each 11-cell bundle containing 11 of both kinds of cell. Very, very strange!{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden'': §II To Add is to Multiply|p=38|ps=; the sequence of regular polygons with harmonious proportions beginning {5}, {7}, {9}, {11} as a family of natural numbers; &quot;The numbers in this optimal family have remarkable properties [that allow] similar figures to be arranged easily or allow a figure to be dissected into a set of similar figures. This is the origin of the Greek idea of geometric progression — that ''multiplication can be accomplished instead by addition''.... For instance, it is well known that the square of the golden ratio is one more than itself (written above as 𝜙&lt;sup&gt;2&lt;/sup&gt; = 𝜙 + 1) and one less than 𝜙 is its reciprocal (1/𝜙 = 𝜙 - 1). The first relation equates multiplication with addition, while the second accomplishes division through subtraction. [These are pentagon {5} bisectional golden ratios, but] the heptagonal [{7} tri-sectional] ratios also behave in surprising ways — adding to multiply, subtracting to divide.... There are analogous constructions for [nonagon {9}] quadrisectional and [hendecagon {11}] pentasectional systems.&quot;}}

The 120-cell has three interchangeable cyclic expansion-contraction cycles, the pentad, hexad and heptad cyclic designs, rooted in the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems, respectively. Each of them constructs the whole sequence of regular 4-polytopes from a single building block. The key to all three cycles is that 5 + 6 = 11, and there is an 11-point quasi-regular 4-polytope in the sequence of convex 4-polytopes. It lies between the 8-point (16-cell) and the 16-point (8-cell), and is itself a &lt;math&gt;B_4&lt;/math&gt; polytope. Its role is to be a unified single building block common to all three cyclic expansion-contraction cycles. It is involved with the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems and expresses in one building block their relations to each other.

We have already seen how the hexad building cycle works. It is plain compounding of 4-orthoplexes at the beginning, increasing to densely intricate non-disjoint symmetry toward the end, making it difficult to see why it works once past the 24-cell. This is how the build works, in terms of sixes. The 16-cell is a 2-hexad (3 pairs of completely orthogonal great square planes). The 24-cell is 3 16-cells and a 4-hexad, 4 different ways: 16 non-disjoint hexads (great hexagon planes). The 120-point (600-cell) is 25 24-cells and a 20-hexad, 10 different ways: 200 non-disjoint hexads (great hexagon planes). The 600-point (120-cell) is 225 (9 times 25) 24-cells. ....

The pentad building cycle doesn't do plain compounding to regular 4-polytopes until the last one, the 120-cell. Fives don't even appear as pentagons until after the 24-cell. We have seen how there are 11 5-cells in the quasi-regular 11-cell, which occurs before the 8-cell, and also how everything can be compounded from irregular 5-cells (4-orthoschemes specialized for each 4-polytope). It is still not evident how a pentad cycle can work to build all the regular 4-polytopes from the regular 4-simplex. This is how is how the build works, in terms of fives. ....

This is how the build works, in terms of elevens. The 11-cell is 4 orthogonal heptads (intersecting in their skew hexagon Petrie polygon). The 24-cell is also 4 heptads (intersecting in 4 orthogonal digon axes, so each heptad contributes 6 unique points to a single fibration of 24 points). Heptads are quarter elevenads, so the 11-cell and 24-cell are both 1 elevenad. ....

See the tabular sequence of convex 4-polytopes in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Build with the blocks|§Build with the blocks]]'', below: table rows ''Inscribed, Pentad, Hexad, Heptad''.

=== 11 of them ===
There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of eleven 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum eleven of them. There is only a single Hopf fibration of the 11 great circles of an 11-cell. 

The 5-cell and the 11-cell are limit cases. The 5-cells are completely disjoint, but the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 10 other instances of itself.

== Cell rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer &amp; Segerman|2013|loc=''Puzzling the 120-cell''}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his section of the Wikipedia article on the 120-cell entitled ''[[120-cell#Visualization|§Visualization]]''.{{Sfn|Ruen &amp; Goucher et al. eds. 120-cell|2024|loc=Goucher's ''[[120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious place to look for the 11-cell's Hopf map is in Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other place it could be hiding. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The quasi-regular convex 11-cell|§The quasi-regular convex 11-cell]]'', and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen: triacontagon|2011|loc=''Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them except his illustrations. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Four orthogonal projections of the 120-cell to a central plane, selected from the table of 3 regular and 11 compound 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|100 icosahedra&lt;br&gt;120 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|5 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 icosahedra&lt;br&gt;6 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|120
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|4
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|50
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|7.2°
|align=center|36°
|align=center|6°
|align=center|7.2°
|}

....

Here we summarize our findings on the 11-cells so far, including in advance some results to be obtained in the next section two sections.

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section....

== The 15 major chords ring everything ==
The natural numbers are each a distinct flavor.

Each integer has its own distinct taste. The first six integers {0..5} are the six strongest, most familiar tastes in nature, instantly distinguished and utterly unique.

The seventh flavor {6} is the first ratio of two of the others. Thereafter follows in strange sequence all the ratios of all the flavors, each marriage of every two flavors, most of which are fractions &lt;math&gt;\{k/d\}&lt;/math&gt;, but some of which are integers &lt;math&gt;\{k\}&lt;/math&gt;.

Some of these ratios are successful flavors in their own right, as distinct and inimitable as the marriage of chocolate and oranges, but the marriage of many flavor pairs is mud. It is a rule of thumb among flavor specialists that nearly any taste can be successfully alloyed with either onions or chocolate, but the direct marriage of these two flavors is notably unsuccessful. Some flavors make famous pairings or infamous ones, and subtle new pairings are always to be discovered that work surprisingly well for some purpose, but the majority of flavor marriages are undistinguished compounds.

In polytope geometry, each chord of a polytope is both is a distinct 1-dimensional object, a chord of the sphere of a distinct length &lt;math&gt;l&lt;/math&gt;, and a distinct natural number &lt;math&gt;h&lt;/math&gt;, a unique flavor. If the polytope is regular, it is a noteworthy and successful flavor. The chord's length &lt;math&gt;l&lt;/math&gt; is a square root, related to the natural number &lt;math&gt;h = k/d&lt;/math&gt; and to a set of polytopes they both represent, by a formula discovered by Steinbach.{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden''; Figure 5. Optimal sections and proportions|p=37|ps=; the regular polygons {5}, {7}, {9} and {11} with their diagonals define respectively: {5} the golden bisection proportional to 𝜙; {7} an analogous trisection; {9} an analogous quadrasection; {11} an analogous pentasection.}} The chord length &lt;math&gt;l&lt;/math&gt; is related to the number of sides of the regular polygon formed by the chord (its numerator &lt;math&gt;k&lt;/math&gt;), and to the winding number or density of its regular skew Petrie polygon (its denominator &lt;math&gt;d&lt;/math&gt;).

The regular 4-polytopes do not contain an infinite variety of these chord flavors, but only 30 of them: only 15 unique pairs of 180° degree complementary chords, which combine to make every regular ''n''-dimensional thing found in nature.{{Efn|The 15 ''major chords'' suffice to construct all the regular ''n''-polytopes. Their 180° complements include 15 ''minor chords'' which occur only in the 120-cell, not in any smaller regular polytope.{{Efn|name=Major and minor chords}} Including the complement of the 180° {{radic|4}} digon {2} chord (the {0} chord {{radic|0}}), and the skew variant of the 90° {{radic|2}} square {4} chord (the {30/7} chord), there are 32 distinct natural numbers which are all the chords of all the regular polytopes in all dimensions.|name=32 chords}} All the regular polytopes (including the largest one on the cover of the box) can be built not only from a box containing a number of blocks of 3 shapes glued together by gravity, they can be also be built from a box containing a number of sticks of 15 lengths glued together by rubber joints.

[[File:15 major chords.png|500px|thumb|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the others occur only in the interior of 120-cell, not as edges of any of the regular 4-polytopes. There are 30 distinct 4-space chordal distances between vertices within the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to identify any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=Major and minor chords}} chords are properly labelled #1 — #15 in the order they fan out by size.{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''; Figure 3|ps=; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with a &quot;fan of chords&quot; diagram like the one here.|p=23}} They join vertex pairs which are 1 — 15 edges apart on a &lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon. A #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; chord forms &lt;small&gt;&lt;math&gt;f+1&lt;/math&gt;&lt;/small&gt; regular geodesic polygons which include a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; disjoint regular {{nowrap|&lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt;-gons}} (e.g. &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; a compound of 5 disjoint hexagons) plus a single [[W:Triacontagon#Triacontagram|regular skew {{nowrap|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram}}]] that wraps around the sphere &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times before closing its circle. Notice the maverick #11 chord, with its non-conforming origin point for its {{nowrap|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;}} polygram.]]

These 15 chords (natural numbers) include the integers 2, 3, 4, 5, 6, 10, 11, 15 and 30. The others are the fractions 5/2, 10/3, 13/7, 15/2, 15/4, and 15/7. Represented among the chord lengths (the square roots) measured in ''unit radii'' (not in {{radic|2}} radii, as metrics are often given elsewhere in this article) are the {{background color|paleturquoise|roots of the first 4 positive integers {{radic|1}}, {{radic|2}}, {{radic|3}} and {{radic|4}}}}, the {{background color|#FFCCCC|root of the very distinguished rational fraction {{radic|5/2}}}}, and the roots of various distinguished irrational fractions as Steinbach discovered: for example, 4 of them are {{background color|yellow|roots of golden numbers (a small integer ± 𝜙)}}. The 15 chords each form their regular polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, which is just &lt;small&gt;&lt;math&gt;\{k\}&lt;/math&gt;&lt;/small&gt; in 8 cases. Arranged in order #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; from shortest edge-length square root to longest, the #1 — #15 chords bridge vertex pairs which are &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; vertices apart on a [[W:Triacontagon#Petrie polygons|&lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon]]. Each chord makes a [[W:Triacontagon#Triacontagram|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; polygram]] of its edge length, a compound of the smaller &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. The 15 distinct polygons form a sequence you would not anticipate exactly unless you have Steinbach's formula:

{| style=&quot;background-color:transparent; white-space:nowrap;&quot;
|-
| {30/1} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|.073}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{30} 
| 120-cell edges
|-
| {30/2}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.191}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15}
|- style=&quot;background: yellow;&quot;|
| {30/3}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.382}} 
| &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi}&lt;/math&gt;&lt;/small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{10}''' 
| 600-cell edges
|-
| {30/4} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.573}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/2}
|- style=&quot;background: paleturquoise;&quot;|
| {30/5}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|1}}'''
| &lt;math&gt;1&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{6}''' 
| 24-cell, 8-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/6}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|1.382}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{5}'''
|- style=&quot;background: paleturquoise;&quot;|
| {30/7}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|2}}'''
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{4}''' 
| 16-cell edges
|- style=&quot;background: #FFCCCC;&quot;|
| {30/8}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.5}} 
| &lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;&lt;small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/4} 
| 5-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/9}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.618}} 
| &lt;math&gt;\phi&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{10/3}
|- style=&quot;background: paleturquoise;&quot;|
| {30/10}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|3}}''' 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{3}'''
|-
| {30/11}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.426}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{11/11}
|- style=&quot;background: yellow;&quot;|
| {30/12}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.618}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{5/2}
|-
| {30/13}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.810}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{13/7}
|-
| {30/14}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.928}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/7}
|- style=&quot;background: paleturquoise;&quot;|
| {30/15}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|4}}''' 
| &lt;math&gt;2&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{2}'''
|}

{| class=wikitable style=&quot;white-space:nowrap; float:right;&quot;
!colspan=6|Natural number &lt;small&gt;&lt;math&gt;h=\{k/d\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Fibers&lt;br&gt;&lt;small&gt;&lt;math&gt;f\{h\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Cells&lt;br&gt;&lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|4-tope&lt;br&gt;&lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Verts&lt;br&gt;&lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|[[W:Triacontagon#Triacontagram|30-gram]]&lt;br&gt;&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;
|-
!Chord
!Arc
!colspan=2|&lt;small&gt;&lt;math&gt;l/\sqrt{1}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;l/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;h&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#1
|15.5~°
|{{radic|0.073}}
|0.270
|{{radic|0.146}}
|&lt;small&gt;&lt;math&gt;30&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1\{30\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;5\{3,3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/1\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#2
|25.2~°
|{{radic|0.191}}
|0.437
|{{radic|0.382}}
|&lt;small&gt;&lt;math&gt;15&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;..\{3,3\}..&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/2\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#3
|36°
|{{radic|0.382}}
|0.618
|{{radic|0.764}}
|&lt;small&gt;&lt;math&gt;10&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;30\{3,3\}20&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/3\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#4
|44.5~°
|{{radic|0.573}}
|0.757
|{{radic|1.146}}
|&lt;small&gt;&lt;math&gt;15/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/4\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5
|60°
|{{radic|1}}
|1
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;6&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/5\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#6
|72°
|{{radic|1.382}}
|1.175
|{{radic|2.624}}
|&lt;small&gt;&lt;math&gt;5&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}} 
|&lt;small&gt;&lt;math&gt;10\{5,3\}12&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/6\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#7
|90°
|{{radic|2}}
|1.414
|{{radic|4}}
|&lt;small&gt;&lt;math&gt;4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7\{4\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;4\{4,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;9\{3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/7\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: #FFCCCC;&quot;|
|#8
|104.5~°
|{{radic|2.5}}
|1.581
|{{radic|5}}
|&lt;small&gt;&lt;math&gt;15/4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/4\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}} 
|&lt;small&gt;&lt;math&gt;15\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/8\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#9
|108°
|{{radic|2.618}}
|1.618
|{{radic|5.236}}
|&lt;small&gt;&lt;math&gt;10/3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10/3\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/9\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#10
|120°
|{{radic|3}}
|1.732
|{{radic|6}}
|&lt;small&gt;&lt;math&gt;3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;10\{3\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/10\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#11
|135.5~°
|{{radic|3.426}}
|1.851~
|{{radic|6.852}}
|&lt;small&gt;&lt;math&gt;11/11&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;11\{11/11\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;11\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#12
|144°
|{{radic|3.618}}
|1.902
|{{radic|7.235}}
|&lt;small&gt;&lt;math&gt;5/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2},3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/12\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#13
|154.8~°
|{{radic|3.810}}
|1.952
|{{radic|7.621}}
|&lt;small&gt;&lt;math&gt;13/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;13\{13/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;7\{5,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/13\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#14
|164.5~°
|{{radic|3.928}}
|1.982
|{{radic|7.857}}
|&lt;small&gt;&lt;math&gt;15/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;137\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/14\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{radic|8}}
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15\{2\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐𝜙&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;8\{3,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/15\}&lt;/math&gt;&lt;/small&gt;
|}

The polygons '''{10}''', '''{6}''', '''{5}''', '''{4}''', '''{3}''', and '''{2}''' are flat polygons that each lie on a great circle in a central plane of the 3-sphere. The polygons {30}, {15}, {15/2}, {15/4}, {10/3}, {11/11}, {5/2}, {13/7}, {15/7} and all the {30/''n''} polygons are skew polygrams spiraling through curved 3-space, each successive edge in a different central plane of the 3-sphere. Projected to the 1-sphere (a circle in the plane), they are [[W:Star polygon#Regular star polygon|regular star polygrams]], the simplest being the pentagram {5/2}. In 4-space they wind around the 3-sphere more than once ''without self-intersecting'', before they close their circuit. They lie on helical geodesic circles or ''isoclines'' of circumference greater than 2𝝅''r''.

Even the 6 integer chords which form flat polygons also form skew polygrams. Each of the 15 chords occurs as a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, a fiber bundle of Clifford parallel polygons that totals 30 vertices per bundle. Each chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; also forms a single skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram that winds through another, shorter chord's bundle of parallel polygons, intersecting them in succession multiple times each, and weaving that whole bundle of parallel fibers together as the cross-thread does on a loom. The polygram cross-thread has to wind multiple times through the whole bundle (&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times around the [[W:Triacontagon#Triacontagram|30-gram]] in which each edge connects every &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt;th vertex) to complete an enumeration of all 30 vertices in the bundle. For example, the #5 chord forms hexagon &lt;small&gt;&lt;math&gt;\{6\}&lt;/math&gt;&lt;/small&gt; flat planes in parallel bundles of 5, written &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt;, and also a skew &lt;small&gt;&lt;math&gt;\{15/2\}&lt;/math&gt;&lt;/small&gt;-gram of #5 edges which winds 2 times through the #3 chord's bundle of 3 decagon &lt;small&gt;&lt;math&gt;\{10\}&lt;/math&gt;&lt;/small&gt; flat planes. Every chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; forms both its &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram and its fiber bundle of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, and the edges of one chord's parallel polygons are also the edges of its skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram weaving through some shorter chord's bundle of parallel polygons. Each numbered chord is a natural number, a ratio of two integers and their polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, either flat or skew. Each chord number defines a &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; skew polygon that is a compound bundle of the &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. All these flat or skew polygons are regular polygons made of their numerator instances of a single chord #, whose length is that square root, Steinbach's function of the natural number.

The relationship between each bundle of flat Clifford parallel great circle polygons of a distinct length, and its corresponding bundle of skew polygrams of a longer length, is precisely that of a discrete simple rotation and its corresponding discrete isoclinic rotation. For example, the flat great circle hexagons of edge length {{radic|1}} lie in the invariant rotation planes of the 24-cell's characteristic 60° isoclinic rotation, in which the vertices move on skew hexagram rotation paths of edge length {{radic|3}}. The 24-cell contains 4 bundles of 4 Clifford parallel (disjoint) hexagon central planes each. The 24 vertices of the 24-cell in each bundle are linked by rotation in the 4 hexagon invariant planes in two different ways: simply and isoclinically. In a simple rotation the vertices of a single invariant hexagon exchange places on that great circle hexagon of edge {{radic|1}}. In an isoclinic rotation in which any one of these 4 disjoint hexagon central planes is invariant, all 4 of them are necessarily invariant planes in the rotation, because they rotate in parallel. In that isoclinic rotation the vertices of all 4 hexagon invariant planes (all 24 vertices of the 24-cell) exchange places concurrently, moving along 4 disjoint helical hexagrams of edge {{radic|3}} in parallel. It is a characteristic only of this distinct rotation in hexagon invariant planes that the helical vertex path or ''isocline'' is a hexagram of period 6, edge length {{radic|3}}, and circumference 720°. Rotations in other kinds of polygon invariant planes have different characteristic isoclines. The period of their isoclinic rotation must be ''the same or greater'' than the period of their simple rotation; in most cases (unlike this one) it is greater.

The 15 chords form not only distinct fiber bundles of polygons, but also distinct bundles of polyhedra: cell rings &lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt; and compound vertex figures &lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;. These combine to form their characteristic regular 4-polytope &lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Coxeter|1973|p=269|loc=Compounds|ps=; &quot;It is remarkable that the vertices of {5, 3, 3} include the vertices of all the other fifteen regular polytopes in four dimensions.&quot;}} All these elements of a regular 4-polytope constructed from one chord are themselves regular polytopes, and each of the 16 regular 4-polytopes is a function of one characteristic chord. If the regular 4-polytope is convex, its characteristic chord is its unit-radius edge. For a non-convex regular 4-polytope, the characteristic chord is its unit-radius ''mid-edge diameter'', the distance from the midpoint of each edge to the midpoint of the completely orthogonal edge. Each of the characteristic chords (each distinct natural number) has its expression as one of the 16 regular 4-polytopes and all the products of the distinct regular 1-polytopes, 2-polytopes and 3-polytopes that construct it, and the whole distinct construction from the natural number to the 4-polytope is an [[W:SO(4)|SO(4) symmetry group]] expression of that 4-polytope's characteristic discrete [[W:SO(4)#Isoclinic rotations|isoclinic rotation]], in the invariant central planes of its characteristic chord, by the characteristic chord arc.

...demonstrate this theory by proper enumeration of the chord lengths in the table...

...color-code the △ ☐ 𝜙 ✩ symbols ... chord colors in diagram should match them ... some of the diagram colors are wrong and must be changed, but otherwise diagram is pretty good ... i think in a colored diagram all chords of the same length should have the same size line dash, the polygons are not obvious enough and the 15 fanning-out chords are readily identified by their labels...

... add back the counts of chords in the box (very small font number under each table entry), and present these two images as the front and back a box of building sticks and rubber joints...What's in the other box...

...say the rational-root major chords {{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|5/2}} are the ''n''-edges: their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge...

...abbreviated text of the [[120-cell#Relationships among interior polytopes|''§Relationships among interior polytopes'']] section from the 120-cell article...to explain the color-coding

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius=1}}

See also the elements, properties and metrics of the sequence of regular 4-polytopes summarized in tabular form in the section ''[[#Build with the blocks|§Build with the blocks]]'', below.

== Alicia Boole Stott's original formulation of dimensional analogy ==
&lt;blockquote&gt;&quot;...a method by which bodies having a certain kind of semi-regularity may be derived from regular bodies in an Euclidean space of any number of dimensions...&quot;{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''; presents cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues|pp=12-45}}&lt;/blockquote&gt;

Alica Boole Stott's 1910 paper reads like an exceptionally well-written patent application. It is a comprehensive stepwise description of her original invention, a general method of dimensional analogy between regular and semiregular polytopes of the same edge length.

[[File:The Roots of Three.png|thumb|An illustration of Boole Stott's expansion of unit-edge triangle to hexagon captures the geometric basis in 3 of the &lt;math&gt;A_n&lt;/math&gt;root system.]]

The engineering-like articulation of her new principles is entirely discrete and group-theoretic. Boole Stott repeatedly refers to sets of polytope elements as ''groups''. All her transformations are discrete instances of one or another special case of her &quot;two inverse operations which may be called ''expansion'' and ''contraction''.&quot; Hers is an exploration of ''n''-dimensional space at its most basic: space defined by the discrete objects which fill it, themselves defined directly as groups of elements with a few operations on them, which transform them into each other. This is geometry which seems to precede Euclid's postulates, and does not depend on them. Instead it corresponds directly to the deeper group theory mathematics which underlies geometry, even though Boole Stott, like the ancients, did not have that mathematics.

Boole Stott enumerated her expansion and contraction operations as transformations between regular and semi-regular polytopes of a single edge length. In her 1910 paper she did not consider any polytopes of multiple edge lengths, or expansions of the subject polytope by a distance other than that exactly required to produce new edges of the original edge length, so of course she did not enumerate Moxness's non-uniform rhombicosidodecahedron (Hull #8). This may be why section 8&lt;sub&gt;3&lt;/sub&gt; is missing from her earlier drawings of the sections of {5,3,3}; she may not have had a construction for it.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The Boole Stott expansion (or contraction) operation is as precisely defined a transformation as Coxeter's reflection, rotation and translation operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions). Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;
where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;
where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;
For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} An instance of the expansion (or contraction) operation is an application of the general operation to a particular subject polytope, operating on all of its elements symmetrically at once. The general operation is restricted, such that one may only expand (or contract) the subject polytope by pushing all its edges, faces or cells directly away from (or toward) its center, without changing the size of the moved elements. This splits each vertex into several (or coalesces several vertices into one), and has corresponding effects on the other elements, producing new edges ''of the same size'' between pairs of separated vertices (or reducing some edges to single vertices). The distance all the elements are moved away from (or toward) the center must be such that the vertices separate producing new edges of only the original edge length (or come together reducing edges to single vertices). It is permitted to combine these three operations, e.g. by pushing the edges ''and'' the faces, or by pushing all three element sets, provided the combination results in a semi-regular polytope of only one edge length. The three operations are all commutative with one another, so combined operations can be considered either as a sequence of expansions (or contractions) linking several related polytopes, or as their concurrent movements, a single transformation of the subject polytope. Since there are 8 combinations of one to three operations there are 8 polytope variants (including the original regular polytope), and all are adjacent to one another by a single transformation. The variants of the original regular polytope are named for which element set(s) are moved. By enumerating all 8 variants of expansion (and contraction) for each of the six regular convex 4-polytopes, Boole Stott produced a table of regular and semi-regular 4-polytopes divided into 12 sets of 8: six expansion sets and six contraction sets.{{Sfn|Boole Stott|1910|loc=''Table of Polytopes in S&lt;sub&gt;4&lt;/sub&gt;''; note that Boole Stott's usage of ''S&lt;sub&gt;4&lt;/sub&gt;'' denotes a Euclidean 4-space ''E&lt;sub&gt;4&lt;/sub&gt;'', or a 3-sphere space embedded in it, ''not'' a 4-sphere space embedded in a Euclidean 5-space|p=30}} Each set includes just one of the six regular convex 4-polytopes, with its distinct unit-radius edge length, and the 7 possible semi-regular expansions (or contractions) of it, which all preserve that characteristic edge length. The polytopes in each set of 8 all have the same total number of elements as their original regular polytope, but each polytope in the set has a distinct combination of cell, face, and vertex figures. The 12 sets are not completely disjoint, and even within one set the 8 operations do not necessarily produce 8 unique polytopes. For example, the regular 5-cell (the 4-simplex polytope) has 5 cells, 10 faces, 10 edges, and 5 vertices (30 elements). Expanding the 5-cell produces 7 related semi-regular 4-polytopes of 30 elements each (two of which happen to be the same polytope). In each expansion, the 5 cells have expanded into semi-regular or regular polyhedra, the 10 faces and 10 edges ''may'' have expanded into prisms, and the 5 original vertices have expanded into regular or semi-regular cells.

== The regular 11-cell 4-polytope ==

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosadodecahedron). Each non-convex Jessen's icosahedron is a set of 6 reflex edges that are {{radic|5}} 5-cell edges, each set symmetrically arranged in 3 parallel pairs, not as far apart ({{radic|3}}) as they are long, in 3 orthogonal central planes. The 12-point (Jessen's sets of 6 reflex 5-cell edges) are inscribed in the 60-point with their endpoints as a vertex of a pentagon face. The 12 pentagon faces are joined to 5 other pentagon faces by the 5-cell edges; not to their nearest neighbor pentagons but to their next-to-nearest neighbor pentagons; and since there are 5 vertices to each pentagon face, there are 10 12-point (Jessen's sets of 6 reflex 5-cell edges). Each pentagon face is joined to another pentagon face by 5 reflex edges of five distinct Jessen's. The 5 reflex edges are Clifford parallel, the 5 edges of a pentagonal prism column that has a twist to it. The contraction of those 5 edges into 1, and those 10 vertices (two pentagons) into 2, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing all their edges as a single Jessen's. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (octahedron) by conflating its parallel pairs of edges, leaving the 11-cell's abstract 6-point (hexad cell). The hexad's situation is that it occurs in bundles of 5, disjoint but close together in the sense of adjacent, like the great circle rings in a Hopf fiber bundle. 

To summarize, the 12-point (2-hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell). We shall now be able to see how Fuller's perfect hexads combine with Todd's perfect pentads to form the perfect heptads of the 11-cell and 137-cell.  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=alias the hemi-cuboctahedron}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces. (The ''heptad'' arrives at the party, anonymously; they are our distinguished anti-celebrity, who most of the other guests have never even heard of.) The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. This 12-point (truncated tetrahedron) is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 100 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp: Truncated tetrahedron|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp: Truncated tetrahedron|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}} Of course Moxness has been there, seen that, already with his graphics software;{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} see his parameterized illustration of it exactly as it occurs in the 11-cell, below in his bestiary of A3, B3, and H3 Archimedian and Catalan polyhedra in the 120-cell.]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:Archimedean_and_Catalan_solid_hulls_with_their_Weyl_orbit_definitions.svg|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of the 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. In this catalog of Archimedean and Catalan polyhedra sighted in the 120-cell by Moxness's software,{{Sfn|Moxness: Archimedean and Catalan hulls|2023|loc=Hull #1 Archimedean Name A3 110 Truncated Tetrahedron A (upper left)}} it is the first entry (upper left), Moxness's Hull # = 1 with 12 vertices of 3D norm {{radic|11}}/2. On a {{radic|2}}-radius 3-sphere, 5 of these with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell. 20 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges are cells in the [[#The rings of the 11-cells|cell rings of the quasi-regular 11-cell]], which also has 12 regular 5-cells as cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

...

Both kinds of cells in the 11-cell have expressions in two different dimensions (on the 2-sphere and 3-sphere), because they are all centered on the center of the 3-sphere, unlike the cells of most 4-polytopes, which are centered on various off-center points within the 4-polytope's interior. 

One kind of 11-cell cell appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (semi-regular truncated cuboctahedron), a.k.a. the truncated tetrahedron. The other kind of cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now, except in the 120-cell. But this is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron, and a regular tetrahedron behind each face. The 11-cell is thus a honeycomb of mixed regular tetrahedra and truncated cuboctahedra (a.k.a. truncated tetrahedra) that meet at their shared 5-cell faces. 

Both the 5-cells (decahedra) and the truncated cuboctahedra (truncated tetrahedra) are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks in their 4-polytopes (off-center).{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} Therefore all the 11-cell's cells are concentric, which indeed makes a different kind of honeycomb, an inside-out cellular honeycomb with all the cells piled on top of each other overlapping in 4-space. They do not overlap in 3-sphere space however; their 3-volumes (but not their 4-contents) are disjoint. They make a proper 4D honeycomb like the cells of any 4-polytope, with volumetrically disjoint 3D cells that meet each other in the proper way at their shared 2D faces.

An 11-cell contains 6 completely disjoint 5-point (1-pentad 5-cell decahedron cells), and 5 volumetrically disjoint 12-point (2-hexad truncated cuboctahedron cells), 11 volumetrically disjoint cells altogether. Of course it is supposed to have 11 identical 11-point (elevenad hemi-icosahedral cells), and that is also true in its abstract sense, which is not the same sense as disjoint cellular containment: each pentad or hexad cell is inscribed in one of 11 larger 60-point (Moxness's rhombicosadodecahedron cells). The abstract pentads and hexads are only hemi-polyhedra, so the 11-cell's pentad cells are 12 (out of 120) disjoint instances of a pentad 5-point, and its hexad cells are 10 (out of 100) disjoint instances of a hexad 6-point, 1/10th of the 120-cell's cellular volume in both cases. The 11-cell's hemi-icosahedral cells are 11 (out of 120) non-disjoint instances of an elevenad 60-point. Each 11-cell shares each of its hexad cells with 9 other 11-cells, and each of its 11 vertices (out of 600) occurs in 9 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.|name=Tetrahedral vertex figures}}

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

...insert image thumb of 11-cell...

...Construct the 11-cell from the heptad....

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...Four concentric 4-polytopes may be mutually orthogonal. In any set of 4 mutually orthogonal regular 5-cells, one pair of faces in each orthogonal pair are completely orthogonal to each other. Thus three edges in one 5-cell in an orthogonal pair are opposite (completely orthogonal) to a face of the orthogonal 5-cell. .... A 5-cell has 5 vertices in its abstract hemi-icosahedral cells, but only 4 vertices in one concrete icosahedral cell; the 5th lies in an icosahedral cell of another 11-cell (not in another icosahedral cell of the same 11-cell).

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

....

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 11-cell ===

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon.... (wrong, 11-cell's Hopf map is the truncated icosahedron)

....

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk: Truncated icosahedron|2018}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

...

Since the rhombicosadodecahedron has 120 5-cell edges, the 120-cell has 120 completely disjoint 5-cells, and the rhombicosadodecahedron shares each vertex with 11 other rhombicosadodecahedra, how a rhombicosadodecahedron is bound to its neighbor 5-cells is constrained. A further constraint is that the abstract hemi-icosahedron has only 15 5-cell edges, so the 120 actual 5-cell edges of the rhombicosadodecahedron are conflated 8-to-1 in the hemi-icosahedron. To be conflated by contraction into a single edge, those 8 edges must lie parallel to each other in the 120-cell; either actually parallel in the same plane, or Clifford parallel in separate planes, as two edges in completely orthogonal planes may be parallel and orthogonal at the same time. Each 5-cell inscribed in the 120-cell contains 5 such tetrahedra occur in parallel pairs in 4 orthogonal planes (none of them completely orthogonal planes). 

.... 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
Each conflated vertex is the 10 vertices of two orthogonal regular 5-cells...
The 60-point contains 6 pairs of these orthogonal regular 5-cell cells...
...its 20 hexad cells are ..., a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere.

== The regular 137-cell 4-polytope ==

(description originally written for the 11-cell, which applies only to a fibration of 11 11-cells)

It is quasi-regular with two kinds of cells, the 12-point (regular icosahedron) and the 5-point (regular 5-cell), which have just one kind of regular face between them. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of regular facets (it is semi-regular with regular facets), it also means that its facets have expressions of two different dimensionalities because they are concentric in the 3-sphere. The 11-cell has two kinds of regular cells. One appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (regular icosahedron), but since it is embedded at the center of the 3-sphere (unlike the cells of most 4-polytopes) it also acts as an inscribed 4-polytope.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} The other cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now except in the 120-cell. This is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron. The 11-cell is a honeycomb of mixed icosahedron and half-icosahedron cells. Like the icosahedra, the decahedra are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks, off-center in their 4-polytopes. All the 11-cell's cells are concentric, which indeed makes an unusual honeycomb of cells, an inside-out honeycomb. It is a proper 4D honeycomb however, with volumetrically disjoint 3D cells that meet each other in the proper way at shared 2D faces. The embedded 12-point (regular icosahedron) cell is an icosahedral pyramid (as a 4-polytope), which is filled by 20 4-point (irregular tetrahedra) that meet at its center; the center point of the 11-cell acts as a 12th vertex, just as the center point of the 24-cell acts as a 25th vertex in some honeycombs. The 5-point (regular 5-cell) cell contains five 4-point (regular tetrahedra) which meet the 12-point (regular icosahedra)'s irregular tetrahedral cells face-to-face, so the 11-cell is quasi-regular in the usual sense of having two varieties of regular 3-polytope cells, but additionally both varieties of of its cells are bi-dimensional: another clue that the 11-cell has something to do with dimensional analogy

....

The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

....

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur to make an 11-cell. 11 of them occur in a 137-cell, all 120 of them occur in the 120-cell, and they occur nowhere else.

How can this regular 4-polytope possibly exist? By Schläfli's criterion {3, 3, ..} exists, but not as a finite polytope.{{Sfn|Coxeter|1973|loc=§7.7 Schläfli's criterion}} Coxeter found {3, 5, 3} as a finite hyperbolic honeycomb, but not in Euclidean 4-space.{{Sfn|Coxeter|1968|title=The Beauty of Geometry: Twelve Geometric Essays|loc=''Regular Honeycombs in Hyperbolic Space''|ps=; &quot;The honeycombs considered by the above authors [Schlegel, Schläfli, Klein, Sommerville] have finite cells and finite vertex figures. ... A further extension allows a cell or vertex figure to be a ''star''-polytope, so that the honeycomb covers the space several times. .... We shall find that there are four regular star-honeycombs in hyperbolic 4-space, as well as two infinite families of them in the hyperbolic plane.&quot;}} We find that {3, 3, ..} is a central expression of the &lt;math&gt;H_4&lt;/math&gt; root system, being the regular convex hull in 4 dimensions of 137 points, 96 pentads and 600 hexads. The minimal expression of &lt;math&gt;H_4&lt;/math&gt; is the regular convex hull {3, 3, 5} of 120 points, 600 tetrads and 75 hexads, and its maximal expression is the regular convex hull {5, 3, 3} of 600 points, 120 pentads and 675 hexads. {3, 3, ..} is a third regular &lt;math&gt;H_4&lt;/math&gt; polytope which fits between them. The properties and metrics of this regular sequence are presented in tabular form below in ''[[#Build with the blocks|§Build with the blocks]]''.

{|class=&quot;wikitable&quot;
!colspan=3|Regular &lt;math&gt;H_4&lt;/math&gt; polytopes
|-
|[[File:600-cell.gif|200px]]
|[[File:5-cell.gif|200px]]
|[[File:120-cell.gif|200px]]
|-
!120-point 600-cell
!137-point 137-cell
!600-point 120-cell
|}

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 137-cell ===
....

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=Apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|2022|loc=''[[Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]''}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie: On Fuller's use of language|2024|loc=''[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry'']]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center, 1948-1949|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with their mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the gender neutral shad can open their six beaks all the way, until they become flat squares and they becomes a cuboctahedron, or they can shut them all tight like a turtle retracting into their octahedron shell. The six mouths always move in unison. This is [[Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do their ''really'' odd trick -- where they flip their 6 jaws 90 degrees in their 6 faces and shut their 6 beaks on the opposite axis of their squares than the one they opened them on -- or whether they will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than their normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make their edge length exactly the same size as their radius, when they open their mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing their mouths in spherical synchrony, their 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron (the hexad), then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=''[[W:SO(4)|SO(4)]]''}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=Isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its orthogonal hexad's planes (wz, wx, wy).{{Efn|name=Six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of (expanded octahederon 16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh &amp; Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== Legendre's 12-point hexad binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a 12-point double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's perfectly realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, this 2-hexad expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre hexad) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction.{{Efn|name=Concentric polyhedra}} In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== ''n''-simplexes ===
....

The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== ''n''-orthoplexes ===
....
[[File:Great 5-cell √5 digons rectangle.png|thumb|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} ''regular'' great hexagons. These are the same 200 dodecagon central planes illustrated above, which also contain 6 120-cell edges and two opposing ''irregular'' great hexagons (truncated {{radic|6}} triangles). The 1200 {{radic|5}} 5-cell face triangles occupy 1200 other, non-central planes.]]

...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's Hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[16--cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[16--cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=Concentric polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[16--cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== ''n''-taliesins ===
The 4-simplex pentad building block combines with the 4-orthoplex hexad building block to make the 4-taliesin heptad building block. The heptad is the essential building block of the ''taliesin'' family of polytopes. The 11-cell and 57-cell hemi-polytopes and the 137-cell and 120-cell regular 4-polytopes are nuclear members of this trans-dimensional family. The 24-cell and 600-cell are extended members of it, because they also contain instances of the heptad building block.

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its spherical symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* [[W:Taliesin Myrddin Namkai-Meche|Taliesin Myrddin Namkai-Meche]], a young Reed College scholar who was stabbed to death by a crazy man in 2017 on a Portland, Oregon light rail train, when he rose to the nonviolent defense of two black teenagers who were being violently assaulted in racist and anti-Muslim language by a right-wing extremist.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in orthogonal [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].
* '''Taliesan polytope''', an (''n+1)''-polytope whose ''n''-polytope dimensional analogue inscribed concentrically in the ''n''-sphere forms a convex honeycomb of itself which is the (''n+1)''-polytope.{{Efn|name=Embedding point}}
* The taliesan polytope's honeycomb, a Hopf fibration of its great circles.
* The ''n''-dimensional family of dimensionally analogous polytopes related to a taliesan polytope by expansion and contraction operations.
* The characteristic [[w:Orthoscheme|orthoscheme]] of a teliesan polytope, an irregular simplex that is the complete gene for the entire expression of trans-dimensional symmetry relationships among the polytopes of the ''n''-dimensional family, as well as the complete gene for the teliesan polytope's rotational symmetry.{{Efn|name=Characteristic orthoscheme}}

=== ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== ''n''-equilaterals ===
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed at four different [[#n-leviathon|chord]] lengths from each other. In the {{radic|2}}-radius polytopes we have using as examples in this paper they are {{radic|2}}, {{radic|4}}, {{radic|6}} and {{radic|8}}, but in a ''unit-radius'' polytope they are: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}.

Each vertex is joined to 8 others by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}. The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== ''n''-pentagonals ===

[[File:600-cell.gif|thumb|The 120-point (600-cell), &lt;math&gt;H^4&lt;/math&gt;.]]
...

[[File:600-cell vertex geometry.png|thumb|Vertex geometry of the 600-cell, showing the 5 regular great circle polygons and the 8 vertex-to-vertex chord lengths with angles of arc. The golden ratio{{Efn|name=golden chords|group=}} governs the fractional roots of every other chord,{{Efn|name=fractional root chords}} and the radial golden triangles which meet at the center.|alt=|400x400px]]
The 120 vertices of the 600-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏&lt;sup&gt;−1&lt;/sup&gt;) beginning with a vertex; see column ''a''}} at eight different [[#n-leviathon|chord]] lengths from each other. These edges and chords of the 600-cell are simply the edges and chords of its five great circle polygons. In ascending order of length in a ''unit-radius'' polytope (not in a {{radic|2}} radius polytopes like our usual examples), they are {{radic|0.𝚫}}, {{radic|1}}, {{radic|1.𝚫}}, {{radic|2}}, {{radic|2.𝚽}}, {{radic|3}}, {{radic|3.𝚽}}, and {{radic|4}}.{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=Decimal fractions of square roots|group=}}

Notice that the four [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#''n-''equilaterals|hypercubic chords]] of the 24-cell ({{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|4}}) alternate with the four new chords of the 600-cell's additional great circles, the decagons and pentagons. The new ''golden chord'' lengths are necessarily square roots of fractions, but very special fractions related to the golden ratio{{Efn|1=The fractional-root ''golden chords'' are irrational fractions that are functions of {{radic|5}}. They exemplify that the [[W:golden ratio|golden ratio]] &lt;big&gt;𝜙&lt;/big&gt; {{=}} {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618 is a circle ratio related to &lt;big&gt;𝜋&lt;/big&gt;:&lt;br&gt;
: {{sfrac|𝜋|5}} = arccos (𝜙/2)
is one decagon edge, the 𝚽 = {{radic|0.𝚫}} = {{radic|0.382~}} ≈ 0.618 chord.
Reciprocally, in this function discovered by Robert Everest expressing &lt;big&gt;𝜙&lt;/big&gt; as a function of &lt;big&gt;𝜋&lt;/big&gt; and the numbers 1, 2, 3 and 5 of the Fibonacci series:&lt;br&gt;
: &lt;big&gt;𝜙&lt;/big&gt; = 1 – 2 cos ({{sfrac|3𝜋|5}})
{{sfrac|3𝜋|5}} is the arc length of the &lt;big&gt;𝜙&lt;/big&gt; = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618 chord.|name=golden chords|group=}} including the two [[W:golden section|golden section]]s of {{radic|5}}, as shown in the diagram.{{Efn|The 600-cell edges are decagon edges of length {{radic|0.𝚫}}, which is 𝚽, the ''smaller'' golden section of {{radic|5}}; the edges are in the inverse [[W:golden ratio|golden ratio]] {{sfrac|1|φ}} to the {{radic|1}} hexagon chords (the 24-cell edges).
The other fractional-root chords exhibit golden relationships as well. The chord of length {{radic|1.𝚫}} is a pentagon edge.
The next fractional-root chord is a decagon diagonal of length {{radic|2.𝚽}} which is &lt;big&gt;𝜙&lt;/big&gt;, the ''larger'' golden section of {{radic|5}}; it is in the golden ratio{{Efn|name=golden chords|group=}} to the {{radic|1}} chord (and the radius).{{Efn|Notice in the diagram how the &lt;big&gt;𝜙&lt;/big&gt; chord (the ''larger'' golden section) sums with the adjacent 𝚽 edge (the ''smaller'' golden section) to {{radic|5}}, as if together they were a {{radic|5}} chord bent to fit inside the {{radic|4}} diameter.}}
The last fractional-root chord is the pentagon diagonal of length {{radic|3.𝚽}}.
The [[W:Pentagon#Side length is given|diagonal of a regular pentagon]] is always in the golden ratio to its edge, and indeed &lt;big&gt;𝜙&lt;/big&gt;{{radic|1.𝚫}} is {{radic|3.𝚽}}.|name=fractional root chords|group=}}

....


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== ''n''-leviathon ===
...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of its &lt;math&gt;10^2&lt;/math&gt; disjoint instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction, before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]].{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}} .... It is only because she did not have the 11-cell to start with that she did not discover the regular 137-cell {3, 3, ..} by her method.

Between Alicia Boole Stott and her identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and her identification of symmetry groups with conservation laws (in her 1st theorem), we owe the deepest mathematics of geometry and physics originally to two pioneering women, colleagues of their contemporaries the greatest men of the academy, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just an [[W:abstract polytope|abstract 4-polytope]], not just a singleton convex 4-polytope, and not just a honeycomb.{{Sfn|Coxeter|1970|loc=''Twisted Honeycombs''}} The 11-point (11-cell) has a concrete ...regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}&lt;nowiki/&gt;} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, 3, 3} as the 137-point (137-cell).

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 11-cell, 137-cell and 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=''Clifford Spinors and Root System Induction: H4 and the Grand Antiprism''}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=Apology}}

== Build with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Duveneck|1978|loc=Carl Jung, quoted in ''Life on Two Levels''|p=ii|ps=.{{Sfn|Jung|1961|ps=: &quot;The best of truths is of no use unless it has become one's most personal inner experience. It is the duty of everyone who takes a solitary path to share with society what he finds on his journey of discovery.&quot;}}}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1954|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius={{radic|2}}}}

== Acknowledgements ==
...

== Notes ==
{{Regular convex 4-polytopes Notelist}}

== Citations ==
{{Reflist}}

== References ==
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{{Refend}}</text>
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      <text bytes="208632" sha1="ezb2cbdwuea8i7xfyz0jp9xh90yjpws" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|June 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a regular 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the &lt;s&gt;120-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 5, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 11 [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] cells. Eleven 11-cells (plural) are the 137-cell, the &lt;s&gt;600-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 3, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 137 [[W:Rhombic triacontahedron|triacontahedron]] cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=''Regularity of Graphs, Complexes and Designs''}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=''A Symmetrical Arrangement of Eleven Hemi-Icosahedra''}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]]
The 4-dimensional regular polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:Simplex|simplex]] (called the [[5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:Orthoplex|orthoplex]] (called the [[16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular convex 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the [[w:Tesseract|16-point (8-cell) tesseract]], the 4-cubic, which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found within any of the other regular convex 4-polytopes, except in the largest and most complex one, the [[120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). However, we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the 16-cell is their very starting point, and the most frequently used tool in the box.

== 5-cells and hemi-icosahedra in the 11-cell ==
[[File:Symmetrical_5-set_Venn_diagram.svg|thumb|The 5-point (10-face) regular 5-cell (the regular 4-simplex). Grünbaum's rotationally symmetrical 5-set Venn diagram{{Sfn|Grünbaum|1975|loc=''Rotationally symmetrical 5-set Venn diagram'', Fig 1 (e)|ps=; partitions the individual elements of the 5-cell.}} is an illustration of the 5-cell labeling each of its &lt;math&gt;2^5&lt;/math&gt; elements.{{Sfn|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled; image includes the Python code to render it, optimising for maximum area of the smallest regions.}}]]
[[File:Hemi-icosahedron.png|thumb|The 6-point (10 face) [[W:hemi-icosahedron|hemi-icosahedron]], an abstraction of the regular icosahedron, has half as many faces, edges and vertices. Each element of the abstract polyhedron represents two real elements found in different places in the 11-cell's realization.{{Sfn|Ruen: hemi-icosahedron|2007}}]]
The most apparent relationship between the pentad 5-cell and the hexad hemi-icosahedron is that they both have 10 triangular faces. When we see a facet congruence between a 4-polytope and a 3-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 6-point 5-simplex. Another hexad is the 6-point 3-orthoplex; thus the hemi-icosahedron is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no chord congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals, 5-cell and 16-cell, who are famously stand-offish with each other. Interesting!

The cell of the 11-cell is an abstract hexad hemi-icosahedron with 5 central planes, most handsomely illustrated by Séquin.{{Sfn|Séquin &amp; Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing rings of polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The 11 hemi-icosahedral cells have 10 triangle faces each; the 5-cell's 5 tetrahedral cells have 10 faces and 10 edges all together. If 11-cell faces are 5-cell faces, then 3 of each 5-cell's 5 vertices are a hemi-icosahedron face, and its other 2 vertices must be some 11-cell edge lying opposite the face. Coxeter determined that the 11-cell does indeed have an edge opposite each face, but it does not belong to the same hemi-icosahedral cell as its opposing face. He found that the 10 edges opposite each hemi-icosahedron's faces are the 10 edges of a single 5-cell, which does not share any vertices, edges or faces with the hemi-icosahedron. Of course, in the 11-point 11-cell there is just one such 5-point 5-cell that is completely disjoint from each 6-point hemi-icosahedron.{{Sfn|Coxeter|1984|p=110|loc=§6. The Petrie polygon [of the 11-cell]|ps=; &quot;We may reasonably call this edge and face ''opposites''. It is easy to find the face opposite to a given edge by looking at the faces to which a given edge belongs. ... Conversely, given a face, we can find the opposite edge by seeing which vertices belong to neither of the hemi-icosahedra which share that face. The ten edges opposite to the ten faces of one hemi-icosahedron are the edges of the complementary &lt;math&gt;a_4&lt;/math&gt; [4-simplex], that is, the joins of all pairs of the five vertices [of the 11-cell] not belonging to the given hemi-icosahedron.&quot;}}

There are 11 disjoint 5-cell 4-polytopes inscribed in each 11-cell, which also contains 11 hemi-icosahedral cells, 55 faces, 55 edges and 11 vertices. The real 11-cell is more complex and harder to see than the abstract 11-cell representing it, because the real hemi-icosahedron is also more complex than the abstract hemi-icosahedron. Seeing the real 11-cell will be easier once we have identified the real hemi-icosahedron, and seen exactly where the 11-cell's real elements reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each hemi-icosahedron's 10 faces come from 10 disjoint 5-cells, and there is just one 5-cell with which it does not share a face.{{Efn|As Coxeter observes (in the previous citation), that unrepresented 5-point (5-cell) is the other 5 vertices of the 11-point (11-cell) that are not vertices of this 6-point (hemi-icosahedron), the hemi-icosahedron's disjoint complement.}} Each 5-cell has 10 faces from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share a face.

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 disjoint 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent hemi-icosahedron faces do not actually meet at an edge; there is a narrow rectangle separating them, which has been abstracted to an edge. The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in two different planes which are completely orthogonal to each other. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must be sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness: 8 concentric hulls|2022|loc=Hull #8 (lower right)|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell of the 11-cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Compounds in the 120-cell|§Compounds in the 120-cell]]'' that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software which uses Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s to render the polyhedra which are found in the interior of ''n''-dimensional polytopes.{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the tetrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is a real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (Hull #8) is a concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this {12} irregular great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing {6} irregular great hexagons (truncated triangles) of alternating 120-cell edges and 5-cell edges. The great dodecagon is a great circle of the hemi-icosahedral cell, and also a great circle of the 120-cell, because unlike the cells of most 4-polytopes, the 11-cell's cells are embedded in 4-space at the exact center of the 4-polytope, making them inscribed 4-polytopes themselves.{{Efn|name=Embedding point}} There are 25 great dodecagons and 120 5-cell edges in Moxness's 60-point (Hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (Hull #8) polyhedron altogether.

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point (Hull #8) is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] similar to the one from the catalog shown here,{{Sfn|Piesk: Rhombicosidodecahedron|2018}} but a slightly shallower truncation with smaller red pentagons (120-cell faces) and narrower rhombs. Rhombicosidodecahedra are made by truncating the [[W:Rhombic triacontahedron|rhombic triacontahedron]], which is the 30-sided polyhedron with only one kind of face, the dual of the 30-point icosidodecahedron. The 120-cell contains 120 of these 11-cell cells.]]
Moxness's 60-point (Hull #8) is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (Hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|The 5-point (regular 5-cell) can be another abstraction of Moxness's 60-point (Hull #8), 12-vertices-into-1, but it is not strictly the result of a contraction operation as described by Boole Stott, since she did not apply her expansion and contraction operations to uniform polytopes with more than one edge length.}} In fact Moxness was not the first to find a rhombicosidodecahedron in the 120-cell; Alicia Boole Stott was. It is the 6th section of {5,3,3} beginning with a cell, which Boole Stott identified as the ''e&lt;sub&gt;2&lt;/sub&gt; expansion'' of the icosahedron (or equivalently of its dual polyhedron the dodecahedron).{{Sfn|Boole Stott|1910|loc=§Examples of the e&lt;sub&gt;2&lt;/sub&gt; expansion|p=7}} But that rhombicosidodecahedron identified by Boole Stott is not Moxness's Hull #8: rather, it is the semi-regular Archimedean solid, with a single edge length and square faces. Moxness's Hull #8 with its two distinct edge lengths and rectangular faces is Coxeter's 8th section of {5,3,3} beginning with a cell, which is missing from the sections illustrated by Boole Stott.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than tetrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. The 120-cell contains 120 of Moxness's 60-point (11-cell cells). They are non-disjoint, with 12 60-points sharing each vertex.

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (Hull #8) is the real hemi-icosahedron cell, with 120 5-cell edges in it as chords lurking under its surface. The rhombicosadodecahedron has enough of those 5-cell edges in its interior to make 20 disjoint tetrahedral cells which belong to 20 distinct 5-cells.

The 600-point (120-cell) may be constructed as a semi-regular honeycomb of two kinds of cells, the 120 non-disjoint 60-point (Moxness's rhombicosadodecahedra) and their neighbors the 120 disjoint 5-point (5-cells), cell-bonded together by shared tetrahedral cells of edge-length {{radic|5}}. Each rhombicosadodecahedron is bonded to 20 islanded 5-cells, sharing with each a 5-cell tetrahedron (4 of its interior 5-cell faces, and 6 of its interior 5-cell edges). The rhombicosadodecahedron is also bonded directly to 10 other rhombicosadodecahedra (since each 11-cell has 11 rhombicosadodecahedron cells), again not by sharing one of its exterior faces, but this time by sharing an interior face, a {{radic|6}} triangle which lies in a central plane.{{Efn|The {{radic|6}} chord is another hidden chord lurking beneath the 60-point rhombicosadodecahedron's surface, like the {{radic|5}} 5-cell edge. Both these hidden chords lie on the same great circle with the 120-cell edges (the pentagon face edges). Three {{radic|6}} chords make a triangle which lies in a central plane. This {{radic|6}} great triangle must not be confused with the {{radic|5}} face triangle of the 5-cell, which does not lie in a central plane.}} The 120 rhombicosadodecahedra are also bonded directly together at ''all'' their exterior faces: at their pentagon faces (which are 120-cell faces of dodecahedral cells); at their small triangular faces (which are 600-cell faces of small tetrahedral cells, 10 of which are inscribed in each dodecahedral cell); and at their rectangular faces (which are faces of 10-point [[w:Pentagonal_prism|pentagonal prism]] cells, where the pentagonal prism is a column inscribed in a dodecahedral cell, joining its opposite pentagon faces). The 120 rhombicosadodecahedra are bonded directly to each other at their own exterior faces; no dodecahedra, small tetrahedra, or pentagonal prisms lie between them, and those objects are not cells of this honeycomb; we mention them here only to help locate the rhombicosadodecahedra within the 120-cell. The only objects which fill holes between the rhombicosadodecahedra are the 120 islanded 5-cells.{{Efn|This honeycomb of the 120-cell resembles the [[600-cell#Icosahedra|honeycomb of the 600-cell consisting of 24 face-bonded regular icosahedra]] with 24 clusters of 5 tetrahedra filling the holes between them. In the 120-cell there are 5 times as many of everything as in the 600-cell, so there are 120 (rhombicosadodecahedron) icosahedral cells with 600 tetrahedra filling the holes between them.}}

It is possible for rhombicosadodecahedra to meet each other at more than one kind of exterior face because they are not meeting there as cells of the same 11-cell; they belong to different 11-cells. The 11 rhombicosadodecahedra in the same 11-cell do not meet at their own exterior faces, they meet at their interior {{radic|6}} triangle faces. As a cellular 4-polytope, the 11-cell is inside-out. Its cells meet each other at interior planes, sharing interior {{radic|6}} triangle faces, instead of in the usual way that 4-polytopes' cells share an exterior face. It is a deeper kind of intertwining to be sure, but the 11 rhombicosadodecahedra in the same 11-cell are still volumetrically disjoint cells, and the 11-cell is a perfectly real 4-polytope with real cells. It is a semi-regular 4-polytope, because it has two different kinds of cells: 11 rhombicosadodecahedra and 11 5-cells. The 11 5-cells are islanded (not bound to each other), and the 11 rhombicosadodecahedra are bound both to 5-cells (at {{radic|5}} 5-cell faces) and to each other (at {{radic|6}} central triangle faces). In the section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#What's in the box|§What's in the box]]'', below, we will look closely at the relation between the {{radic|5}} chord and the {{radic|6}} chord, which is the relation between the pentad and the hexad, and at the general phenomenon of hemi-polytopes which meet each other at a central plane. Then in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Cell rings of the 11-cells|§Cell rings of the 11-cells]]'' we will look at the two-cell cellular structure of the 11-cell in more detail.

Considered as a single complex of overlapping cells rather than as distinct honeycombs of volumetrically disjoint cells, the 120-cell contains 60-point (rhombicosadodecahedral cells), islanded 5-point (5-cells), 20-point (dodecahedral cells), and 10-point ([[w:Pentagonal_prism|pentagonal prism]] cells). The 60-points meet each other in pairs at their pentagon faces, meet regular dodecahedra at those pentagon faces also, meet each other in quads at their 600-cell faces where they share an interior 4-point (600-cell tetrahedral cell) and its four faces (just as they share a larger interior 5-cell tetrahedron and its faces), and meet each other and the 10-point (pentagonal prisms) at their rectangle faces. We might try to imagine such a complex with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with parts truncated from 30 neighboring truncated dodecahedra pentagonal prisms). There are 120 of each of these 3 kinds of dodecahedron. 

Moxness's 60-point (rhombicosadodecahedron cell) is a central object of the 120-cell, and contains many wonders. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra by expansion and contraction operations, as Alicia Boole Stott discovered before 1910,{{Sfn|Polo-Blanco: ''Theory and history of geometric models of Alicia Boole Stott''|2007|loc=§5.3.2 1910 paper on semi-regular polytopes|pp=152-158|ps=; summarizes Boole Stott's method and results from {{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}}, including her tables of expansion-contraction dimensional analogies and a few of her illustrations.}} and those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, rediscovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter &amp; Petrie|1938|p=4|loc=''[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]''}}|name=Concentric polyhedra}} Before we move on from the 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice one more thing about the hidden {{radic|5}} chords lurking below its surface. The 12 little pentagon faces are spanned by sets of 6 {{radic|5}} chords (5-cell edges). The 6 chords of each set are disjoint (they don't touch or form 5-cell faces), but they are symmetrically arranged in 3 parallel pairs, {{radic|3}} apart, which lie in 3 orthogonal central planes. It turns out they are the 6 reflex edges of a 12-point non-convex polyhedron called the [[w:Jessen's_icosahedron|Jessen's icosahedron]], inscribed in the 60-point cell. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again and work our way outwards, before eventually in ''[[#The perfection of Fuller's cyclic design|§The perfection of Fuller's cyclic design]]'' we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cell)s ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks, showing its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Pentahemidemicube.png|120px]]
|[[File:Tetrahemihexahedron_rotation.gif|120px]]
|[[File:Pentahemicosahedron.png|120px]]
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts needed to build the 120-cell (the picture on the cover of the box), and everything else that's invisible in that picture which fits inside the 120-cell: all the astonishing regular 4-polytopes. The pentad building block is half a pair of orthogonal 5-point (regular 5-cells), the hexad building block is half an 8-point (16-cell), and the heptad building block is a quarter of an 11-point (11-cell). Everything can be built from these blocks by embedding them at the center of the 3-sphere, inscribing them as 4-polytopes to make compounds as with the 16-cell building block, rather than by assembling them face-to-face as 3-polytopes to make honeycombs in which they are cells or vertex figures.{{Efn|name=Embedding point}}

The pentad building block is the 5-point '''pentahemidemicube''', an abstract polyhedron with 10 equilateral triangle faces, 10 {{radic|5}} edges, and 5 vertices.{{Sfn|Christie: Pentahemidemicube|2024|loc=File:Pentahemidemicube.png|ps=; &quot;The 5-point ''pentahemidemicube'' is an abstract polyhedron with 10 equilateral triangle faces, 10 edges, and 5 vertices. It has a real presentation as the regular 5-cell.&quot;}} It has a real presentation as the 5-point (regular 5-cell 4-polytope), a regular tetrahedron expanded by addition of a fifth vertex equidistant from all the others in a fourth dimension. The pentad block is a tetrahedral pyramid (a 5-cell) inscribed in a cube: a tetrahedron inscribed in a cube (a demicube) expanded by addition of a fifth vertex, one of the cube's other 4 vertices. Each pentad (5-cell) building block lies in the 120-cell orthogonal to three other pentad (5-cell) building blocks. They are four completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges. Each regular 5-cell in the 120-cell is a pentad (pentahemidemicube) building block, in one of 30 orthogonal sets of 4. There are 120 pentad building blocks in the box.

Every vertex of the 16-cell, and therefore every vertex of the 8-cell, 24-cell, 600-cell and 120-cell, is surrounded by a 6-point (octahedral vertex figure). The hexad building block is that 6-point octahedron embedded at the center of the 4-polytope instead of at a vertex, that is, inscribed in the 3-sphere as a 4-polytope, the 6-point '''[[w:Tetrahemihexahedron|tetrahemihexahedron]]''' or '''hemi-cuboctahedron'''.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; &quot;The tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U&lt;sub&gt;4&lt;/sub&gt;. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices.&quot;}} The hexads have 4 red triangle 5-cell faces, bounded by 12 {{radic|5}} edges. Their 3 yellow square faces lie in central planes, because the building block is an abstract 4-polytope. The hexad is an irregular abstract polyhedron that also has 6 {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

Two orthogonal 5-point (pentad) building blocks intersect in a 4-point (inscribed tetrahedron) to form a 6-point (hexad) building block. Two orthogonal 6-point (hexad) building blocks intersect in a 4-point (central square) to form an 8-point (16-cell) building block, the most usable building block, from which everything larger can be built by compounding.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 120 pentads to pair with a completely orthogonal hexad, and although there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed) they only result in 120 pairs two different ways, so there are 120 11-cells. The 11-cell is 6 pentads' and 5 hexads' smallest joint compound, and the 120-cell is their largest, 120 pentads and 100 hexads.

The heptad{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''|pages=22–31}} building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|4}} edges and 9 {{radic|5}} edges), and 7 vertices.{{Sfn|Christie: Pentahemicosahedron|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;}} It is a compound of a 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]) and a 6-point (hexad [[W:hemi-icosahedron|hemi-icosahedron]]) intersecting in a 4-point (tetrahedron), that is, of three orthogonal 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]&lt;nowiki/&gt;s) intersecting in their common 4-point (tetrahedron). It is also the abstract cube contracted by removal of one vertex, which reduces one of the cube's two inscribed tetrahedra (demicubes) to a single remaining triangular face. As with all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Four &lt;small&gt;&lt;math&gt;(11-7)&lt;/math&gt;&lt;/small&gt; orthogonal 7-point (pentahemicosahedron) blocks can be joined at a common Petrie polygon (a skew hexagon which contains their orthogonal 6-edge paths) to construct a real 11-point ([[W:11-cell|11-cell]] semi-regular 4-polytope), which magically contains six &lt;small&gt;&lt;math&gt;(11-5)&lt;/math&gt;&lt;/small&gt; 5-point (1-pentad regular decahedra) as cells and five &lt;small&gt;&lt;math&gt;(11-6)&lt;/math&gt;&lt;/small&gt; 12-point (2-hexad semi-regular cuboctahedra) as cells. There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like [[W:Euclid's postulates|Euclid's postulates]], and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as honeycombs of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles. It possesses the complete symmetry of the polytope without any redundancy, because it contains one of each of the polytope's characteristic root elements.{{Efn|A [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be partitioned radially by its planes of symmetry (Coxeter's &quot;reflecting circles&quot;) into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated rotationally to generate its regular 4-polytope because it is the complete [[W:gene|gene]] for it, containing all of its elements and capturing all of its symmetry without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring that triggers the reflections.|name=Characteristic orthoscheme}} It is the gene for the polytope, which can be replicated to construct the polytope.

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its &lt;math&gt;A_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the regular compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;small&gt;&lt;math&gt;120\times 120 = 14400&lt;/math&gt;&lt;/small&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[16--cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its &lt;math&gt;B_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 384. The 120-cell is the convex hull of the regular compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 24-point (24-cell) is subdivided into 1152 instances of its [[24-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;F_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 1152. The 120-cell is the convex hull of the regular compound of 25 disjoint 24-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;25\times 1152 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has three equivalent origins in distinct root systems, though most are obviously constructed from one root, and not so obviously from the others. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything larger than the pentad can be constructed by compounding hexads, and we shall see that everything as large as a 24-cell has a construction from heptads which are a product of a pentad and a hexad. Surely every uniform polytope can be constructed by some function of expansions and contractions from any of three root systems, however indirect the recipe. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes, related by an equals sign between them, then give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics by [[W:Noether's theorem|Noether's theorem]].

=== Pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 octahedral cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell contains the 600 octahedral cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It also contains the 600 cubical cells of its 75 8-cells. It is a 600-cell in three ways, although the octahedra and cubes are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell and the 8-cell, respectively.|name=600 cells three ways}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (1-pentad 5-cell), out of the 8-point (2-hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (1-pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope that contains 5-point (1-pentad 5-cell) building blocks beside the largest one, it just isn't a ''regular'' 4-polytope. It is semi-regular with two kinds of cells, the 4-point (regular tetrahedron) and the 12-point (truncated cuboctahedron), which have just one kind of regular face between them, a 5-cell face. That semi-regular 4-polytope is the 11-cell.

We said earlier that the 11-cell had 11 disjoint 5-cells and 11 hemi-icosahedral cells in it, and that its construction from them might be hard to see, but now we are going to account numerically, at least, for how it can have 6 pentad cells plus 5 hexad cells magically adding to 11 of the same kind of elevenad cell. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, 11. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (5-cell regular decahedron) cells, and the 12 6-point (hexad) building blocks into 100 12-point (regular icosahedron) cells, and then magically adding one whole 5-point (5-cell regular decahedron) cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 12-point (regular icosahedron) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such (cell bundles of 6 + 5 = 11 cells), with each 11-cell bundle containing 11 of both kinds of cell. Very, very strange!{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden'': §II To Add is to Multiply|p=38|ps=; the sequence of regular polygons with harmonious proportions beginning {5}, {7}, {9}, {11} as a family of natural numbers; &quot;The numbers in this optimal family have remarkable properties [that allow] similar figures to be arranged easily or allow a figure to be dissected into a set of similar figures. This is the origin of the Greek idea of geometric progression — that ''multiplication can be accomplished instead by addition''.... For instance, it is well known that the square of the golden ratio is one more than itself (written above as 𝜙&lt;sup&gt;2&lt;/sup&gt; = 𝜙 + 1) and one less than 𝜙 is its reciprocal (1/𝜙 = 𝜙 - 1). The first relation equates multiplication with addition, while the second accomplishes division through subtraction. [These are pentagon {5} bisectional golden ratios, but] the heptagonal [{7} tri-sectional] ratios also behave in surprising ways — adding to multiply, subtracting to divide.... There are analogous constructions for [nonagon {9}] quadrisectional and [hendecagon {11}] pentasectional systems.&quot;}}

The 120-cell has three interchangeable cyclic expansion-contraction cycles, the pentad, hexad and heptad cyclic designs, rooted in the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems, respectively. Each of them constructs the whole sequence of regular 4-polytopes from a single building block. The key to all three cycles is that 5 + 6 = 11, and there is an 11-point quasi-regular 4-polytope in the sequence of convex 4-polytopes. It lies between the 8-point (16-cell) and the 16-point (8-cell), and is itself a &lt;math&gt;B_4&lt;/math&gt; polytope. Its role is to be a unified single building block common to all three cyclic expansion-contraction cycles. It is involved with the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems and expresses in one building block their relations to each other.

We have already seen how the hexad building cycle works. It is plain compounding of 4-orthoplexes at the beginning, increasing to densely intricate non-disjoint symmetry toward the end, making it difficult to see why it works once past the 24-cell. This is how the build works, in terms of sixes. The 16-cell is a 2-hexad (3 pairs of completely orthogonal great square planes). The 24-cell is 3 16-cells and a 4-hexad, 4 different ways: 16 non-disjoint hexads (great hexagon planes). The 120-point (600-cell) is 25 24-cells and a 20-hexad, 10 different ways: 200 non-disjoint hexads (great hexagon planes). The 600-point (120-cell) is 225 (9 times 25) 24-cells. ....

The pentad building cycle doesn't do plain compounding to regular 4-polytopes until the last one, the 120-cell. Fives don't even appear as pentagons until after the 24-cell. We have seen how there are 11 5-cells in the quasi-regular 11-cell, which occurs before the 8-cell, and also how everything can be compounded from irregular 5-cells (4-orthoschemes specialized for each 4-polytope). It is still not evident how a pentad cycle can work to build all the regular 4-polytopes from the regular 4-simplex. This is how is how the build works, in terms of fives. ....

This is how the build works, in terms of elevens. The 11-cell is 4 orthogonal heptads (intersecting in their skew hexagon Petrie polygon). The 24-cell is also 4 heptads (intersecting in 4 orthogonal digon axes, so each heptad contributes 6 unique points to a single fibration of 24 points). Heptads are quarter elevenads, so the 11-cell and 24-cell are both 1 elevenad. ....

See the tabular sequence of convex 4-polytopes in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Build with the blocks|§Build with the blocks]]'', below: table rows ''Inscribed, Pentad, Hexad, Heptad''.

=== 11 of them ===
There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of eleven 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum eleven of them. There is only a single Hopf fibration of the 11 great circles of an 11-cell. 

The 5-cell and the 11-cell are limit cases. The 5-cells are completely disjoint, but the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 10 other instances of itself.

== Cell rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer &amp; Segerman|2013|loc=''Puzzling the 120-cell''}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his section of the Wikipedia article on the 120-cell entitled ''[[120-cell#Visualization|§Visualization]]''.{{Sfn|Ruen &amp; Goucher et al. eds. 120-cell|2024|loc=Goucher's ''[[120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious place to look for the 11-cell's Hopf map is in Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other place it could be hiding. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The quasi-regular convex 11-cell|§The quasi-regular convex 11-cell]]'', and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen: triacontagon|2011|loc=''Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them except his illustrations. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Four orthogonal projections of the 120-cell to a central plane, selected from the table of 3 regular and 11 compound 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|100 icosahedra&lt;br&gt;120 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|5 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 icosahedra&lt;br&gt;6 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|120
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|4
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|50
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|7.2°
|align=center|36°
|align=center|6°
|align=center|7.2°
|}

....

Here we summarize our findings on the 11-cells so far, including in advance some results to be obtained in the next section two sections.

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section....

== The 15 major chords ring everything ==
The natural numbers are each a distinct flavor.

Each integer has its own distinct taste. The first six integers {0..5} are the six strongest, most familiar tastes in nature, instantly distinguished and utterly unique.

The seventh flavor {6} is the first ratio of two of the others. Thereafter follows in strange sequence all the ratios of all the flavors, each marriage of every two flavors, most of which are fractions &lt;math&gt;\{k/d\}&lt;/math&gt;, but some of which are integers &lt;math&gt;\{k\}&lt;/math&gt;.

Some of these ratios are successful flavors in their own right, as distinct and inimitable as the marriage of chocolate and oranges, but the marriage of many flavor pairs is mud. It is a rule of thumb among flavor specialists that nearly any taste can be successfully alloyed with either onions or chocolate, but the direct marriage of these two flavors is notably unsuccessful. Some flavors make famous pairings or infamous ones, and subtle new pairings are always to be discovered that work surprisingly well for some purpose, but the majority of flavor marriages are undistinguished compounds.

In polytope geometry, each chord of a polytope is both is a distinct 1-dimensional object, a chord of the sphere of a distinct length &lt;math&gt;l&lt;/math&gt;, and a distinct natural number &lt;math&gt;h&lt;/math&gt;, a unique flavor. If the polytope is regular, it is a noteworthy and successful flavor. The chord's length &lt;math&gt;l&lt;/math&gt; is a square root, related to the natural number &lt;math&gt;h = k/d&lt;/math&gt; and to a set of polytopes they both represent, by a formula discovered by Steinbach.{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden''; Figure 5. Optimal sections and proportions|p=37|ps=; the regular polygons {5}, {7}, {9} and {11} with their diagonals define respectively: {5} the golden bisection proportional to 𝜙; {7} an analogous trisection; {9} an analogous quadrasection; {11} an analogous pentasection.}} The chord length &lt;math&gt;l&lt;/math&gt; is related to the number of sides of the regular polygon formed by the chord (its numerator &lt;math&gt;k&lt;/math&gt;), and to the winding number or density of its regular skew Petrie polygon (its denominator &lt;math&gt;d&lt;/math&gt;).

The regular 4-polytopes do not contain an infinite variety of these chord flavors, but only 30 of them: only 15 unique pairs of 180° degree complementary chords, which combine to make every regular ''n''-dimensional thing found in nature.{{Efn|The 15 ''major chords'' suffice to construct all the regular ''n''-polytopes. Their 180° complements include 15 ''minor chords'' which occur only in the 120-cell, not in any smaller regular polytope.{{Efn|name=Major and minor chords}} Including the complement of the 180° {{radic|4}} digon {2} chord (the {0} chord {{radic|0}}), and the skew variant of the 90° {{radic|2}} square {4} chord (the {30/7} chord), there are 32 distinct natural numbers which are all the chords of all the regular polytopes in all dimensions.|name=32 chords}} All the regular polytopes (including the largest one on the cover of the box) can be built not only from a box containing a number of blocks of 3 shapes glued together by gravity, they can be also be built from a box containing a number of sticks of 15 lengths glued together by rubber joints.

[[File:15 major chords.png|500px|thumb|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the others occur only in the interior of 120-cell, not as edges of any of the regular 4-polytopes. There are 30 distinct 4-space chordal distances between vertices within the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to identify any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=Major and minor chords}} chords are properly labelled #1 — #15 in the order they fan out by size.{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''; Figure 3|ps=; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with a &quot;fan of chords&quot; diagram like the one here.|p=23}} They join vertex pairs which are 1 — 15 edges apart on a &lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon. A #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; chord forms &lt;small&gt;&lt;math&gt;f+1&lt;/math&gt;&lt;/small&gt; regular geodesic polygons which include a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; disjoint regular {{nowrap|&lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt;-gons}} (e.g. &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; a compound of 5 disjoint hexagons) plus a single [[W:Triacontagon#Triacontagram|regular skew {{nowrap|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram}}]] that wraps around the sphere &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times before closing its circle. Notice the maverick #11 chord, with its non-conforming origin point for its {{nowrap|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;}} polygram.]]

These 15 chords (natural numbers) include the integers 2, 3, 4, 5, 6, 10, 11, 15 and 30. The others are the fractions 5/2, 10/3, 13/7, 15/2, 15/4, and 15/7. Represented among the chord lengths (the square roots) measured in ''unit radii'' (not in {{radic|2}} radii, as metrics are often given elsewhere in this article) are the {{background color|paleturquoise|roots of the first 4 positive integers {{radic|1}}, {{radic|2}}, {{radic|3}} and {{radic|4}}}}, the {{background color|#FFCCCC|root of the very distinguished rational fraction {{radic|5/2}}}}, and the roots of various distinguished irrational fractions as Steinbach discovered: for example, 4 of them are {{background color|yellow|roots of golden numbers (a small integer ± 𝜙)}}. The 15 chords each form their regular polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, which is just &lt;small&gt;&lt;math&gt;\{k\}&lt;/math&gt;&lt;/small&gt; in 8 cases. Arranged in order #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; from shortest edge-length square root to longest, the #1 — #15 chords bridge vertex pairs which are &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; vertices apart on a [[W:Triacontagon#Petrie polygons|&lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon]]. Each chord makes a [[W:Triacontagon#Triacontagram|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; polygram]] of its edge length, a compound of the smaller &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. The 15 distinct polygons form a sequence you would not anticipate exactly unless you have Steinbach's formula:

{| style=&quot;background-color:transparent; white-space:nowrap;&quot;
|-
| {30/1} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|.073}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{30} 
| 120-cell edges
|-
| {30/2}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.191}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15}
|- style=&quot;background: yellow;&quot;|
| {30/3}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.382}} 
| &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi}&lt;/math&gt;&lt;/small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{10}''' 
| 600-cell edges
|-
| {30/4} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.573}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/2}
|- style=&quot;background: paleturquoise;&quot;|
| {30/5}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|1}}'''
| &lt;math&gt;1&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{6}''' 
| 24-cell, 8-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/6}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|1.382}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{5}'''
|- style=&quot;background: paleturquoise;&quot;|
| {30/7}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|2}}'''
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{4}''' 
| 16-cell edges
|- style=&quot;background: #FFCCCC;&quot;|
| {30/8}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.5}} 
| &lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;&lt;small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/4} 
| 5-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/9}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.618}} 
| &lt;math&gt;\phi&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{10/3}
|- style=&quot;background: paleturquoise;&quot;|
| {30/10}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|3}}''' 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{3}'''
|-
| {30/11}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.426}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{11/11}
|- style=&quot;background: yellow;&quot;|
| {30/12}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.618}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{5/2}
|-
| {30/13}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.810}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{13/7}
|-
| {30/14}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.928}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/7}
|- style=&quot;background: paleturquoise;&quot;|
| {30/15}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|4}}''' 
| &lt;math&gt;2&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{2}'''
|}

{| class=wikitable style=&quot;white-space:nowrap; float:right;&quot;
!colspan=6|Natural number &lt;small&gt;&lt;math&gt;h=\{k/d\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Fibers&lt;br&gt;&lt;small&gt;&lt;math&gt;f\{h\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Cells&lt;br&gt;&lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|4-tope&lt;br&gt;&lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Verts&lt;br&gt;&lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|[[W:Triacontagon#Triacontagram|30-gram]]&lt;br&gt;&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;
|-
!Chord
!Arc
!colspan=2|&lt;small&gt;&lt;math&gt;l/\sqrt{1}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;l/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;h&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#1
|15.5~°
|{{radic|0.073}}
|0.270
|{{radic|0.146}}
|&lt;small&gt;&lt;math&gt;30&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1\{30\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;5\{3,3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/1\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#2
|25.2~°
|{{radic|0.191}}
|0.437
|{{radic|0.382}}
|&lt;small&gt;&lt;math&gt;15&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;..\{3,3\}..&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/2\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#3
|36°
|{{radic|0.382}}
|0.618
|{{radic|0.764}}
|&lt;small&gt;&lt;math&gt;10&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;30\{3,3\}20&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/3\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#4
|44.5~°
|{{radic|0.573}}
|0.757
|{{radic|1.146}}
|&lt;small&gt;&lt;math&gt;15/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/4\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5
|60°
|{{radic|1}}
|1
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;6&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/5\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#6
|72°
|{{radic|1.382}}
|1.175
|{{radic|2.624}}
|&lt;small&gt;&lt;math&gt;5&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}} 
|&lt;small&gt;&lt;math&gt;10\{5,3\}12&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/6\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#7
|90°
|{{radic|2}}
|1.414
|{{radic|4}}
|&lt;small&gt;&lt;math&gt;4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7\{4\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;4\{4,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;9\{3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/7\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: #FFCCCC;&quot;|
|#8
|104.5~°
|{{radic|2.5}}
|1.581
|{{radic|5}}
|&lt;small&gt;&lt;math&gt;15/4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/4\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}} 
|&lt;small&gt;&lt;math&gt;15\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/8\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#9
|108°
|{{radic|2.618}}
|1.618
|{{radic|5.236}}
|&lt;small&gt;&lt;math&gt;10/3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10/3\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/9\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#10
|120°
|{{radic|3}}
|1.732
|{{radic|6}}
|&lt;small&gt;&lt;math&gt;3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;10\{3\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/10\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#11
|135.5~°
|{{radic|3.426}}
|1.851~
|{{radic|6.852}}
|&lt;small&gt;&lt;math&gt;11/11&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;11\{11/11\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;11\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#12
|144°
|{{radic|3.618}}
|1.902
|{{radic|7.235}}
|&lt;small&gt;&lt;math&gt;5/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2},3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/12\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#13
|154.8~°
|{{radic|3.810}}
|1.952
|{{radic|7.621}}
|&lt;small&gt;&lt;math&gt;13/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;13\{13/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;7\{5,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/13\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#14
|164.5~°
|{{radic|3.928}}
|1.982
|{{radic|7.857}}
|&lt;small&gt;&lt;math&gt;15/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;137\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/14\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{radic|8}}
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15\{2\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐𝜙&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;8\{3,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/15\}&lt;/math&gt;&lt;/small&gt;
|}

The polygons '''{10}''', '''{6}''', '''{5}''', '''{4}''', '''{3}''', and '''{2}''' are flat polygons that each lie on a great circle in a central plane of the 3-sphere. The polygons {30}, {15}, {15/2}, {15/4}, {10/3}, {11/11}, {5/2}, {13/7}, {15/7} and all the {30/''n''} polygons are skew polygrams spiraling through curved 3-space, each successive edge in a different central plane of the 3-sphere. Projected to the 1-sphere (a circle in the plane), they are [[W:Star polygon#Regular star polygon|regular star polygrams]], the simplest being the pentagram {5/2}. In 4-space they wind around the 3-sphere more than once ''without self-intersecting'', before they close their circuit. They lie on helical geodesic circles or ''isoclines'' of circumference greater than 2𝝅''r''.

Even the 6 integer chords which form flat polygons also form skew polygrams. Each of the 15 chords occurs as a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, a fiber bundle of Clifford parallel polygons that totals 30 vertices per bundle. Each chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; also forms a single skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram that winds through another, shorter chord's bundle of parallel polygons, intersecting them in succession multiple times each, and weaving that whole bundle of parallel fibers together as the cross-thread does on a loom. The polygram cross-thread has to wind multiple times through the whole bundle (&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times around the [[W:Triacontagon#Triacontagram|30-gram]] in which each edge connects every &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt;th vertex) to complete an enumeration of all 30 vertices in the bundle. For example, the #5 chord forms hexagon &lt;small&gt;&lt;math&gt;\{6\}&lt;/math&gt;&lt;/small&gt; flat planes in parallel bundles of 5, written &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt;, and also a skew &lt;small&gt;&lt;math&gt;\{15/2\}&lt;/math&gt;&lt;/small&gt;-gram of #5 edges which winds 2 times through the #3 chord's bundle of 3 decagon &lt;small&gt;&lt;math&gt;\{10\}&lt;/math&gt;&lt;/small&gt; flat planes. Every chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; forms both its &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram and its fiber bundle of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, and the edges of one chord's parallel polygons are also the edges of its skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram weaving through some shorter chord's bundle of parallel polygons. Each numbered chord is a natural number, a ratio of two integers and their polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, either flat or skew. Each chord number defines a &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; skew polygon that is a compound bundle of the &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. All these flat or skew polygons are regular polygons made of their numerator instances of a single chord #, whose length is that square root, Steinbach's function of the natural number.

The relationship between each bundle of flat Clifford parallel great circle polygons of a distinct length, and its corresponding bundle of skew polygrams of a longer length, is precisely that of a discrete simple rotation and its corresponding discrete isoclinic rotation. For example, the flat great circle hexagons of edge length {{radic|1}} lie in the invariant rotation planes of the 24-cell's characteristic 60° isoclinic rotation, in which the vertices move on skew hexagram rotation paths of edge length {{radic|3}}. The 24-cell contains 4 bundles of 4 Clifford parallel (disjoint) hexagon central planes each. The 24 vertices of the 24-cell in each bundle are linked by rotation in the 4 hexagon invariant planes in two different ways: simply and isoclinically. In a simple rotation the vertices of a single invariant hexagon exchange places on that great circle hexagon of edge {{radic|1}}. In an isoclinic rotation in which any one of these 4 disjoint hexagon central planes is invariant, all 4 of them are necessarily invariant planes in the rotation, because they rotate in parallel. In that isoclinic rotation the vertices of all 4 hexagon invariant planes (all 24 vertices of the 24-cell) exchange places concurrently, moving along 4 disjoint helical hexagrams of edge {{radic|3}} in parallel. It is a characteristic only of this distinct rotation in hexagon invariant planes that the helical vertex path or ''isocline'' is a hexagram of period 6, edge length {{radic|3}}, and circumference 720°. Rotations in other kinds of polygon invariant planes have different characteristic isoclines. The period of their isoclinic rotation must be ''the same or greater'' than the period of their simple rotation; in most cases (unlike this one) it is greater.

The 15 chords form not only distinct fiber bundles of polygons, but also distinct bundles of polyhedra: cell rings &lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt; and compound vertex figures &lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;. These combine to form their characteristic regular 4-polytope &lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Coxeter|1973|p=269|loc=Compounds|ps=; &quot;It is remarkable that the vertices of {5, 3, 3} include the vertices of all the other fifteen regular polytopes in four dimensions.&quot;}} All these elements of a regular 4-polytope constructed from one chord are themselves regular polytopes, and each of the 16 regular 4-polytopes is a function of one characteristic chord. If the regular 4-polytope is convex, its characteristic chord is its unit-radius edge. For a non-convex regular 4-polytope, the characteristic chord is its unit-radius ''mid-edge diameter'', the distance from the midpoint of each edge to the midpoint of the completely orthogonal edge. Each of the characteristic chords (each distinct natural number) has its expression as one of the 16 regular 4-polytopes and all the products of the distinct regular 1-polytopes, 2-polytopes and 3-polytopes that construct it, and the whole distinct construction from the natural number to the 4-polytope is an [[W:SO(4)|SO(4) symmetry group]] expression of that 4-polytope's characteristic discrete [[W:SO(4)#Isoclinic rotations|isoclinic rotation]], in the invariant central planes of its characteristic chord, by the characteristic chord arc.

...demonstrate this theory by proper enumeration of the chord lengths in the table...

...color-code the △ ☐ 𝜙 ✩ symbols ... chord colors in diagram should match them ... some of the diagram colors are wrong and must be changed, but otherwise diagram is pretty good ... i think in a colored diagram all chords of the same length should have the same size line dash, the polygons are not obvious enough and the 15 fanning-out chords are readily identified by their labels...

... add back the counts of chords in the box (very small font number under each table entry), and present these two images as the front and back a box of building sticks and rubber joints...What's in the other box...

...say the rational-root major chords {{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|5/2}} are the ''n''-edges: their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge...

...abbreviated text of the [[120-cell#Relationships among interior polytopes|''§Relationships among interior polytopes'']] section from the 120-cell article...to explain the color-coding

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius=1}}

See also the elements, properties and metrics of the sequence of regular 4-polytopes summarized in tabular form in the section ''[[#Build with the blocks|§Build with the blocks]]'', below.

== Alicia Boole Stott's original formulation of dimensional analogy ==
&lt;blockquote&gt;&quot;...a method by which bodies having a certain kind of semi-regularity may be derived from regular bodies in an Euclidean space of any number of dimensions...&quot;{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''; presents cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues|pp=12-45}}&lt;/blockquote&gt;

Alica Boole Stott's 1910 paper reads like an exceptionally well-written patent application. It is a comprehensive stepwise description of her original invention, a general method of dimensional analogy between regular and semiregular polytopes of the same edge length.

[[File:The Roots of Three.png|thumb|An illustration of Boole Stott's expansion of unit-edge triangle to hexagon captures the geometric basis in 3 of the &lt;math&gt;A_n&lt;/math&gt;root system.]]

The engineering-like articulation of her new principles is entirely discrete and group-theoretic. Boole Stott repeatedly refers to sets of polytope elements as ''groups''. All her transformations are discrete instances of one or another special case of her &quot;two inverse operations which may be called ''expansion'' and ''contraction''.&quot; Hers is an exploration of ''n''-dimensional space at its most basic: space defined by the discrete objects which fill it, themselves defined directly as groups of elements with a few operations on them, which transform them into each other. This is geometry which seems to precede Euclid's postulates, and does not depend on them. Instead it corresponds directly to the deeper group theory mathematics which underlies geometry, even though Boole Stott, like the ancients, did not have that mathematics.

Boole Stott enumerated her expansion and contraction operations as transformations between regular and semi-regular polytopes of a single edge length. In her 1910 paper she did not consider any polytopes of multiple edge lengths, or expansions of the subject polytope by a distance other than that exactly required to produce new edges of the original edge length, so of course she did not enumerate Moxness's non-uniform rhombicosidodecahedron (Hull #8). This may be why section 8&lt;sub&gt;3&lt;/sub&gt; is missing from her earlier drawings of the sections of {5,3,3}; she may not have had a construction for it.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The Boole Stott expansion (or contraction) operation is as precisely defined a transformation as Coxeter's reflection, rotation and translation operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions). Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;
where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;
where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;
For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} An instance of the expansion (or contraction) operation is an application of the general operation to a particular subject polytope, operating on all of its elements symmetrically at once. The general operation is restricted, such that one may only expand (or contract) the subject polytope by pushing all its edges, faces or cells directly away from (or toward) its center, without changing the size of the moved elements. This splits each vertex into several (or coalesces several vertices into one), and has corresponding effects on the other elements, producing new edges ''of the same size'' between pairs of separated vertices (or reducing some edges to single vertices). The distance all the elements are moved away from (or toward) the center must be such that the vertices separate producing new edges of only the original edge length (or come together reducing edges to single vertices). It is permitted to combine these three operations, e.g. by pushing the edges ''and'' the faces, or by pushing all three element sets, provided the combination results in a semi-regular polytope of only one edge length. The three operations are all commutative with one another, so combined operations can be considered either as a sequence of expansions (or contractions) linking several related polytopes, or as their concurrent movements, a single transformation of the subject polytope. Since there are 8 combinations of one to three operations there are 8 polytope variants (including the original regular polytope), and all are adjacent to one another by a single transformation. The variants of the original regular polytope are named for which element set(s) are moved. By enumerating all 8 variants of expansion (and contraction) for each of the six regular convex 4-polytopes, Boole Stott produced a table of regular and semi-regular 4-polytopes divided into 12 sets of 8: six expansion sets and six contraction sets.{{Sfn|Boole Stott|1910|loc=''Table of Polytopes in S&lt;sub&gt;4&lt;/sub&gt;''; note that Boole Stott's usage of ''S&lt;sub&gt;4&lt;/sub&gt;'' denotes a Euclidean 4-space ''E&lt;sub&gt;4&lt;/sub&gt;'', or a 3-sphere space embedded in it, ''not'' a 4-sphere space embedded in a Euclidean 5-space|p=30}} Each set includes just one of the six regular convex 4-polytopes, with its distinct unit-radius edge length, and the 7 possible semi-regular expansions (or contractions) of it, which all preserve that characteristic edge length. The polytopes in each set of 8 all have the same total number of elements as their original regular polytope, but each polytope in the set has a distinct combination of cell, face, and vertex figures. The 12 sets are not completely disjoint, and even within one set the 8 operations do not necessarily produce 8 unique polytopes. For example, the regular 5-cell (the 4-simplex polytope) has 5 cells, 10 faces, 10 edges, and 5 vertices (30 elements). Expanding the 5-cell produces 7 related semi-regular 4-polytopes of 30 elements each (two of which happen to be the same polytope). In each expansion, the 5 cells have expanded into semi-regular or regular polyhedra, the 10 faces and 10 edges ''may'' have expanded into prisms, and the 5 original vertices have expanded into regular or semi-regular cells.

== The regular 11-cell 4-polytope ==

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosadodecahedron). Each non-convex Jessen's icosahedron is a set of 6 reflex edges that are {{radic|5}} 5-cell edges, each set symmetrically arranged in 3 parallel pairs, not as far apart ({{radic|3}}) as they are long, in 3 orthogonal central planes. The 12-point (Jessen's sets of 6 reflex 5-cell edges) are inscribed in the 60-point with their endpoints as a vertex of a pentagon face. The 12 pentagon faces are joined to 5 other pentagon faces by the 5-cell edges; not to their nearest neighbor pentagons but to their next-to-nearest neighbor pentagons; and since there are 5 vertices to each pentagon face, there are 10 12-point (Jessen's sets of 6 reflex 5-cell edges). Each pentagon face is joined to another pentagon face by 5 reflex edges of five distinct Jessen's. The 5 reflex edges are Clifford parallel, the 5 edges of a pentagonal prism column that has a twist to it. The contraction of those 5 edges into 1, and those 10 vertices (two pentagons) into 2, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing all their edges as a single Jessen's. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (octahedron) by conflating its parallel pairs of edges, leaving the 11-cell's abstract 6-point (hexad cell). The hexad's situation is that it occurs in bundles of 5, disjoint but close together in the sense of adjacent, like the great circle rings in a Hopf fiber bundle. 

To summarize, the 12-point (2-hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell). We shall now be able to see how Fuller's perfect hexads combine with Todd's perfect pentads to form the perfect heptads of the 11-cell and 137-cell.  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=alias the hemi-cuboctahedron}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces. (The ''heptad'' arrives at the party, anonymously; they are our distinguished anti-celebrity, who most of the other guests have never even heard of.) The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. This 12-point (truncated tetrahedron) is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 100 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp: Truncated tetrahedron|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp: Truncated tetrahedron|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}} Of course Moxness has been there, seen that, already with his graphics software;{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} see his parameterized illustration of it exactly as it occurs in the 11-cell, below in his bestiary of A3, B3, and H3 Archimedian and Catalan polyhedra in the 120-cell.]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:Archimedean_and_Catalan_solid_hulls_with_their_Weyl_orbit_definitions.svg|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of the 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. In this catalog of Archimedean and Catalan polyhedra sighted in the 120-cell by Moxness's software,{{Sfn|Moxness: Archimedean and Catalan hulls|2023|loc=Hull #1 Archimedean Name A3 110 Truncated Tetrahedron A (upper left)}} it is the first entry (upper left), Moxness's Hull # = 1 with 12 vertices of 3D norm {{radic|11}}/2. On a {{radic|2}}-radius 3-sphere, 5 of these with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell. 20 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges are cells in the [[#The rings of the 11-cells|cell rings of the quasi-regular 11-cell]], which also has 12 regular 5-cells as cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

...

Both kinds of cells in the 11-cell have expressions in two different dimensions (on the 2-sphere and 3-sphere), because they are all centered on the center of the 3-sphere, unlike the cells of most 4-polytopes, which are centered on various off-center points within the 4-polytope's interior. 

One kind of 11-cell cell appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (semi-regular truncated cuboctahedron), a.k.a. the truncated tetrahedron. The other kind of cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now, except in the 120-cell. But this is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron, and a regular tetrahedron behind each face. The 11-cell is thus a honeycomb of mixed regular tetrahedra and truncated cuboctahedra (a.k.a. truncated tetrahedra) that meet at their shared 5-cell faces. 

Both the 5-cells (decahedra) and the truncated cuboctahedra (truncated tetrahedra) are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks in their 4-polytopes (off-center).{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} Therefore all the 11-cell's cells are concentric, which indeed makes a different kind of honeycomb, an inside-out cellular honeycomb with all the cells piled on top of each other overlapping in 4-space. They do not overlap in 3-sphere space however; their 3-volumes (but not their 4-contents) are disjoint. They make a proper 4D honeycomb like the cells of any 4-polytope, with volumetrically disjoint 3D cells that meet each other in the proper way at their shared 2D faces.

An 11-cell contains 6 completely disjoint 5-point (1-pentad 5-cell decahedron cells), and 5 volumetrically disjoint 12-point (2-hexad truncated cuboctahedron cells), 11 volumetrically disjoint cells altogether. Of course it is supposed to have 11 identical 11-point (elevenad hemi-icosahedral cells), and that is also true in its abstract sense, which is not the same sense as disjoint cellular containment: each pentad or hexad cell is inscribed in one of 11 larger 60-point (Moxness's rhombicosadodecahedron cells). The abstract pentads and hexads are only hemi-polyhedra, so the 11-cell's pentad cells are 12 (out of 120) disjoint instances of a pentad 5-point, and its hexad cells are 10 (out of 100) disjoint instances of a hexad 6-point, 1/10th of the 120-cell's cellular volume in both cases. The 11-cell's hemi-icosahedral cells are 11 (out of 120) non-disjoint instances of an elevenad 60-point. Each 11-cell shares each of its hexad cells with 9 other 11-cells, and each of its 11 vertices (out of 600) occurs in 9 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.|name=Tetrahedral vertex figures}}

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

...insert image thumb of 11-cell...

...Construct the 11-cell from the heptad....

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...Four concentric 4-polytopes may be mutually orthogonal. In any set of 4 mutually orthogonal regular 5-cells, one pair of faces in each orthogonal pair are completely orthogonal to each other. Thus three edges in one 5-cell in an orthogonal pair are opposite (completely orthogonal) to a face of the orthogonal 5-cell. .... A 5-cell has 5 vertices in its abstract hemi-icosahedral cells, but only 4 vertices in one concrete icosahedral cell; the 5th lies in an icosahedral cell of another 11-cell (not in another icosahedral cell of the same 11-cell).

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

....

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 11-cell ===

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon.... (wrong, 11-cell's Hopf map is the truncated icosahedron)

....

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk: Truncated icosahedron|2018}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

...

Since the rhombicosadodecahedron has 120 5-cell edges, the 120-cell has 120 completely disjoint 5-cells, and the rhombicosadodecahedron shares each vertex with 11 other rhombicosadodecahedra, how a rhombicosadodecahedron is bound to its neighbor 5-cells is constrained. A further constraint is that the abstract hemi-icosahedron has only 15 5-cell edges, so the 120 actual 5-cell edges of the rhombicosadodecahedron are conflated 8-to-1 in the hemi-icosahedron. To be conflated by contraction into a single edge, those 8 edges must lie parallel to each other in the 120-cell; either actually parallel in the same plane, or Clifford parallel in separate planes, as two edges in completely orthogonal planes may be parallel and orthogonal at the same time. Each 5-cell inscribed in the 120-cell contains 5 such tetrahedra occur in parallel pairs in 4 orthogonal planes (none of them completely orthogonal planes). 

.... 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
Each conflated vertex is the 10 vertices of two orthogonal regular 5-cells...
The 60-point contains 6 pairs of these orthogonal regular 5-cell cells...
...its 20 hexad cells are ..., a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere.

== The regular 137-cell 4-polytope ==

(description originally written for the 11-cell, which applies only to a fibration of 11 11-cells)

It is quasi-regular with two kinds of cells, the 12-point (regular icosahedron) and the 5-point (regular 5-cell), which have just one kind of regular face between them. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of regular facets (it is semi-regular with regular facets), it also means that its facets have expressions of two different dimensionalities because they are concentric in the 3-sphere. The 11-cell has two kinds of regular cells. One appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (regular icosahedron), but since it is embedded at the center of the 3-sphere (unlike the cells of most 4-polytopes) it also acts as an inscribed 4-polytope.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} The other cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now except in the 120-cell. This is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron. The 11-cell is a honeycomb of mixed icosahedron and half-icosahedron cells. Like the icosahedra, the decahedra are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks, off-center in their 4-polytopes. All the 11-cell's cells are concentric, which indeed makes an unusual honeycomb of cells, an inside-out honeycomb. It is a proper 4D honeycomb however, with volumetrically disjoint 3D cells that meet each other in the proper way at shared 2D faces. The embedded 12-point (regular icosahedron) cell is an icosahedral pyramid (as a 4-polytope), which is filled by 20 4-point (irregular tetrahedra) that meet at its center; the center point of the 11-cell acts as a 12th vertex, just as the center point of the 24-cell acts as a 25th vertex in some honeycombs. The 5-point (regular 5-cell) cell contains five 4-point (regular tetrahedra) which meet the 12-point (regular icosahedra)'s irregular tetrahedral cells face-to-face, so the 11-cell is quasi-regular in the usual sense of having two varieties of regular 3-polytope cells, but additionally both varieties of of its cells are bi-dimensional: another clue that the 11-cell has something to do with dimensional analogy

....

The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

....

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur to make an 11-cell. 11 of them occur in a 137-cell, all 120 of them occur in the 120-cell, and they occur nowhere else.

How can this regular 4-polytope possibly exist? By Schläfli's criterion {3, 3, ..} exists, but not as a finite polytope.{{Sfn|Coxeter|1973|loc=§7.7 Schläfli's criterion}} Coxeter found {3, 5, 3} as a finite hyperbolic honeycomb, but not in Euclidean 4-space.{{Sfn|Coxeter|1968|title=The Beauty of Geometry: Twelve Geometric Essays|loc=''Regular Honeycombs in Hyperbolic Space''|ps=; &quot;The honeycombs considered by the above authors [Schlegel, Schläfli, Klein, Sommerville] have finite cells and finite vertex figures. ... A further extension allows a cell or vertex figure to be a ''star''-polytope, so that the honeycomb covers the space several times. .... We shall find that there are four regular star-honeycombs in hyperbolic 4-space, as well as two infinite families of them in the hyperbolic plane.&quot;}} We find that {3, 3, ..} is a central expression of the &lt;math&gt;H_4&lt;/math&gt; root system, being the regular convex hull in 4 dimensions of 137 points, 96 pentads and 600 hexads. The minimal expression of &lt;math&gt;H_4&lt;/math&gt; is the regular convex hull {3, 3, 5} of 120 points, 600 tetrads and 75 hexads, and its maximal expression is the regular convex hull {5, 3, 3} of 600 points, 120 pentads and 675 hexads. {3, 3, ..} is a third regular &lt;math&gt;H_4&lt;/math&gt; polytope which fits between them. The properties and metrics of this regular sequence are presented in tabular form below in ''[[#Build with the blocks|§Build with the blocks]]''.

{|class=&quot;wikitable&quot;
!colspan=3|Regular &lt;math&gt;H_4&lt;/math&gt; polytopes
|-
|[[File:600-cell.gif|200px]]
|[[File:5-cell.gif|200px]]
|[[File:120-cell.gif|200px]]
|-
!120-point 600-cell
!137-point 137-cell
!600-point 120-cell
|}

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 137-cell ===
....

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=Apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|2022|loc=''[[Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]''}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie: On Fuller's use of language|2024|loc=''[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry'']]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center, 1948-1949|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with their mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the gender neutral shad can open their six beaks all the way, until they become flat squares and they becomes a cuboctahedron, or they can shut them all tight like a turtle retracting into their octahedron shell. The six mouths always move in unison. This is [[Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do their ''really'' odd trick -- where they flip their 6 jaws 90 degrees in their 6 faces and shut their 6 beaks on the opposite axis of their squares than the one they opened them on -- or whether they will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than their normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make their edge length exactly the same size as their radius, when they open their mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing their mouths in spherical synchrony, their 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron (the hexad), then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=''[[W:SO(4)|SO(4)]]''}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=Isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its orthogonal hexad's planes (wz, wx, wy).{{Efn|name=Six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of (expanded octahederon 16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh &amp; Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== Legendre's 12-point hexad binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a 12-point double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's perfectly realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, this 2-hexad expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre hexad) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction.{{Efn|name=Concentric polyhedra}} In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== ''n''-simplexes ===
....

The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== ''n''-orthoplexes ===
....
[[File:Great 5-cell √5 digons rectangle.png|thumb|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} ''regular'' great hexagons. These are the same 200 dodecagon central planes illustrated above, which also contain 6 120-cell edges and two opposing ''irregular'' great hexagons (truncated {{radic|6}} triangles). The 1200 {{radic|5}} 5-cell face triangles occupy 1200 other, non-central planes.]]

...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's Hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[16--cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[16--cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=Concentric polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[16--cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== ''n''-taliesins ===
The 4-simplex pentad building block combines with the 4-orthoplex hexad building block to make the 4-taliesin heptad building block. The heptad is the essential building block of the ''taliesin'' family of polytopes. The 11-cell and 57-cell hemi-polytopes and the 137-cell and 120-cell regular 4-polytopes are nuclear members of this trans-dimensional family. The 24-cell and 600-cell are extended members of it, because they also contain instances of the heptad building block.

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its spherical symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* [[W:Taliesin Myrddin Namkai-Meche|Taliesin Myrddin Namkai-Meche]], a young Reed College scholar who was stabbed to death by a crazy man in 2017 on a Portland, Oregon light rail train, when he rose to the nonviolent defense of two black teenagers who were being violently assaulted in racist and anti-Muslim language by a right-wing extremist.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in orthogonal [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].
* '''Taliesan polytope''', an (''n+1)''-polytope whose ''n''-polytope dimensional analogue inscribed concentrically in the ''n''-sphere forms a convex honeycomb of itself which is the (''n+1)''-polytope.{{Efn|name=Embedding point}}
* The taliesan polytope's honeycomb, a Hopf fibration of its great circles.
* The ''n''-dimensional family of dimensionally analogous polytopes related to a taliesan polytope by expansion and contraction operations.
* The characteristic [[w:Orthoscheme|orthoscheme]] of a teliesan polytope, an irregular simplex that is the complete gene for the entire expression of trans-dimensional symmetry relationships among the polytopes of the ''n''-dimensional family, as well as the complete gene for the teliesan polytope's rotational symmetry.{{Efn|name=Characteristic orthoscheme}}

=== ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== ''n''-equilaterals ===
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed at four different [[#n-leviathon|chord]] lengths from each other. In the {{radic|2}}-radius polytopes we have using as examples in this paper they are {{radic|2}}, {{radic|4}}, {{radic|6}} and {{radic|8}}, but in a ''unit-radius'' polytope they are: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}.

Each vertex is joined to 8 others by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}. The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== ''n''-pentagonals ===

[[File:600-cell.gif|thumb|The 120-point (600-cell), &lt;math&gt;H^4&lt;/math&gt;.]]
...

[[File:600-cell vertex geometry.png|thumb|Vertex geometry of the 600-cell, showing the 5 regular great circle polygons and the 8 vertex-to-vertex chord lengths with angles of arc. The golden ratio{{Efn|name=golden chords|group=}} governs the fractional roots of every other chord,{{Efn|name=fractional root chords}} and the radial golden triangles which meet at the center.|alt=|400x400px]]
The 120 vertices of the 600-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏&lt;sup&gt;−1&lt;/sup&gt;) beginning with a vertex; see column ''a''}} at eight different [[#n-leviathon|chord]] lengths from each other. These edges and chords of the 600-cell are simply the edges and chords of its five great circle polygons. In ascending order of length in a ''unit-radius'' polytope (not in a {{radic|2}} radius polytopes like our usual examples), they are {{radic|0.𝚫}}, {{radic|1}}, {{radic|1.𝚫}}, {{radic|2}}, {{radic|2.𝚽}}, {{radic|3}}, {{radic|3.𝚽}}, and {{radic|4}}.{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=Decimal fractions of square roots|group=}}

Notice that the four [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#''n-''equilaterals|hypercubic chords]] of the 24-cell ({{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|4}}) alternate with the four new chords of the 600-cell's additional great circles, the decagons and pentagons. The new ''golden chord'' lengths are necessarily square roots of fractions, but very special fractions related to the golden ratio{{Efn|1=The fractional-root ''golden chords'' are irrational fractions that are functions of {{radic|5}}. They exemplify that the [[W:golden ratio|golden ratio]] &lt;big&gt;𝜙&lt;/big&gt; {{=}} {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618 is a circle ratio related to &lt;big&gt;𝜋&lt;/big&gt;:&lt;br&gt;
: {{sfrac|𝜋|5}} = arccos (𝜙/2)
is one decagon edge, the 𝚽 = {{radic|0.𝚫}} = {{radic|0.382~}} ≈ 0.618 chord.
Reciprocally, in this function discovered by Robert Everest expressing &lt;big&gt;𝜙&lt;/big&gt; as a function of &lt;big&gt;𝜋&lt;/big&gt; and the numbers 1, 2, 3 and 5 of the Fibonacci series:&lt;br&gt;
: &lt;big&gt;𝜙&lt;/big&gt; = 1 – 2 cos ({{sfrac|3𝜋|5}})
{{sfrac|3𝜋|5}} is the arc length of the &lt;big&gt;𝜙&lt;/big&gt; = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618 chord.|name=golden chords|group=}} including the two [[W:golden section|golden section]]s of {{radic|5}}, as shown in the diagram.{{Efn|The 600-cell edges are decagon edges of length {{radic|0.𝚫}}, which is 𝚽, the ''smaller'' golden section of {{radic|5}}; the edges are in the inverse [[W:golden ratio|golden ratio]] {{sfrac|1|φ}} to the {{radic|1}} hexagon chords (the 24-cell edges).
The other fractional-root chords exhibit golden relationships as well. The chord of length {{radic|1.𝚫}} is a pentagon edge.
The next fractional-root chord is a decagon diagonal of length {{radic|2.𝚽}} which is &lt;big&gt;𝜙&lt;/big&gt;, the ''larger'' golden section of {{radic|5}}; it is in the golden ratio{{Efn|name=golden chords|group=}} to the {{radic|1}} chord (and the radius).{{Efn|Notice in the diagram how the &lt;big&gt;𝜙&lt;/big&gt; chord (the ''larger'' golden section) sums with the adjacent 𝚽 edge (the ''smaller'' golden section) to {{radic|5}}, as if together they were a {{radic|5}} chord bent to fit inside the {{radic|4}} diameter.}}
The last fractional-root chord is the pentagon diagonal of length {{radic|3.𝚽}}.
The [[W:Pentagon#Side length is given|diagonal of a regular pentagon]] is always in the golden ratio to its edge, and indeed &lt;big&gt;𝜙&lt;/big&gt;{{radic|1.𝚫}} is {{radic|3.𝚽}}.|name=fractional root chords|group=}}

....


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== ''n''-leviathon ===
...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of its &lt;math&gt;10^2&lt;/math&gt; disjoint instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction, before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]].{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}} .... It is only because she did not have the 11-cell to start with that she did not discover the regular 137-cell {3, 3, ..} by her method.

Between Alicia Boole Stott and her identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and her identification of symmetry groups with conservation laws (in her 1st theorem), we owe the deepest mathematics of geometry and physics originally to two pioneering women, colleagues of their contemporaries the greatest men of the academy, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just an [[W:abstract polytope|abstract 4-polytope]], not just a singleton convex 4-polytope, and not just a honeycomb.{{Sfn|Coxeter|1970|loc=''Twisted Honeycombs''}} The 11-point (11-cell) has a concrete ...regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}&lt;nowiki/&gt;} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, 3, 3} as the 137-point (137-cell).

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 11-cell, 137-cell and 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=''Clifford Spinors and Root System Induction: H4 and the Grand Antiprism''}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=Apology}}

== Build with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Duveneck|1978|loc=Carl Jung, quoted in ''Life on Two Levels''|p=ii|ps=.{{Sfn|Jung|1961|ps=: &quot;The best of truths is of no use unless it has become one's most personal inner experience. It is the duty of everyone who takes a solitary path to share with society what he finds on his journey of discovery.&quot;}}}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1954|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius={{radic|2}}}}

== Acknowledgements ==
...

== Notes ==
{{Regular convex 4-polytopes Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Citation|last=Boole Stott|first=Alicia|author-link=W:Alicia Boole Stott|title=Geometrical deduction of semiregular from regular polytopes and space fillings|date=1910|pages=12-45|publisher=Johannes Muller|place=Amsterdam|url=https://dwc.knaw.nl/DL/publications/PU00011492.pdf}}
* {{Citation|last=Schoute|first=Pieter Hendrik|author-link=W:Pieter Hendrik Schoute|title=Analytical treatment of the polytopes regularly derived from the regular  polytopes|date=1911|pages=114-197|publisher=Johannes Muller|place=Amsterdam|url=https://www.google.com/books/edition/Analytical_Treatment_of_the_Polytopes_Re/G_oxAQAAMAAJ?hl=en&amp;gbpv=0}}
* {{Citation | last=Grünbaum | first=Branko | author-link=W:Branko Grünbaum | year=1976 | title=Regularity of Graphs, Complexes and Designs | journal=Colloques Internationaux C.N.R.S. | publisher=Orsay | volume=260 | pages=191-197 | url=https://faculty.washington.edu/moishe/branko/BG111.Regularity%20of%20graphs,etc.pdf | ref=}}
* {{Citation | last=Grünbaum | first=Branko | author-link=W:Branko Grünbaum | year=1975 | title=Venn Diagrams and Independent Families of Sets | journal=Mathematics Magazine | volume=48 | issue=1 | url=https://maa.org/sites/default/files/pdf/upload_library/22/Ford/BrankoGrunbaum.pdf | doi=10.1080/0025570X.1975.11976431 }}
* {{Citation | last=Coxeter | first=H.S.M. | author-link=W:Harold Scott MacDonald Coxeter | year=1984 | title=A Symmetrical Arrangement of Eleven Hemi-Icosahedra | journal=Annals of Discrete Mathematics (20): Convexity and graph theory | series=North-Holland Mathematics Studies | publisher=North-Holland | volume=87 | pages=103-114 | doi=10.1016/S0304-0208(08)72814-7 | url=https://www.sciencedirect.com/science/article/pii/S0304020808728147
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      <text bytes="208627" sha1="e60laa7r5pym8j1yiijoci350os3121" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|June 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a regular 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the &lt;s&gt;120-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 5, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 11 [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] cells. Eleven 11-cells (plural) are the 137-cell, the &lt;s&gt;600-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 3, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 137 [[W:Rhombic triacontahedron|triacontahedron]] cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=''Regularity of Graphs, Complexes and Designs''}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=''A Symmetrical Arrangement of Eleven Hemi-Icosahedra''}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]]
The 4-dimensional regular polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:Simplex|simplex]] (called the [[5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:Orthoplex|orthoplex]] (called the [[16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular convex 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the [[w:Tesseract|16-point (8-cell) tesseract]], the 4-cubic, which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found within any of the other regular convex 4-polytopes, except in the largest and most complex one, the [[120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). However, we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the 16-cell is their very starting point, and the most frequently used tool in the box.

== 5-cells and hemi-icosahedra in the 11-cell ==
[[File:Symmetrical_5-set_Venn_diagram.svg|thumb|The 5-point (10-face) regular 5-cell (the regular 4-simplex). Grünbaum's rotationally symmetrical 5-set Venn diagram{{Sfn|Grünbaum|1975|loc=''Rotationally symmetrical 5-set Venn diagram'', Fig 1 (e)|ps=; partitions the individual elements of the 5-cell.}} is an illustration of the 5-cell labeling each of its &lt;math&gt;2^5&lt;/math&gt; elements.{{Sfn|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled; image includes the Python code to render it, optimising for maximum area of the smallest regions.}}]]
[[File:Hemi-icosahedron.png|thumb|The 6-point (10 face) [[W:hemi-icosahedron|hemi-icosahedron]], an abstraction of the regular icosahedron, has half as many faces, edges and vertices. Each element of the abstract polyhedron represents two real elements found in different places in the 11-cell's realization.{{Sfn|Ruen: hemi-icosahedron|2007}}]]
The most apparent relationship between the pentad 5-cell and the hexad hemi-icosahedron is that they both have 10 triangular faces. When we see a facet congruence between a 4-polytope and a 3-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 6-point 5-simplex. Another hexad is the 6-point 3-orthoplex; thus the hemi-icosahedron is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no chord congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals, 5-cell and 16-cell, who are famously stand-offish with each other. Interesting!

The cell of the 11-cell is an abstract hexad hemi-icosahedron with 5 central planes, most handsomely illustrated by Séquin.{{Sfn|Séquin &amp; Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing rings of polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The 11 hemi-icosahedral cells have 10 triangle faces each; the 5-cell's 5 tetrahedral cells have 10 faces and 10 edges all together. If 11-cell faces are 5-cell faces, then 3 of each 5-cell's 5 vertices are a hemi-icosahedron face, and its other 2 vertices must be some 11-cell edge lying opposite the face. Coxeter determined that the 11-cell does indeed have an edge opposite each face, but it does not belong to the same hemi-icosahedral cell as its opposing face. He found that the 10 edges opposite each hemi-icosahedron's faces are the 10 edges of a single 5-cell, which does not share any vertices, edges or faces with the hemi-icosahedron. Of course, in the 11-point 11-cell there is just one such 5-point 5-cell that is completely disjoint from each 6-point hemi-icosahedron.{{Sfn|Coxeter|1984|p=110|loc=§6. The Petrie polygon [of the 11-cell]|ps=; &quot;We may reasonably call this edge and face ''opposites''. It is easy to find the face opposite to a given edge by looking at the faces to which a given edge belongs. ... Conversely, given a face, we can find the opposite edge by seeing which vertices belong to neither of the hemi-icosahedra which share that face. The ten edges opposite to the ten faces of one hemi-icosahedron are the edges of the complementary &lt;math&gt;a_4&lt;/math&gt; [4-simplex], that is, the joins of all pairs of the five vertices [of the 11-cell] not belonging to the given hemi-icosahedron.&quot;}}

There are 11 disjoint 5-cell 4-polytopes inscribed in each 11-cell, which also contains 11 hemi-icosahedral cells, 55 faces, 55 edges and 11 vertices. The real 11-cell is more complex and harder to see than the abstract 11-cell representing it, because the real hemi-icosahedron is also more complex than the abstract hemi-icosahedron. Seeing the real 11-cell will be easier once we have identified the real hemi-icosahedron, and exactly where the 11-cell's real elements reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each hemi-icosahedron's 10 faces come from 10 disjoint 5-cells, and there is just one 5-cell with which it does not share a face.{{Efn|As Coxeter observes (in the previous citation), that unrepresented 5-point (5-cell) is the other 5 vertices of the 11-point (11-cell) that are not vertices of this 6-point (hemi-icosahedron), the hemi-icosahedron's disjoint complement.}} Each 5-cell has 10 faces from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share a face.

In the 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 disjoint 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent hemi-icosahedron faces do not actually meet at an edge; there is a narrow rectangle separating them, which has been abstracted to an edge. The 5-cells are all completely disjoint from each other.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two faces in two different planes which are completely orthogonal to each other. Each 11-cell face bonds to two opposing 5-cells, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must be sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness: 8 concentric hulls|2022|loc=Hull #8 (lower right)|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell of the 11-cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Compounds in the 120-cell|§Compounds in the 120-cell]]'' that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software which uses Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s to render the polyhedra which are found in the interior of ''n''-dimensional polytopes.{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the tetrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is a real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (Hull #8) is a concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this {12} irregular great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing {6} irregular great hexagons (truncated triangles) of alternating 120-cell edges and 5-cell edges. The great dodecagon is a great circle of the hemi-icosahedral cell, and also a great circle of the 120-cell, because unlike the cells of most 4-polytopes, the 11-cell's cells are embedded in 4-space at the exact center of the 4-polytope, making them inscribed 4-polytopes themselves.{{Efn|name=Embedding point}} There are 25 great dodecagons and 120 5-cell edges in Moxness's 60-point (Hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (Hull #8) polyhedron altogether.

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point (Hull #8) is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] similar to the one from the catalog shown here,{{Sfn|Piesk: Rhombicosidodecahedron|2018}} but a slightly shallower truncation with smaller red pentagons (120-cell faces) and narrower rhombs. Rhombicosidodecahedra are made by truncating the [[W:Rhombic triacontahedron|rhombic triacontahedron]], which is the 30-sided polyhedron with only one kind of face, the dual of the 30-point icosidodecahedron. The 120-cell contains 120 of these 11-cell cells.]]
Moxness's 60-point (Hull #8) is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (Hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|The 5-point (regular 5-cell) can be another abstraction of Moxness's 60-point (Hull #8), 12-vertices-into-1, but it is not strictly the result of a contraction operation as described by Boole Stott, since she did not apply her expansion and contraction operations to uniform polytopes with more than one edge length.}} In fact Moxness was not the first to find a rhombicosidodecahedron in the 120-cell; Alicia Boole Stott was. It is the 6th section of {5,3,3} beginning with a cell, which Boole Stott identified as the ''e&lt;sub&gt;2&lt;/sub&gt; expansion'' of the icosahedron (or equivalently of its dual polyhedron the dodecahedron).{{Sfn|Boole Stott|1910|loc=§Examples of the e&lt;sub&gt;2&lt;/sub&gt; expansion|p=7}} But that rhombicosidodecahedron identified by Boole Stott is not Moxness's Hull #8: rather, it is the semi-regular Archimedean solid, with a single edge length and square faces. Moxness's Hull #8 with its two distinct edge lengths and rectangular faces is Coxeter's 8th section of {5,3,3} beginning with a cell, which is missing from the sections illustrated by Boole Stott.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than tetrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. The 120-cell contains 120 of Moxness's 60-point (11-cell cells). They are non-disjoint, with 12 60-points sharing each vertex.

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (Hull #8) is the real hemi-icosahedron cell, with 120 5-cell edges in it as chords lurking under its surface. The rhombicosadodecahedron has enough of those 5-cell edges in its interior to make 20 disjoint tetrahedral cells which belong to 20 distinct 5-cells.

The 600-point (120-cell) may be constructed as a semi-regular honeycomb of two kinds of cells, the 120 non-disjoint 60-point (Moxness's rhombicosadodecahedra) and their neighbors the 120 disjoint 5-point (5-cells), cell-bonded together by shared tetrahedral cells of edge-length {{radic|5}}. Each rhombicosadodecahedron is bonded to 20 islanded 5-cells, sharing with each a 5-cell tetrahedron (4 of its interior 5-cell faces, and 6 of its interior 5-cell edges). The rhombicosadodecahedron is also bonded directly to 10 other rhombicosadodecahedra (since each 11-cell has 11 rhombicosadodecahedron cells), again not by sharing one of its exterior faces, but this time by sharing an interior face, a {{radic|6}} triangle which lies in a central plane.{{Efn|The {{radic|6}} chord is another hidden chord lurking beneath the 60-point rhombicosadodecahedron's surface, like the {{radic|5}} 5-cell edge. Both these hidden chords lie on the same great circle with the 120-cell edges (the pentagon face edges). Three {{radic|6}} chords make a triangle which lies in a central plane. This {{radic|6}} great triangle must not be confused with the {{radic|5}} face triangle of the 5-cell, which does not lie in a central plane.}} The 120 rhombicosadodecahedra are also bonded directly together at ''all'' their exterior faces: at their pentagon faces (which are 120-cell faces of dodecahedral cells); at their small triangular faces (which are 600-cell faces of small tetrahedral cells, 10 of which are inscribed in each dodecahedral cell); and at their rectangular faces (which are faces of 10-point [[w:Pentagonal_prism|pentagonal prism]] cells, where the pentagonal prism is a column inscribed in a dodecahedral cell, joining its opposite pentagon faces). The 120 rhombicosadodecahedra are bonded directly to each other at their own exterior faces; no dodecahedra, small tetrahedra, or pentagonal prisms lie between them, and those objects are not cells of this honeycomb; we mention them here only to help locate the rhombicosadodecahedra within the 120-cell. The only objects which fill holes between the rhombicosadodecahedra are the 120 islanded 5-cells.{{Efn|This honeycomb of the 120-cell resembles the [[600-cell#Icosahedra|honeycomb of the 600-cell consisting of 24 face-bonded regular icosahedra]] with 24 clusters of 5 tetrahedra filling the holes between them. In the 120-cell there are 5 times as many of everything as in the 600-cell, so there are 120 (rhombicosadodecahedron) icosahedral cells with 600 tetrahedra filling the holes between them.}}

It is possible for rhombicosadodecahedra to meet each other at more than one kind of exterior face because they are not meeting there as cells of the same 11-cell; they belong to different 11-cells. The 11 rhombicosadodecahedra in the same 11-cell do not meet at their own exterior faces, they meet at their interior {{radic|6}} triangle faces. As a cellular 4-polytope, the 11-cell is inside-out. Its cells meet each other at interior planes, sharing interior {{radic|6}} triangle faces, instead of in the usual way that 4-polytopes' cells share an exterior face. It is a deeper kind of intertwining to be sure, but the 11 rhombicosadodecahedra in the same 11-cell are still volumetrically disjoint cells, and the 11-cell is a perfectly real 4-polytope with real cells. It is a semi-regular 4-polytope, because it has two different kinds of cells: 11 rhombicosadodecahedra and 11 5-cells. The 11 5-cells are islanded (not bound to each other), and the 11 rhombicosadodecahedra are bound both to 5-cells (at {{radic|5}} 5-cell faces) and to each other (at {{radic|6}} central triangle faces). In the section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#What's in the box|§What's in the box]]'', below, we will look closely at the relation between the {{radic|5}} chord and the {{radic|6}} chord, which is the relation between the pentad and the hexad, and at the general phenomenon of hemi-polytopes which meet each other at a central plane. Then in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Cell rings of the 11-cells|§Cell rings of the 11-cells]]'' we will look at the two-cell cellular structure of the 11-cell in more detail.

Considered as a single complex of overlapping cells rather than as distinct honeycombs of volumetrically disjoint cells, the 120-cell contains 60-point (rhombicosadodecahedral cells), islanded 5-point (5-cells), 20-point (dodecahedral cells), and 10-point ([[w:Pentagonal_prism|pentagonal prism]] cells). The 60-points meet each other in pairs at their pentagon faces, meet regular dodecahedra at those pentagon faces also, meet each other in quads at their 600-cell faces where they share an interior 4-point (600-cell tetrahedral cell) and its four faces (just as they share a larger interior 5-cell tetrahedron and its faces), and meet each other and the 10-point (pentagonal prisms) at their rectangle faces. We might try to imagine such a complex with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with parts truncated from 30 neighboring truncated dodecahedra pentagonal prisms). There are 120 of each of these 3 kinds of dodecahedron. 

Moxness's 60-point (rhombicosadodecahedron cell) is a central object of the 120-cell, and contains many wonders. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra by expansion and contraction operations, as Alicia Boole Stott discovered before 1910,{{Sfn|Polo-Blanco: ''Theory and history of geometric models of Alicia Boole Stott''|2007|loc=§5.3.2 1910 paper on semi-regular polytopes|pp=152-158|ps=; summarizes Boole Stott's method and results from {{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}}, including her tables of expansion-contraction dimensional analogies and a few of her illustrations.}} and those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, rediscovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter &amp; Petrie|1938|p=4|loc=''[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]''}}|name=Concentric polyhedra}} Before we move on from the 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice one more thing about the hidden {{radic|5}} chords lurking below its surface. The 12 little pentagon faces are spanned by sets of 6 {{radic|5}} chords (5-cell edges). The 6 chords of each set are disjoint (they don't touch or form 5-cell faces), but they are symmetrically arranged in 3 parallel pairs, {{radic|3}} apart, which lie in 3 orthogonal central planes. It turns out they are the 6 reflex edges of a 12-point non-convex polyhedron called the [[w:Jessen's_icosahedron|Jessen's icosahedron]], inscribed in the 60-point cell. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again and work our way outwards, before eventually in ''[[#The perfection of Fuller's cyclic design|§The perfection of Fuller's cyclic design]]'' we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cell)s ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks, showing its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Pentahemidemicube.png|120px]]
|[[File:Tetrahemihexahedron_rotation.gif|120px]]
|[[File:Pentahemicosahedron.png|120px]]
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts needed to build the 120-cell (the picture on the cover of the box), and everything else that's invisible in that picture which fits inside the 120-cell: all the astonishing regular 4-polytopes. The pentad building block is half a pair of orthogonal 5-point (regular 5-cells), the hexad building block is half an 8-point (16-cell), and the heptad building block is a quarter of an 11-point (11-cell). Everything can be built from these blocks by embedding them at the center of the 3-sphere, inscribing them as 4-polytopes to make compounds as with the 16-cell building block, rather than by assembling them face-to-face as 3-polytopes to make honeycombs in which they are cells or vertex figures.{{Efn|name=Embedding point}}

The pentad building block is the 5-point '''pentahemidemicube''', an abstract polyhedron with 10 equilateral triangle faces, 10 {{radic|5}} edges, and 5 vertices.{{Sfn|Christie: Pentahemidemicube|2024|loc=File:Pentahemidemicube.png|ps=; &quot;The 5-point ''pentahemidemicube'' is an abstract polyhedron with 10 equilateral triangle faces, 10 edges, and 5 vertices. It has a real presentation as the regular 5-cell.&quot;}} It has a real presentation as the 5-point (regular 5-cell 4-polytope), a regular tetrahedron expanded by addition of a fifth vertex equidistant from all the others in a fourth dimension. The pentad block is a tetrahedral pyramid (a 5-cell) inscribed in a cube: a tetrahedron inscribed in a cube (a demicube) expanded by addition of a fifth vertex, one of the cube's other 4 vertices. Each pentad (5-cell) building block lies in the 120-cell orthogonal to three other pentad (5-cell) building blocks. They are four completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges. Each regular 5-cell in the 120-cell is a pentad (pentahemidemicube) building block, in one of 30 orthogonal sets of 4. There are 120 pentad building blocks in the box.

Every vertex of the 16-cell, and therefore every vertex of the 8-cell, 24-cell, 600-cell and 120-cell, is surrounded by a 6-point (octahedral vertex figure). The hexad building block is that 6-point octahedron embedded at the center of the 4-polytope instead of at a vertex, that is, inscribed in the 3-sphere as a 4-polytope, the 6-point '''[[w:Tetrahemihexahedron|tetrahemihexahedron]]''' or '''hemi-cuboctahedron'''.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; &quot;The tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U&lt;sub&gt;4&lt;/sub&gt;. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices.&quot;}} The hexads have 4 red triangle 5-cell faces, bounded by 12 {{radic|5}} edges. Their 3 yellow square faces lie in central planes, because the building block is an abstract 4-polytope. The hexad is an irregular abstract polyhedron that also has 6 {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

Two orthogonal 5-point (pentad) building blocks intersect in a 4-point (inscribed tetrahedron) to form a 6-point (hexad) building block. Two orthogonal 6-point (hexad) building blocks intersect in a 4-point (central square) to form an 8-point (16-cell) building block, the most usable building block, from which everything larger can be built by compounding.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 120 pentads to pair with a completely orthogonal hexad, and although there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed) they only result in 120 pairs two different ways, so there are 120 11-cells. The 11-cell is 6 pentads' and 5 hexads' smallest joint compound, and the 120-cell is their largest, 120 pentads and 100 hexads.

The heptad{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''|pages=22–31}} building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|4}} edges and 9 {{radic|5}} edges), and 7 vertices.{{Sfn|Christie: Pentahemicosahedron|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;}} It is a compound of a 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]) and a 6-point (hexad [[W:hemi-icosahedron|hemi-icosahedron]]) intersecting in a 4-point (tetrahedron), that is, of three orthogonal 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]&lt;nowiki/&gt;s) intersecting in their common 4-point (tetrahedron). It is also the abstract cube contracted by removal of one vertex, which reduces one of the cube's two inscribed tetrahedra (demicubes) to a single remaining triangular face. As with all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Four &lt;small&gt;&lt;math&gt;(11-7)&lt;/math&gt;&lt;/small&gt; orthogonal 7-point (pentahemicosahedron) blocks can be joined at a common Petrie polygon (a skew hexagon which contains their orthogonal 6-edge paths) to construct a real 11-point ([[W:11-cell|11-cell]] semi-regular 4-polytope), which magically contains six &lt;small&gt;&lt;math&gt;(11-5)&lt;/math&gt;&lt;/small&gt; 5-point (1-pentad regular decahedra) as cells and five &lt;small&gt;&lt;math&gt;(11-6)&lt;/math&gt;&lt;/small&gt; 12-point (2-hexad semi-regular cuboctahedra) as cells. There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like [[W:Euclid's postulates|Euclid's postulates]], and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as honeycombs of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles. It possesses the complete symmetry of the polytope without any redundancy, because it contains one of each of the polytope's characteristic root elements.{{Efn|A [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be partitioned radially by its planes of symmetry (Coxeter's &quot;reflecting circles&quot;) into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated rotationally to generate its regular 4-polytope because it is the complete [[W:gene|gene]] for it, containing all of its elements and capturing all of its symmetry without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring that triggers the reflections.|name=Characteristic orthoscheme}} It is the gene for the polytope, which can be replicated to construct the polytope.

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its &lt;math&gt;A_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the regular compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;small&gt;&lt;math&gt;120\times 120 = 14400&lt;/math&gt;&lt;/small&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[16--cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its &lt;math&gt;B_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 384. The 120-cell is the convex hull of the regular compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 24-point (24-cell) is subdivided into 1152 instances of its [[24-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;F_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 1152. The 120-cell is the convex hull of the regular compound of 25 disjoint 24-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;25\times 1152 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has three equivalent origins in distinct root systems, though most are obviously constructed from one root, and not so obviously from the others. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything larger than the pentad can be constructed by compounding hexads, and we shall see that everything as large as a 24-cell has a construction from heptads which are a product of a pentad and a hexad. Surely every uniform polytope can be constructed by some function of expansions and contractions from any of three root systems, however indirect the recipe. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes, related by an equals sign between them, then give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics by [[W:Noether's theorem|Noether's theorem]].

=== Pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 octahedral cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell contains the 600 octahedral cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It also contains the 600 cubical cells of its 75 8-cells. It is a 600-cell in three ways, although the octahedra and cubes are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell and the 8-cell, respectively.|name=600 cells three ways}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (1-pentad 5-cell), out of the 8-point (2-hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (1-pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope that contains 5-point (1-pentad 5-cell) building blocks beside the largest one, it just isn't a ''regular'' 4-polytope. It is semi-regular with two kinds of cells, the 4-point (regular tetrahedron) and the 12-point (truncated cuboctahedron), which have just one kind of regular face between them, a 5-cell face. That semi-regular 4-polytope is the 11-cell.

We said earlier that the 11-cell had 11 disjoint 5-cells and 11 hemi-icosahedral cells in it, and that its construction from them might be hard to see, but now we are going to account numerically, at least, for how it can have 6 pentad cells plus 5 hexad cells magically adding to 11 of the same kind of elevenad cell. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, 11. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (5-cell regular decahedron) cells, and the 12 6-point (hexad) building blocks into 100 12-point (regular icosahedron) cells, and then magically adding one whole 5-point (5-cell regular decahedron) cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 12-point (regular icosahedron) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such (cell bundles of 6 + 5 = 11 cells), with each 11-cell bundle containing 11 of both kinds of cell. Very, very strange!{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden'': §II To Add is to Multiply|p=38|ps=; the sequence of regular polygons with harmonious proportions beginning {5}, {7}, {9}, {11} as a family of natural numbers; &quot;The numbers in this optimal family have remarkable properties [that allow] similar figures to be arranged easily or allow a figure to be dissected into a set of similar figures. This is the origin of the Greek idea of geometric progression — that ''multiplication can be accomplished instead by addition''.... For instance, it is well known that the square of the golden ratio is one more than itself (written above as 𝜙&lt;sup&gt;2&lt;/sup&gt; = 𝜙 + 1) and one less than 𝜙 is its reciprocal (1/𝜙 = 𝜙 - 1). The first relation equates multiplication with addition, while the second accomplishes division through subtraction. [These are pentagon {5} bisectional golden ratios, but] the heptagonal [{7} tri-sectional] ratios also behave in surprising ways — adding to multiply, subtracting to divide.... There are analogous constructions for [nonagon {9}] quadrisectional and [hendecagon {11}] pentasectional systems.&quot;}}

The 120-cell has three interchangeable cyclic expansion-contraction cycles, the pentad, hexad and heptad cyclic designs, rooted in the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems, respectively. Each of them constructs the whole sequence of regular 4-polytopes from a single building block. The key to all three cycles is that 5 + 6 = 11, and there is an 11-point quasi-regular 4-polytope in the sequence of convex 4-polytopes. It lies between the 8-point (16-cell) and the 16-point (8-cell), and is itself a &lt;math&gt;B_4&lt;/math&gt; polytope. Its role is to be a unified single building block common to all three cyclic expansion-contraction cycles. It is involved with the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems and expresses in one building block their relations to each other.

We have already seen how the hexad building cycle works. It is plain compounding of 4-orthoplexes at the beginning, increasing to densely intricate non-disjoint symmetry toward the end, making it difficult to see why it works once past the 24-cell. This is how the build works, in terms of sixes. The 16-cell is a 2-hexad (3 pairs of completely orthogonal great square planes). The 24-cell is 3 16-cells and a 4-hexad, 4 different ways: 16 non-disjoint hexads (great hexagon planes). The 120-point (600-cell) is 25 24-cells and a 20-hexad, 10 different ways: 200 non-disjoint hexads (great hexagon planes). The 600-point (120-cell) is 225 (9 times 25) 24-cells. ....

The pentad building cycle doesn't do plain compounding to regular 4-polytopes until the last one, the 120-cell. Fives don't even appear as pentagons until after the 24-cell. We have seen how there are 11 5-cells in the quasi-regular 11-cell, which occurs before the 8-cell, and also how everything can be compounded from irregular 5-cells (4-orthoschemes specialized for each 4-polytope). It is still not evident how a pentad cycle can work to build all the regular 4-polytopes from the regular 4-simplex. This is how is how the build works, in terms of fives. ....

This is how the build works, in terms of elevens. The 11-cell is 4 orthogonal heptads (intersecting in their skew hexagon Petrie polygon). The 24-cell is also 4 heptads (intersecting in 4 orthogonal digon axes, so each heptad contributes 6 unique points to a single fibration of 24 points). Heptads are quarter elevenads, so the 11-cell and 24-cell are both 1 elevenad. ....

See the tabular sequence of convex 4-polytopes in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Build with the blocks|§Build with the blocks]]'', below: table rows ''Inscribed, Pentad, Hexad, Heptad''.

=== 11 of them ===
There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of eleven 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum eleven of them. There is only a single Hopf fibration of the 11 great circles of an 11-cell. 

The 5-cell and the 11-cell are limit cases. The 5-cells are completely disjoint, but the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 10 other instances of itself.

== Cell rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer &amp; Segerman|2013|loc=''Puzzling the 120-cell''}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his section of the Wikipedia article on the 120-cell entitled ''[[120-cell#Visualization|§Visualization]]''.{{Sfn|Ruen &amp; Goucher et al. eds. 120-cell|2024|loc=Goucher's ''[[120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious place to look for the 11-cell's Hopf map is in Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other place it could be hiding. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The quasi-regular convex 11-cell|§The quasi-regular convex 11-cell]]'', and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen: triacontagon|2011|loc=''Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them except his illustrations. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Four orthogonal projections of the 120-cell to a central plane, selected from the table of 3 regular and 11 compound 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|100 icosahedra&lt;br&gt;120 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|5 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 icosahedra&lt;br&gt;6 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|120
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|4
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|50
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|7.2°
|align=center|36°
|align=center|6°
|align=center|7.2°
|}

....

Here we summarize our findings on the 11-cells so far, including in advance some results to be obtained in the next section two sections.

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section....

== The 15 major chords ring everything ==
The natural numbers are each a distinct flavor.

Each integer has its own distinct taste. The first six integers {0..5} are the six strongest, most familiar tastes in nature, instantly distinguished and utterly unique.

The seventh flavor {6} is the first ratio of two of the others. Thereafter follows in strange sequence all the ratios of all the flavors, each marriage of every two flavors, most of which are fractions &lt;math&gt;\{k/d\}&lt;/math&gt;, but some of which are integers &lt;math&gt;\{k\}&lt;/math&gt;.

Some of these ratios are successful flavors in their own right, as distinct and inimitable as the marriage of chocolate and oranges, but the marriage of many flavor pairs is mud. It is a rule of thumb among flavor specialists that nearly any taste can be successfully alloyed with either onions or chocolate, but the direct marriage of these two flavors is notably unsuccessful. Some flavors make famous pairings or infamous ones, and subtle new pairings are always to be discovered that work surprisingly well for some purpose, but the majority of flavor marriages are undistinguished compounds.

In polytope geometry, each chord of a polytope is both is a distinct 1-dimensional object, a chord of the sphere of a distinct length &lt;math&gt;l&lt;/math&gt;, and a distinct natural number &lt;math&gt;h&lt;/math&gt;, a unique flavor. If the polytope is regular, it is a noteworthy and successful flavor. The chord's length &lt;math&gt;l&lt;/math&gt; is a square root, related to the natural number &lt;math&gt;h = k/d&lt;/math&gt; and to a set of polytopes they both represent, by a formula discovered by Steinbach.{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden''; Figure 5. Optimal sections and proportions|p=37|ps=; the regular polygons {5}, {7}, {9} and {11} with their diagonals define respectively: {5} the golden bisection proportional to 𝜙; {7} an analogous trisection; {9} an analogous quadrasection; {11} an analogous pentasection.}} The chord length &lt;math&gt;l&lt;/math&gt; is related to the number of sides of the regular polygon formed by the chord (its numerator &lt;math&gt;k&lt;/math&gt;), and to the winding number or density of its regular skew Petrie polygon (its denominator &lt;math&gt;d&lt;/math&gt;).

The regular 4-polytopes do not contain an infinite variety of these chord flavors, but only 30 of them: only 15 unique pairs of 180° degree complementary chords, which combine to make every regular ''n''-dimensional thing found in nature.{{Efn|The 15 ''major chords'' suffice to construct all the regular ''n''-polytopes. Their 180° complements include 15 ''minor chords'' which occur only in the 120-cell, not in any smaller regular polytope.{{Efn|name=Major and minor chords}} Including the complement of the 180° {{radic|4}} digon {2} chord (the {0} chord {{radic|0}}), and the skew variant of the 90° {{radic|2}} square {4} chord (the {30/7} chord), there are 32 distinct natural numbers which are all the chords of all the regular polytopes in all dimensions.|name=32 chords}} All the regular polytopes (including the largest one on the cover of the box) can be built not only from a box containing a number of blocks of 3 shapes glued together by gravity, they can be also be built from a box containing a number of sticks of 15 lengths glued together by rubber joints.

[[File:15 major chords.png|500px|thumb|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the others occur only in the interior of 120-cell, not as edges of any of the regular 4-polytopes. There are 30 distinct 4-space chordal distances between vertices within the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to identify any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=Major and minor chords}} chords are properly labelled #1 — #15 in the order they fan out by size.{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''; Figure 3|ps=; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with a &quot;fan of chords&quot; diagram like the one here.|p=23}} They join vertex pairs which are 1 — 15 edges apart on a &lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon. A #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; chord forms &lt;small&gt;&lt;math&gt;f+1&lt;/math&gt;&lt;/small&gt; regular geodesic polygons which include a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; disjoint regular {{nowrap|&lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt;-gons}} (e.g. &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; a compound of 5 disjoint hexagons) plus a single [[W:Triacontagon#Triacontagram|regular skew {{nowrap|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram}}]] that wraps around the sphere &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times before closing its circle. Notice the maverick #11 chord, with its non-conforming origin point for its {{nowrap|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;}} polygram.]]

These 15 chords (natural numbers) include the integers 2, 3, 4, 5, 6, 10, 11, 15 and 30. The others are the fractions 5/2, 10/3, 13/7, 15/2, 15/4, and 15/7. Represented among the chord lengths (the square roots) measured in ''unit radii'' (not in {{radic|2}} radii, as metrics are often given elsewhere in this article) are the {{background color|paleturquoise|roots of the first 4 positive integers {{radic|1}}, {{radic|2}}, {{radic|3}} and {{radic|4}}}}, the {{background color|#FFCCCC|root of the very distinguished rational fraction {{radic|5/2}}}}, and the roots of various distinguished irrational fractions as Steinbach discovered: for example, 4 of them are {{background color|yellow|roots of golden numbers (a small integer ± 𝜙)}}. The 15 chords each form their regular polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, which is just &lt;small&gt;&lt;math&gt;\{k\}&lt;/math&gt;&lt;/small&gt; in 8 cases. Arranged in order #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; from shortest edge-length square root to longest, the #1 — #15 chords bridge vertex pairs which are &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; vertices apart on a [[W:Triacontagon#Petrie polygons|&lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon]]. Each chord makes a [[W:Triacontagon#Triacontagram|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; polygram]] of its edge length, a compound of the smaller &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. The 15 distinct polygons form a sequence you would not anticipate exactly unless you have Steinbach's formula:

{| style=&quot;background-color:transparent; white-space:nowrap;&quot;
|-
| {30/1} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|.073}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{30} 
| 120-cell edges
|-
| {30/2}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.191}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15}
|- style=&quot;background: yellow;&quot;|
| {30/3}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.382}} 
| &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi}&lt;/math&gt;&lt;/small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{10}''' 
| 600-cell edges
|-
| {30/4} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.573}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/2}
|- style=&quot;background: paleturquoise;&quot;|
| {30/5}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|1}}'''
| &lt;math&gt;1&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{6}''' 
| 24-cell, 8-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/6}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|1.382}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{5}'''
|- style=&quot;background: paleturquoise;&quot;|
| {30/7}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|2}}'''
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{4}''' 
| 16-cell edges
|- style=&quot;background: #FFCCCC;&quot;|
| {30/8}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.5}} 
| &lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;&lt;small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/4} 
| 5-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/9}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.618}} 
| &lt;math&gt;\phi&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{10/3}
|- style=&quot;background: paleturquoise;&quot;|
| {30/10}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|3}}''' 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{3}'''
|-
| {30/11}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.426}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{11/11}
|- style=&quot;background: yellow;&quot;|
| {30/12}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.618}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{5/2}
|-
| {30/13}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.810}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{13/7}
|-
| {30/14}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.928}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/7}
|- style=&quot;background: paleturquoise;&quot;|
| {30/15}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|4}}''' 
| &lt;math&gt;2&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{2}'''
|}

{| class=wikitable style=&quot;white-space:nowrap; float:right;&quot;
!colspan=6|Natural number &lt;small&gt;&lt;math&gt;h=\{k/d\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Fibers&lt;br&gt;&lt;small&gt;&lt;math&gt;f\{h\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Cells&lt;br&gt;&lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|4-tope&lt;br&gt;&lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Verts&lt;br&gt;&lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|[[W:Triacontagon#Triacontagram|30-gram]]&lt;br&gt;&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;
|-
!Chord
!Arc
!colspan=2|&lt;small&gt;&lt;math&gt;l/\sqrt{1}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;l/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;h&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#1
|15.5~°
|{{radic|0.073}}
|0.270
|{{radic|0.146}}
|&lt;small&gt;&lt;math&gt;30&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1\{30\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;5\{3,3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/1\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#2
|25.2~°
|{{radic|0.191}}
|0.437
|{{radic|0.382}}
|&lt;small&gt;&lt;math&gt;15&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;..\{3,3\}..&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/2\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#3
|36°
|{{radic|0.382}}
|0.618
|{{radic|0.764}}
|&lt;small&gt;&lt;math&gt;10&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;30\{3,3\}20&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/3\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#4
|44.5~°
|{{radic|0.573}}
|0.757
|{{radic|1.146}}
|&lt;small&gt;&lt;math&gt;15/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/4\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5
|60°
|{{radic|1}}
|1
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;6&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/5\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#6
|72°
|{{radic|1.382}}
|1.175
|{{radic|2.624}}
|&lt;small&gt;&lt;math&gt;5&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}} 
|&lt;small&gt;&lt;math&gt;10\{5,3\}12&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/6\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#7
|90°
|{{radic|2}}
|1.414
|{{radic|4}}
|&lt;small&gt;&lt;math&gt;4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7\{4\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;4\{4,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;9\{3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/7\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: #FFCCCC;&quot;|
|#8
|104.5~°
|{{radic|2.5}}
|1.581
|{{radic|5}}
|&lt;small&gt;&lt;math&gt;15/4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/4\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}} 
|&lt;small&gt;&lt;math&gt;15\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/8\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#9
|108°
|{{radic|2.618}}
|1.618
|{{radic|5.236}}
|&lt;small&gt;&lt;math&gt;10/3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10/3\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/9\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#10
|120°
|{{radic|3}}
|1.732
|{{radic|6}}
|&lt;small&gt;&lt;math&gt;3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;10\{3\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/10\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#11
|135.5~°
|{{radic|3.426}}
|1.851~
|{{radic|6.852}}
|&lt;small&gt;&lt;math&gt;11/11&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;11\{11/11\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;11\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#12
|144°
|{{radic|3.618}}
|1.902
|{{radic|7.235}}
|&lt;small&gt;&lt;math&gt;5/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2},3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/12\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#13
|154.8~°
|{{radic|3.810}}
|1.952
|{{radic|7.621}}
|&lt;small&gt;&lt;math&gt;13/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;13\{13/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;7\{5,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/13\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#14
|164.5~°
|{{radic|3.928}}
|1.982
|{{radic|7.857}}
|&lt;small&gt;&lt;math&gt;15/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;137\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/14\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{radic|8}}
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15\{2\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐𝜙&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;8\{3,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/15\}&lt;/math&gt;&lt;/small&gt;
|}

The polygons '''{10}''', '''{6}''', '''{5}''', '''{4}''', '''{3}''', and '''{2}''' are flat polygons that each lie on a great circle in a central plane of the 3-sphere. The polygons {30}, {15}, {15/2}, {15/4}, {10/3}, {11/11}, {5/2}, {13/7}, {15/7} and all the {30/''n''} polygons are skew polygrams spiraling through curved 3-space, each successive edge in a different central plane of the 3-sphere. Projected to the 1-sphere (a circle in the plane), they are [[W:Star polygon#Regular star polygon|regular star polygrams]], the simplest being the pentagram {5/2}. In 4-space they wind around the 3-sphere more than once ''without self-intersecting'', before they close their circuit. They lie on helical geodesic circles or ''isoclines'' of circumference greater than 2𝝅''r''.

Even the 6 integer chords which form flat polygons also form skew polygrams. Each of the 15 chords occurs as a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, a fiber bundle of Clifford parallel polygons that totals 30 vertices per bundle. Each chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; also forms a single skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram that winds through another, shorter chord's bundle of parallel polygons, intersecting them in succession multiple times each, and weaving that whole bundle of parallel fibers together as the cross-thread does on a loom. The polygram cross-thread has to wind multiple times through the whole bundle (&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times around the [[W:Triacontagon#Triacontagram|30-gram]] in which each edge connects every &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt;th vertex) to complete an enumeration of all 30 vertices in the bundle. For example, the #5 chord forms hexagon &lt;small&gt;&lt;math&gt;\{6\}&lt;/math&gt;&lt;/small&gt; flat planes in parallel bundles of 5, written &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt;, and also a skew &lt;small&gt;&lt;math&gt;\{15/2\}&lt;/math&gt;&lt;/small&gt;-gram of #5 edges which winds 2 times through the #3 chord's bundle of 3 decagon &lt;small&gt;&lt;math&gt;\{10\}&lt;/math&gt;&lt;/small&gt; flat planes. Every chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; forms both its &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram and its fiber bundle of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, and the edges of one chord's parallel polygons are also the edges of its skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram weaving through some shorter chord's bundle of parallel polygons. Each numbered chord is a natural number, a ratio of two integers and their polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, either flat or skew. Each chord number defines a &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; skew polygon that is a compound bundle of the &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. All these flat or skew polygons are regular polygons made of their numerator instances of a single chord #, whose length is that square root, Steinbach's function of the natural number.

The relationship between each bundle of flat Clifford parallel great circle polygons of a distinct length, and its corresponding bundle of skew polygrams of a longer length, is precisely that of a discrete simple rotation and its corresponding discrete isoclinic rotation. For example, the flat great circle hexagons of edge length {{radic|1}} lie in the invariant rotation planes of the 24-cell's characteristic 60° isoclinic rotation, in which the vertices move on skew hexagram rotation paths of edge length {{radic|3}}. The 24-cell contains 4 bundles of 4 Clifford parallel (disjoint) hexagon central planes each. The 24 vertices of the 24-cell in each bundle are linked by rotation in the 4 hexagon invariant planes in two different ways: simply and isoclinically. In a simple rotation the vertices of a single invariant hexagon exchange places on that great circle hexagon of edge {{radic|1}}. In an isoclinic rotation in which any one of these 4 disjoint hexagon central planes is invariant, all 4 of them are necessarily invariant planes in the rotation, because they rotate in parallel. In that isoclinic rotation the vertices of all 4 hexagon invariant planes (all 24 vertices of the 24-cell) exchange places concurrently, moving along 4 disjoint helical hexagrams of edge {{radic|3}} in parallel. It is a characteristic only of this distinct rotation in hexagon invariant planes that the helical vertex path or ''isocline'' is a hexagram of period 6, edge length {{radic|3}}, and circumference 720°. Rotations in other kinds of polygon invariant planes have different characteristic isoclines. The period of their isoclinic rotation must be ''the same or greater'' than the period of their simple rotation; in most cases (unlike this one) it is greater.

The 15 chords form not only distinct fiber bundles of polygons, but also distinct bundles of polyhedra: cell rings &lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt; and compound vertex figures &lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;. These combine to form their characteristic regular 4-polytope &lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Coxeter|1973|p=269|loc=Compounds|ps=; &quot;It is remarkable that the vertices of {5, 3, 3} include the vertices of all the other fifteen regular polytopes in four dimensions.&quot;}} All these elements of a regular 4-polytope constructed from one chord are themselves regular polytopes, and each of the 16 regular 4-polytopes is a function of one characteristic chord. If the regular 4-polytope is convex, its characteristic chord is its unit-radius edge. For a non-convex regular 4-polytope, the characteristic chord is its unit-radius ''mid-edge diameter'', the distance from the midpoint of each edge to the midpoint of the completely orthogonal edge. Each of the characteristic chords (each distinct natural number) has its expression as one of the 16 regular 4-polytopes and all the products of the distinct regular 1-polytopes, 2-polytopes and 3-polytopes that construct it, and the whole distinct construction from the natural number to the 4-polytope is an [[W:SO(4)|SO(4) symmetry group]] expression of that 4-polytope's characteristic discrete [[W:SO(4)#Isoclinic rotations|isoclinic rotation]], in the invariant central planes of its characteristic chord, by the characteristic chord arc.

...demonstrate this theory by proper enumeration of the chord lengths in the table...

...color-code the △ ☐ 𝜙 ✩ symbols ... chord colors in diagram should match them ... some of the diagram colors are wrong and must be changed, but otherwise diagram is pretty good ... i think in a colored diagram all chords of the same length should have the same size line dash, the polygons are not obvious enough and the 15 fanning-out chords are readily identified by their labels...

... add back the counts of chords in the box (very small font number under each table entry), and present these two images as the front and back a box of building sticks and rubber joints...What's in the other box...

...say the rational-root major chords {{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|5/2}} are the ''n''-edges: their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge...

...abbreviated text of the [[120-cell#Relationships among interior polytopes|''§Relationships among interior polytopes'']] section from the 120-cell article...to explain the color-coding

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius=1}}

See also the elements, properties and metrics of the sequence of regular 4-polytopes summarized in tabular form in the section ''[[#Build with the blocks|§Build with the blocks]]'', below.

== Alicia Boole Stott's original formulation of dimensional analogy ==
&lt;blockquote&gt;&quot;...a method by which bodies having a certain kind of semi-regularity may be derived from regular bodies in an Euclidean space of any number of dimensions...&quot;{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''; presents cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues|pp=12-45}}&lt;/blockquote&gt;

Alica Boole Stott's 1910 paper reads like an exceptionally well-written patent application. It is a comprehensive stepwise description of her original invention, a general method of dimensional analogy between regular and semiregular polytopes of the same edge length.

[[File:The Roots of Three.png|thumb|An illustration of Boole Stott's expansion of unit-edge triangle to hexagon captures the geometric basis in 3 of the &lt;math&gt;A_n&lt;/math&gt;root system.]]

The engineering-like articulation of her new principles is entirely discrete and group-theoretic. Boole Stott repeatedly refers to sets of polytope elements as ''groups''. All her transformations are discrete instances of one or another special case of her &quot;two inverse operations which may be called ''expansion'' and ''contraction''.&quot; Hers is an exploration of ''n''-dimensional space at its most basic: space defined by the discrete objects which fill it, themselves defined directly as groups of elements with a few operations on them, which transform them into each other. This is geometry which seems to precede Euclid's postulates, and does not depend on them. Instead it corresponds directly to the deeper group theory mathematics which underlies geometry, even though Boole Stott, like the ancients, did not have that mathematics.

Boole Stott enumerated her expansion and contraction operations as transformations between regular and semi-regular polytopes of a single edge length. In her 1910 paper she did not consider any polytopes of multiple edge lengths, or expansions of the subject polytope by a distance other than that exactly required to produce new edges of the original edge length, so of course she did not enumerate Moxness's non-uniform rhombicosidodecahedron (Hull #8). This may be why section 8&lt;sub&gt;3&lt;/sub&gt; is missing from her earlier drawings of the sections of {5,3,3}; she may not have had a construction for it.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The Boole Stott expansion (or contraction) operation is as precisely defined a transformation as Coxeter's reflection, rotation and translation operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions). Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;
where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;
where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;
For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} An instance of the expansion (or contraction) operation is an application of the general operation to a particular subject polytope, operating on all of its elements symmetrically at once. The general operation is restricted, such that one may only expand (or contract) the subject polytope by pushing all its edges, faces or cells directly away from (or toward) its center, without changing the size of the moved elements. This splits each vertex into several (or coalesces several vertices into one), and has corresponding effects on the other elements, producing new edges ''of the same size'' between pairs of separated vertices (or reducing some edges to single vertices). The distance all the elements are moved away from (or toward) the center must be such that the vertices separate producing new edges of only the original edge length (or come together reducing edges to single vertices). It is permitted to combine these three operations, e.g. by pushing the edges ''and'' the faces, or by pushing all three element sets, provided the combination results in a semi-regular polytope of only one edge length. The three operations are all commutative with one another, so combined operations can be considered either as a sequence of expansions (or contractions) linking several related polytopes, or as their concurrent movements, a single transformation of the subject polytope. Since there are 8 combinations of one to three operations there are 8 polytope variants (including the original regular polytope), and all are adjacent to one another by a single transformation. The variants of the original regular polytope are named for which element set(s) are moved. By enumerating all 8 variants of expansion (and contraction) for each of the six regular convex 4-polytopes, Boole Stott produced a table of regular and semi-regular 4-polytopes divided into 12 sets of 8: six expansion sets and six contraction sets.{{Sfn|Boole Stott|1910|loc=''Table of Polytopes in S&lt;sub&gt;4&lt;/sub&gt;''; note that Boole Stott's usage of ''S&lt;sub&gt;4&lt;/sub&gt;'' denotes a Euclidean 4-space ''E&lt;sub&gt;4&lt;/sub&gt;'', or a 3-sphere space embedded in it, ''not'' a 4-sphere space embedded in a Euclidean 5-space|p=30}} Each set includes just one of the six regular convex 4-polytopes, with its distinct unit-radius edge length, and the 7 possible semi-regular expansions (or contractions) of it, which all preserve that characteristic edge length. The polytopes in each set of 8 all have the same total number of elements as their original regular polytope, but each polytope in the set has a distinct combination of cell, face, and vertex figures. The 12 sets are not completely disjoint, and even within one set the 8 operations do not necessarily produce 8 unique polytopes. For example, the regular 5-cell (the 4-simplex polytope) has 5 cells, 10 faces, 10 edges, and 5 vertices (30 elements). Expanding the 5-cell produces 7 related semi-regular 4-polytopes of 30 elements each (two of which happen to be the same polytope). In each expansion, the 5 cells have expanded into semi-regular or regular polyhedra, the 10 faces and 10 edges ''may'' have expanded into prisms, and the 5 original vertices have expanded into regular or semi-regular cells.

== The regular 11-cell 4-polytope ==

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosadodecahedron). Each non-convex Jessen's icosahedron is a set of 6 reflex edges that are {{radic|5}} 5-cell edges, each set symmetrically arranged in 3 parallel pairs, not as far apart ({{radic|3}}) as they are long, in 3 orthogonal central planes. The 12-point (Jessen's sets of 6 reflex 5-cell edges) are inscribed in the 60-point with their endpoints as a vertex of a pentagon face. The 12 pentagon faces are joined to 5 other pentagon faces by the 5-cell edges; not to their nearest neighbor pentagons but to their next-to-nearest neighbor pentagons; and since there are 5 vertices to each pentagon face, there are 10 12-point (Jessen's sets of 6 reflex 5-cell edges). Each pentagon face is joined to another pentagon face by 5 reflex edges of five distinct Jessen's. The 5 reflex edges are Clifford parallel, the 5 edges of a pentagonal prism column that has a twist to it. The contraction of those 5 edges into 1, and those 10 vertices (two pentagons) into 2, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing all their edges as a single Jessen's. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (octahedron) by conflating its parallel pairs of edges, leaving the 11-cell's abstract 6-point (hexad cell). The hexad's situation is that it occurs in bundles of 5, disjoint but close together in the sense of adjacent, like the great circle rings in a Hopf fiber bundle. 

To summarize, the 12-point (2-hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell). We shall now be able to see how Fuller's perfect hexads combine with Todd's perfect pentads to form the perfect heptads of the 11-cell and 137-cell.  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=alias the hemi-cuboctahedron}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces. (The ''heptad'' arrives at the party, anonymously; they are our distinguished anti-celebrity, who most of the other guests have never even heard of.) The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. This 12-point (truncated tetrahedron) is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 100 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp: Truncated tetrahedron|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp: Truncated tetrahedron|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}} Of course Moxness has been there, seen that, already with his graphics software;{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} see his parameterized illustration of it exactly as it occurs in the 11-cell, below in his bestiary of A3, B3, and H3 Archimedian and Catalan polyhedra in the 120-cell.]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:Archimedean_and_Catalan_solid_hulls_with_their_Weyl_orbit_definitions.svg|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of the 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. In this catalog of Archimedean and Catalan polyhedra sighted in the 120-cell by Moxness's software,{{Sfn|Moxness: Archimedean and Catalan hulls|2023|loc=Hull #1 Archimedean Name A3 110 Truncated Tetrahedron A (upper left)}} it is the first entry (upper left), Moxness's Hull # = 1 with 12 vertices of 3D norm {{radic|11}}/2. On a {{radic|2}}-radius 3-sphere, 5 of these with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell. 20 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges are cells in the [[#The rings of the 11-cells|cell rings of the quasi-regular 11-cell]], which also has 12 regular 5-cells as cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

...

Both kinds of cells in the 11-cell have expressions in two different dimensions (on the 2-sphere and 3-sphere), because they are all centered on the center of the 3-sphere, unlike the cells of most 4-polytopes, which are centered on various off-center points within the 4-polytope's interior. 

One kind of 11-cell cell appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (semi-regular truncated cuboctahedron), a.k.a. the truncated tetrahedron. The other kind of cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now, except in the 120-cell. But this is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron, and a regular tetrahedron behind each face. The 11-cell is thus a honeycomb of mixed regular tetrahedra and truncated cuboctahedra (a.k.a. truncated tetrahedra) that meet at their shared 5-cell faces. 

Both the 5-cells (decahedra) and the truncated cuboctahedra (truncated tetrahedra) are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks in their 4-polytopes (off-center).{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} Therefore all the 11-cell's cells are concentric, which indeed makes a different kind of honeycomb, an inside-out cellular honeycomb with all the cells piled on top of each other overlapping in 4-space. They do not overlap in 3-sphere space however; their 3-volumes (but not their 4-contents) are disjoint. They make a proper 4D honeycomb like the cells of any 4-polytope, with volumetrically disjoint 3D cells that meet each other in the proper way at their shared 2D faces.

An 11-cell contains 6 completely disjoint 5-point (1-pentad 5-cell decahedron cells), and 5 volumetrically disjoint 12-point (2-hexad truncated cuboctahedron cells), 11 volumetrically disjoint cells altogether. Of course it is supposed to have 11 identical 11-point (elevenad hemi-icosahedral cells), and that is also true in its abstract sense, which is not the same sense as disjoint cellular containment: each pentad or hexad cell is inscribed in one of 11 larger 60-point (Moxness's rhombicosadodecahedron cells). The abstract pentads and hexads are only hemi-polyhedra, so the 11-cell's pentad cells are 12 (out of 120) disjoint instances of a pentad 5-point, and its hexad cells are 10 (out of 100) disjoint instances of a hexad 6-point, 1/10th of the 120-cell's cellular volume in both cases. The 11-cell's hemi-icosahedral cells are 11 (out of 120) non-disjoint instances of an elevenad 60-point. Each 11-cell shares each of its hexad cells with 9 other 11-cells, and each of its 11 vertices (out of 600) occurs in 9 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.|name=Tetrahedral vertex figures}}

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

...insert image thumb of 11-cell...

...Construct the 11-cell from the heptad....

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...Four concentric 4-polytopes may be mutually orthogonal. In any set of 4 mutually orthogonal regular 5-cells, one pair of faces in each orthogonal pair are completely orthogonal to each other. Thus three edges in one 5-cell in an orthogonal pair are opposite (completely orthogonal) to a face of the orthogonal 5-cell. .... A 5-cell has 5 vertices in its abstract hemi-icosahedral cells, but only 4 vertices in one concrete icosahedral cell; the 5th lies in an icosahedral cell of another 11-cell (not in another icosahedral cell of the same 11-cell).

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

....

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 11-cell ===

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon.... (wrong, 11-cell's Hopf map is the truncated icosahedron)

....

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk: Truncated icosahedron|2018}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

...

Since the rhombicosadodecahedron has 120 5-cell edges, the 120-cell has 120 completely disjoint 5-cells, and the rhombicosadodecahedron shares each vertex with 11 other rhombicosadodecahedra, how a rhombicosadodecahedron is bound to its neighbor 5-cells is constrained. A further constraint is that the abstract hemi-icosahedron has only 15 5-cell edges, so the 120 actual 5-cell edges of the rhombicosadodecahedron are conflated 8-to-1 in the hemi-icosahedron. To be conflated by contraction into a single edge, those 8 edges must lie parallel to each other in the 120-cell; either actually parallel in the same plane, or Clifford parallel in separate planes, as two edges in completely orthogonal planes may be parallel and orthogonal at the same time. Each 5-cell inscribed in the 120-cell contains 5 such tetrahedra occur in parallel pairs in 4 orthogonal planes (none of them completely orthogonal planes). 

.... 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
Each conflated vertex is the 10 vertices of two orthogonal regular 5-cells...
The 60-point contains 6 pairs of these orthogonal regular 5-cell cells...
...its 20 hexad cells are ..., a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere.

== The regular 137-cell 4-polytope ==

(description originally written for the 11-cell, which applies only to a fibration of 11 11-cells)

It is quasi-regular with two kinds of cells, the 12-point (regular icosahedron) and the 5-point (regular 5-cell), which have just one kind of regular face between them. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of regular facets (it is semi-regular with regular facets), it also means that its facets have expressions of two different dimensionalities because they are concentric in the 3-sphere. The 11-cell has two kinds of regular cells. One appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (regular icosahedron), but since it is embedded at the center of the 3-sphere (unlike the cells of most 4-polytopes) it also acts as an inscribed 4-polytope.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} The other cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now except in the 120-cell. This is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron. The 11-cell is a honeycomb of mixed icosahedron and half-icosahedron cells. Like the icosahedra, the decahedra are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks, off-center in their 4-polytopes. All the 11-cell's cells are concentric, which indeed makes an unusual honeycomb of cells, an inside-out honeycomb. It is a proper 4D honeycomb however, with volumetrically disjoint 3D cells that meet each other in the proper way at shared 2D faces. The embedded 12-point (regular icosahedron) cell is an icosahedral pyramid (as a 4-polytope), which is filled by 20 4-point (irregular tetrahedra) that meet at its center; the center point of the 11-cell acts as a 12th vertex, just as the center point of the 24-cell acts as a 25th vertex in some honeycombs. The 5-point (regular 5-cell) cell contains five 4-point (regular tetrahedra) which meet the 12-point (regular icosahedra)'s irregular tetrahedral cells face-to-face, so the 11-cell is quasi-regular in the usual sense of having two varieties of regular 3-polytope cells, but additionally both varieties of of its cells are bi-dimensional: another clue that the 11-cell has something to do with dimensional analogy

....

The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

....

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur to make an 11-cell. 11 of them occur in a 137-cell, all 120 of them occur in the 120-cell, and they occur nowhere else.

How can this regular 4-polytope possibly exist? By Schläfli's criterion {3, 3, ..} exists, but not as a finite polytope.{{Sfn|Coxeter|1973|loc=§7.7 Schläfli's criterion}} Coxeter found {3, 5, 3} as a finite hyperbolic honeycomb, but not in Euclidean 4-space.{{Sfn|Coxeter|1968|title=The Beauty of Geometry: Twelve Geometric Essays|loc=''Regular Honeycombs in Hyperbolic Space''|ps=; &quot;The honeycombs considered by the above authors [Schlegel, Schläfli, Klein, Sommerville] have finite cells and finite vertex figures. ... A further extension allows a cell or vertex figure to be a ''star''-polytope, so that the honeycomb covers the space several times. .... We shall find that there are four regular star-honeycombs in hyperbolic 4-space, as well as two infinite families of them in the hyperbolic plane.&quot;}} We find that {3, 3, ..} is a central expression of the &lt;math&gt;H_4&lt;/math&gt; root system, being the regular convex hull in 4 dimensions of 137 points, 96 pentads and 600 hexads. The minimal expression of &lt;math&gt;H_4&lt;/math&gt; is the regular convex hull {3, 3, 5} of 120 points, 600 tetrads and 75 hexads, and its maximal expression is the regular convex hull {5, 3, 3} of 600 points, 120 pentads and 675 hexads. {3, 3, ..} is a third regular &lt;math&gt;H_4&lt;/math&gt; polytope which fits between them. The properties and metrics of this regular sequence are presented in tabular form below in ''[[#Build with the blocks|§Build with the blocks]]''.

{|class=&quot;wikitable&quot;
!colspan=3|Regular &lt;math&gt;H_4&lt;/math&gt; polytopes
|-
|[[File:600-cell.gif|200px]]
|[[File:5-cell.gif|200px]]
|[[File:120-cell.gif|200px]]
|-
!120-point 600-cell
!137-point 137-cell
!600-point 120-cell
|}

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 137-cell ===
....

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=Apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|2022|loc=''[[Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]''}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie: On Fuller's use of language|2024|loc=''[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry'']]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center, 1948-1949|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with their mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the gender neutral shad can open their six beaks all the way, until they become flat squares and they becomes a cuboctahedron, or they can shut them all tight like a turtle retracting into their octahedron shell. The six mouths always move in unison. This is [[Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do their ''really'' odd trick -- where they flip their 6 jaws 90 degrees in their 6 faces and shut their 6 beaks on the opposite axis of their squares than the one they opened them on -- or whether they will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than their normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make their edge length exactly the same size as their radius, when they open their mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing their mouths in spherical synchrony, their 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron (the hexad), then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=''[[W:SO(4)|SO(4)]]''}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=Isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its orthogonal hexad's planes (wz, wx, wy).{{Efn|name=Six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of (expanded octahederon 16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh &amp; Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== Legendre's 12-point hexad binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a 12-point double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's perfectly realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, this 2-hexad expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre hexad) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction.{{Efn|name=Concentric polyhedra}} In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== ''n''-simplexes ===
....

The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== ''n''-orthoplexes ===
....
[[File:Great 5-cell √5 digons rectangle.png|thumb|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} ''regular'' great hexagons. These are the same 200 dodecagon central planes illustrated above, which also contain 6 120-cell edges and two opposing ''irregular'' great hexagons (truncated {{radic|6}} triangles). The 1200 {{radic|5}} 5-cell face triangles occupy 1200 other, non-central planes.]]

...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's Hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[16--cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[16--cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=Concentric polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[16--cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== ''n''-taliesins ===
The 4-simplex pentad building block combines with the 4-orthoplex hexad building block to make the 4-taliesin heptad building block. The heptad is the essential building block of the ''taliesin'' family of polytopes. The 11-cell and 57-cell hemi-polytopes and the 137-cell and 120-cell regular 4-polytopes are nuclear members of this trans-dimensional family. The 24-cell and 600-cell are extended members of it, because they also contain instances of the heptad building block.

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its spherical symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* [[W:Taliesin Myrddin Namkai-Meche|Taliesin Myrddin Namkai-Meche]], a young Reed College scholar who was stabbed to death by a crazy man in 2017 on a Portland, Oregon light rail train, when he rose to the nonviolent defense of two black teenagers who were being violently assaulted in racist and anti-Muslim language by a right-wing extremist.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in orthogonal [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].
* '''Taliesan polytope''', an (''n+1)''-polytope whose ''n''-polytope dimensional analogue inscribed concentrically in the ''n''-sphere forms a convex honeycomb of itself which is the (''n+1)''-polytope.{{Efn|name=Embedding point}}
* The taliesan polytope's honeycomb, a Hopf fibration of its great circles.
* The ''n''-dimensional family of dimensionally analogous polytopes related to a taliesan polytope by expansion and contraction operations.
* The characteristic [[w:Orthoscheme|orthoscheme]] of a teliesan polytope, an irregular simplex that is the complete gene for the entire expression of trans-dimensional symmetry relationships among the polytopes of the ''n''-dimensional family, as well as the complete gene for the teliesan polytope's rotational symmetry.{{Efn|name=Characteristic orthoscheme}}

=== ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== ''n''-equilaterals ===
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed at four different [[#n-leviathon|chord]] lengths from each other. In the {{radic|2}}-radius polytopes we have using as examples in this paper they are {{radic|2}}, {{radic|4}}, {{radic|6}} and {{radic|8}}, but in a ''unit-radius'' polytope they are: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}.

Each vertex is joined to 8 others by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}. The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== ''n''-pentagonals ===

[[File:600-cell.gif|thumb|The 120-point (600-cell), &lt;math&gt;H^4&lt;/math&gt;.]]
...

[[File:600-cell vertex geometry.png|thumb|Vertex geometry of the 600-cell, showing the 5 regular great circle polygons and the 8 vertex-to-vertex chord lengths with angles of arc. The golden ratio{{Efn|name=golden chords|group=}} governs the fractional roots of every other chord,{{Efn|name=fractional root chords}} and the radial golden triangles which meet at the center.|alt=|400x400px]]
The 120 vertices of the 600-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏&lt;sup&gt;−1&lt;/sup&gt;) beginning with a vertex; see column ''a''}} at eight different [[#n-leviathon|chord]] lengths from each other. These edges and chords of the 600-cell are simply the edges and chords of its five great circle polygons. In ascending order of length in a ''unit-radius'' polytope (not in a {{radic|2}} radius polytopes like our usual examples), they are {{radic|0.𝚫}}, {{radic|1}}, {{radic|1.𝚫}}, {{radic|2}}, {{radic|2.𝚽}}, {{radic|3}}, {{radic|3.𝚽}}, and {{radic|4}}.{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=Decimal fractions of square roots|group=}}

Notice that the four [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#''n-''equilaterals|hypercubic chords]] of the 24-cell ({{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|4}}) alternate with the four new chords of the 600-cell's additional great circles, the decagons and pentagons. The new ''golden chord'' lengths are necessarily square roots of fractions, but very special fractions related to the golden ratio{{Efn|1=The fractional-root ''golden chords'' are irrational fractions that are functions of {{radic|5}}. They exemplify that the [[W:golden ratio|golden ratio]] &lt;big&gt;𝜙&lt;/big&gt; {{=}} {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618 is a circle ratio related to &lt;big&gt;𝜋&lt;/big&gt;:&lt;br&gt;
: {{sfrac|𝜋|5}} = arccos (𝜙/2)
is one decagon edge, the 𝚽 = {{radic|0.𝚫}} = {{radic|0.382~}} ≈ 0.618 chord.
Reciprocally, in this function discovered by Robert Everest expressing &lt;big&gt;𝜙&lt;/big&gt; as a function of &lt;big&gt;𝜋&lt;/big&gt; and the numbers 1, 2, 3 and 5 of the Fibonacci series:&lt;br&gt;
: &lt;big&gt;𝜙&lt;/big&gt; = 1 – 2 cos ({{sfrac|3𝜋|5}})
{{sfrac|3𝜋|5}} is the arc length of the &lt;big&gt;𝜙&lt;/big&gt; = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618 chord.|name=golden chords|group=}} including the two [[W:golden section|golden section]]s of {{radic|5}}, as shown in the diagram.{{Efn|The 600-cell edges are decagon edges of length {{radic|0.𝚫}}, which is 𝚽, the ''smaller'' golden section of {{radic|5}}; the edges are in the inverse [[W:golden ratio|golden ratio]] {{sfrac|1|φ}} to the {{radic|1}} hexagon chords (the 24-cell edges).
The other fractional-root chords exhibit golden relationships as well. The chord of length {{radic|1.𝚫}} is a pentagon edge.
The next fractional-root chord is a decagon diagonal of length {{radic|2.𝚽}} which is &lt;big&gt;𝜙&lt;/big&gt;, the ''larger'' golden section of {{radic|5}}; it is in the golden ratio{{Efn|name=golden chords|group=}} to the {{radic|1}} chord (and the radius).{{Efn|Notice in the diagram how the &lt;big&gt;𝜙&lt;/big&gt; chord (the ''larger'' golden section) sums with the adjacent 𝚽 edge (the ''smaller'' golden section) to {{radic|5}}, as if together they were a {{radic|5}} chord bent to fit inside the {{radic|4}} diameter.}}
The last fractional-root chord is the pentagon diagonal of length {{radic|3.𝚽}}.
The [[W:Pentagon#Side length is given|diagonal of a regular pentagon]] is always in the golden ratio to its edge, and indeed &lt;big&gt;𝜙&lt;/big&gt;{{radic|1.𝚫}} is {{radic|3.𝚽}}.|name=fractional root chords|group=}}

....


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== ''n''-leviathon ===
...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of its &lt;math&gt;10^2&lt;/math&gt; disjoint instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction, before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]].{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}} .... It is only because she did not have the 11-cell to start with that she did not discover the regular 137-cell {3, 3, ..} by her method.

Between Alicia Boole Stott and her identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and her identification of symmetry groups with conservation laws (in her 1st theorem), we owe the deepest mathematics of geometry and physics originally to two pioneering women, colleagues of their contemporaries the greatest men of the academy, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just an [[W:abstract polytope|abstract 4-polytope]], not just a singleton convex 4-polytope, and not just a honeycomb.{{Sfn|Coxeter|1970|loc=''Twisted Honeycombs''}} The 11-point (11-cell) has a concrete ...regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}&lt;nowiki/&gt;} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, 3, 3} as the 137-point (137-cell).

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 11-cell, 137-cell and 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=''Clifford Spinors and Root System Induction: H4 and the Grand Antiprism''}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=Apology}}

== Build with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Duveneck|1978|loc=Carl Jung, quoted in ''Life on Two Levels''|p=ii|ps=.{{Sfn|Jung|1961|ps=: &quot;The best of truths is of no use unless it has become one's most personal inner experience. It is the duty of everyone who takes a solitary path to share with society what he finds on his journey of discovery.&quot;}}}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1954|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius={{radic|2}}}}

== Acknowledgements ==
...

== Notes ==
{{Regular convex 4-polytopes Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
* {{Citation|last=Boole Stott|first=Alicia|author-link=W:Alicia Boole Stott|title=Geometrical deduction of semiregular from regular polytopes and space fillings|date=1910|pages=12-45|publisher=Johannes Muller|place=Amsterdam|url=https://dwc.knaw.nl/DL/publications/PU00011492.pdf}}
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      <text bytes="208676" sha1="jv3q34f3658l80yotndfj1xidwjfhgu" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|June 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a regular 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the &lt;s&gt;120-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 5, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 11 [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] cells. Eleven 11-cells (plural) are the 137-cell, the &lt;s&gt;600-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 3, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 137 [[W:Rhombic triacontahedron|triacontahedron]] cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=''Regularity of Graphs, Complexes and Designs''}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=''A Symmetrical Arrangement of Eleven Hemi-Icosahedra''}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]]
The 4-dimensional regular polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:Simplex|simplex]] (called the [[5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:Orthoplex|orthoplex]] (called the [[16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular convex 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the [[w:Tesseract|16-point (8-cell) tesseract]], the 4-cubic, which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found within any of the other regular convex 4-polytopes, except in the largest and most complex one, the [[120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). However, we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the 16-cell is their very starting point, and the most frequently used tool in the box.

== 5-cells and hemi-icosahedra in the 11-cell ==
[[File:Symmetrical_5-set_Venn_diagram.svg|thumb|The 5-point (10-face) regular 5-cell (the regular 4-simplex). Grünbaum's rotationally symmetrical 5-set Venn diagram{{Sfn|Grünbaum|1975|loc=''Rotationally symmetrical 5-set Venn diagram'', Fig 1 (e)|ps=; partitions the individual elements of the 5-cell.}} is an illustration of the 5-cell labeling each of its &lt;math&gt;2^5&lt;/math&gt; elements.{{Sfn|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled; image includes the Python code to render it, optimising for maximum area of the smallest regions.}}]]
[[File:Hemi-icosahedron.png|thumb|The 6-point (10 face) [[W:hemi-icosahedron|hemi-icosahedron]], an abstraction of the regular icosahedron, has half as many faces, edges and vertices. Each element of the abstract polyhedron represents two real elements found in different places in the 11-cell's realization.{{Sfn|Ruen: hemi-icosahedron|2007}}]]
The most apparent relationship between the pentad 5-cell and the hexad hemi-icosahedron is that they both have 10 triangular faces. When we see a facet congruence between a 4-polytope and a 3-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 6-point 5-simplex. Another hexad is the 6-point 3-orthoplex; thus the hemi-icosahedron is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no chord congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals, 5-cell and 16-cell, who are famously stand-offish with each other. Interesting!

The cell of the 11-cell is an abstract hexad hemi-icosahedron with 5 central planes, most handsomely illustrated by Séquin.{{Sfn|Séquin &amp; Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing rings of polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The 11 hemi-icosahedral cells have 10 triangle faces each; the 5-cell's 5 tetrahedral cells have 10 faces and 10 edges all together. If 11-cell faces are 5-cell faces, then 3 of each 5-cell's 5 vertices are a hemi-icosahedron face, and its other 2 vertices must be some 11-cell edge lying opposite the face. Coxeter determined that the 11-cell does indeed have an edge opposite each face, but it does not belong to the same hemi-icosahedral cell as its opposing face. He found that the 10 edges opposite each hemi-icosahedron's faces are the 10 edges of a single 5-cell, which does not share any vertices, edges or faces with the hemi-icosahedron. Of course, in the 11-point 11-cell there is just one such 5-point 5-cell that is completely disjoint from each 6-point hemi-icosahedron.{{Sfn|Coxeter|1984|p=110|loc=§6. The Petrie polygon [of the 11-cell]|ps=; &quot;We may reasonably call this edge and face ''opposites''. It is easy to find the face opposite to a given edge by looking at the faces to which a given edge belongs. ... Conversely, given a face, we can find the opposite edge by seeing which vertices belong to neither of the hemi-icosahedra which share that face. The ten edges opposite to the ten faces of one hemi-icosahedron are the edges of the complementary &lt;math&gt;a_4&lt;/math&gt; [4-simplex], that is, the joins of all pairs of the five vertices [of the 11-cell] not belonging to the given hemi-icosahedron.&quot;}}

There are 11 disjoint 5-cell 4-polytopes inscribed in each 11-cell, which also contains 11 hemi-icosahedral cells, 55 faces, 55 edges and 11 vertices. The real 11-cell is more complex and harder to see than the abstract 11-cell representing it, because the real hemi-icosahedron is also more complex than the abstract hemi-icosahedron. Seeing the real 11-cell will be easier once we have identified the real hemi-icosahedron, and exactly where the 11-cell's real elements reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each hemi-icosahedron's 10 faces come from 10 disjoint 5-cells, and there is just one 5-cell with which it does not share a face.{{Efn|As Coxeter observes (in the previous citation), that unrepresented 5-point (5-cell) is the other 5 vertices of the 11-point (11-cell) that are not vertices of this 6-point (hemi-icosahedron), the hemi-icosahedron's disjoint complement.}} Each 5-cell has 10 faces from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share a face.

In the abstract 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 ''disjoint'' 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent hemi-icosahedron faces do not actually meet at an edge; there is a narrow rectangle separating them, which has been abstracted to an edge. The 5-cells do not touch each other; they are all completely disjoint.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two actual faces in two different planes completely orthogonal to each other. Each duplex 11-cell face bonds to two 5-cells in different places, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 duplex 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must be sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness: 8 concentric hulls|2022|loc=Hull #8 (lower right)|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell of the 11-cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Compounds in the 120-cell|§Compounds in the 120-cell]]'' that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software which uses Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s to render the polyhedra which are found in the interior of ''n''-dimensional polytopes.{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the tetrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is a real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (Hull #8) is a concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this {12} irregular great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing {6} irregular great hexagons (truncated triangles) of alternating 120-cell edges and 5-cell edges. The great dodecagon is a great circle of the hemi-icosahedral cell, and also a great circle of the 120-cell, because unlike the cells of most 4-polytopes, the 11-cell's cells are embedded in 4-space at the exact center of the 4-polytope, making them inscribed 4-polytopes themselves.{{Efn|name=Embedding point}} There are 25 great dodecagons and 120 5-cell edges in Moxness's 60-point (Hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (Hull #8) polyhedron altogether.

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point (Hull #8) is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] similar to the one from the catalog shown here,{{Sfn|Piesk: Rhombicosidodecahedron|2018}} but a slightly shallower truncation with smaller red pentagons (120-cell faces) and narrower rhombs. Rhombicosidodecahedra are made by truncating the [[W:Rhombic triacontahedron|rhombic triacontahedron]], which is the 30-sided polyhedron with only one kind of face, the dual of the 30-point icosidodecahedron. The 120-cell contains 120 of these 11-cell cells.]]
Moxness's 60-point (Hull #8) is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (Hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|The 5-point (regular 5-cell) can be another abstraction of Moxness's 60-point (Hull #8), 12-vertices-into-1, but it is not strictly the result of a contraction operation as described by Boole Stott, since she did not apply her expansion and contraction operations to uniform polytopes with more than one edge length.}} In fact Moxness was not the first to find a rhombicosidodecahedron in the 120-cell; Alicia Boole Stott was. It is the 6th section of {5,3,3} beginning with a cell, which Boole Stott identified as the ''e&lt;sub&gt;2&lt;/sub&gt; expansion'' of the icosahedron (or equivalently of its dual polyhedron the dodecahedron).{{Sfn|Boole Stott|1910|loc=§Examples of the e&lt;sub&gt;2&lt;/sub&gt; expansion|p=7}} But that rhombicosidodecahedron identified by Boole Stott is not Moxness's Hull #8: rather, it is the semi-regular Archimedean solid, with a single edge length and square faces. Moxness's Hull #8 with its two distinct edge lengths and rectangular faces is Coxeter's 8th section of {5,3,3} beginning with a cell, which is missing from the sections illustrated by Boole Stott.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than tetrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. The 120-cell contains 120 of Moxness's 60-point (11-cell cells). They are non-disjoint, with 12 60-points sharing each vertex.

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (Hull #8) is the real hemi-icosahedron cell, with 120 5-cell edges in it as chords lurking under its surface. The rhombicosadodecahedron has enough of those 5-cell edges in its interior to make 20 disjoint tetrahedral cells which belong to 20 distinct 5-cells.

The 600-point (120-cell) may be constructed as a semi-regular honeycomb of two kinds of cells, the 120 non-disjoint 60-point (Moxness's rhombicosadodecahedra) and their neighbors the 120 disjoint 5-point (5-cells), cell-bonded together by shared tetrahedral cells of edge-length {{radic|5}}. Each rhombicosadodecahedron is bonded to 20 islanded 5-cells, sharing with each a 5-cell tetrahedron (4 of its interior 5-cell faces, and 6 of its interior 5-cell edges). The rhombicosadodecahedron is also bonded directly to 10 other rhombicosadodecahedra (since each 11-cell has 11 rhombicosadodecahedron cells), again not by sharing one of its exterior faces, but this time by sharing an interior face, a {{radic|6}} triangle which lies in a central plane.{{Efn|The {{radic|6}} chord is another hidden chord lurking beneath the 60-point rhombicosadodecahedron's surface, like the {{radic|5}} 5-cell edge. Both these hidden chords lie on the same great circle with the 120-cell edges (the pentagon face edges). Three {{radic|6}} chords make a triangle which lies in a central plane. This {{radic|6}} great triangle must not be confused with the {{radic|5}} face triangle of the 5-cell, which does not lie in a central plane.}} The 120 rhombicosadodecahedra are also bonded directly together at ''all'' their exterior faces: at their pentagon faces (which are 120-cell faces of dodecahedral cells); at their small triangular faces (which are 600-cell faces of small tetrahedral cells, 10 of which are inscribed in each dodecahedral cell); and at their rectangular faces (which are faces of 10-point [[w:Pentagonal_prism|pentagonal prism]] cells, where the pentagonal prism is a column inscribed in a dodecahedral cell, joining its opposite pentagon faces). The 120 rhombicosadodecahedra are bonded directly to each other at their own exterior faces; no dodecahedra, small tetrahedra, or pentagonal prisms lie between them, and those objects are not cells of this honeycomb; we mention them here only to help locate the rhombicosadodecahedra within the 120-cell. The only objects which fill holes between the rhombicosadodecahedra are the 120 islanded 5-cells.{{Efn|This honeycomb of the 120-cell resembles the [[600-cell#Icosahedra|honeycomb of the 600-cell consisting of 24 face-bonded regular icosahedra]] with 24 clusters of 5 tetrahedra filling the holes between them. In the 120-cell there are 5 times as many of everything as in the 600-cell, so there are 120 (rhombicosadodecahedron) icosahedral cells with 600 tetrahedra filling the holes between them.}}

It is possible for rhombicosadodecahedra to meet each other at more than one kind of exterior face because they are not meeting there as cells of the same 11-cell; they belong to different 11-cells. The 11 rhombicosadodecahedra in the same 11-cell do not meet at their own exterior faces, they meet at their interior {{radic|6}} triangle faces. As a cellular 4-polytope, the 11-cell is inside-out. Its cells meet each other at interior planes, sharing interior {{radic|6}} triangle faces, instead of in the usual way that 4-polytopes' cells share an exterior face. It is a deeper kind of intertwining to be sure, but the 11 rhombicosadodecahedra in the same 11-cell are still volumetrically disjoint cells, and the 11-cell is a perfectly real 4-polytope with real cells. It is a semi-regular 4-polytope, because it has two different kinds of cells: 11 rhombicosadodecahedra and 11 5-cells. The 11 5-cells are islanded (not bound to each other), and the 11 rhombicosadodecahedra are bound both to 5-cells (at {{radic|5}} 5-cell faces) and to each other (at {{radic|6}} central triangle faces). In the section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#What's in the box|§What's in the box]]'', below, we will look closely at the relation between the {{radic|5}} chord and the {{radic|6}} chord, which is the relation between the pentad and the hexad, and at the general phenomenon of hemi-polytopes which meet each other at a central plane. Then in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Cell rings of the 11-cells|§Cell rings of the 11-cells]]'' we will look at the two-cell cellular structure of the 11-cell in more detail.

Considered as a single complex of overlapping cells rather than as distinct honeycombs of volumetrically disjoint cells, the 120-cell contains 60-point (rhombicosadodecahedral cells), islanded 5-point (5-cells), 20-point (dodecahedral cells), and 10-point ([[w:Pentagonal_prism|pentagonal prism]] cells). The 60-points meet each other in pairs at their pentagon faces, meet regular dodecahedra at those pentagon faces also, meet each other in quads at their 600-cell faces where they share an interior 4-point (600-cell tetrahedral cell) and its four faces (just as they share a larger interior 5-cell tetrahedron and its faces), and meet each other and the 10-point (pentagonal prisms) at their rectangle faces. We might try to imagine such a complex with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with parts truncated from 30 neighboring truncated dodecahedra pentagonal prisms). There are 120 of each of these 3 kinds of dodecahedron. 

Moxness's 60-point (rhombicosadodecahedron cell) is a central object of the 120-cell, and contains many wonders. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra by expansion and contraction operations, as Alicia Boole Stott discovered before 1910,{{Sfn|Polo-Blanco: ''Theory and history of geometric models of Alicia Boole Stott''|2007|loc=§5.3.2 1910 paper on semi-regular polytopes|pp=152-158|ps=; summarizes Boole Stott's method and results from {{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}}, including her tables of expansion-contraction dimensional analogies and a few of her illustrations.}} and those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, rediscovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter &amp; Petrie|1938|p=4|loc=''[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]''}}|name=Concentric polyhedra}} Before we move on from the 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice one more thing about the hidden {{radic|5}} chords lurking below its surface. The 12 little pentagon faces are spanned by sets of 6 {{radic|5}} chords (5-cell edges). The 6 chords of each set are disjoint (they don't touch or form 5-cell faces), but they are symmetrically arranged in 3 parallel pairs, {{radic|3}} apart, which lie in 3 orthogonal central planes. It turns out they are the 6 reflex edges of a 12-point non-convex polyhedron called the [[w:Jessen's_icosahedron|Jessen's icosahedron]], inscribed in the 60-point cell. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again and work our way outwards, before eventually in ''[[#The perfection of Fuller's cyclic design|§The perfection of Fuller's cyclic design]]'' we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cell)s ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks, showing its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Pentahemidemicube.png|120px]]
|[[File:Tetrahemihexahedron_rotation.gif|120px]]
|[[File:Pentahemicosahedron.png|120px]]
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts needed to build the 120-cell (the picture on the cover of the box), and everything else that's invisible in that picture which fits inside the 120-cell: all the astonishing regular 4-polytopes. The pentad building block is half a pair of orthogonal 5-point (regular 5-cells), the hexad building block is half an 8-point (16-cell), and the heptad building block is a quarter of an 11-point (11-cell). Everything can be built from these blocks by embedding them at the center of the 3-sphere, inscribing them as 4-polytopes to make compounds as with the 16-cell building block, rather than by assembling them face-to-face as 3-polytopes to make honeycombs in which they are cells or vertex figures.{{Efn|name=Embedding point}}

The pentad building block is the 5-point '''pentahemidemicube''', an abstract polyhedron with 10 equilateral triangle faces, 10 {{radic|5}} edges, and 5 vertices.{{Sfn|Christie: Pentahemidemicube|2024|loc=File:Pentahemidemicube.png|ps=; &quot;The 5-point ''pentahemidemicube'' is an abstract polyhedron with 10 equilateral triangle faces, 10 edges, and 5 vertices. It has a real presentation as the regular 5-cell.&quot;}} It has a real presentation as the 5-point (regular 5-cell 4-polytope), a regular tetrahedron expanded by addition of a fifth vertex equidistant from all the others in a fourth dimension. The pentad block is a tetrahedral pyramid (a 5-cell) inscribed in a cube: a tetrahedron inscribed in a cube (a demicube) expanded by addition of a fifth vertex, one of the cube's other 4 vertices. Each pentad (5-cell) building block lies in the 120-cell orthogonal to three other pentad (5-cell) building blocks. They are four completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges. Each regular 5-cell in the 120-cell is a pentad (pentahemidemicube) building block, in one of 30 orthogonal sets of 4. There are 120 pentad building blocks in the box.

Every vertex of the 16-cell, and therefore every vertex of the 8-cell, 24-cell, 600-cell and 120-cell, is surrounded by a 6-point (octahedral vertex figure). The hexad building block is that 6-point octahedron embedded at the center of the 4-polytope instead of at a vertex, that is, inscribed in the 3-sphere as a 4-polytope, the 6-point '''[[w:Tetrahemihexahedron|tetrahemihexahedron]]''' or '''hemi-cuboctahedron'''.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; &quot;The tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U&lt;sub&gt;4&lt;/sub&gt;. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices.&quot;}} The hexads have 4 red triangle 5-cell faces, bounded by 12 {{radic|5}} edges. Their 3 yellow square faces lie in central planes, because the building block is an abstract 4-polytope. The hexad is an irregular abstract polyhedron that also has 6 {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

Two orthogonal 5-point (pentad) building blocks intersect in a 4-point (inscribed tetrahedron) to form a 6-point (hexad) building block. Two orthogonal 6-point (hexad) building blocks intersect in a 4-point (central square) to form an 8-point (16-cell) building block, the most usable building block, from which everything larger can be built by compounding.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 120 pentads to pair with a completely orthogonal hexad, and although there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed) they only result in 120 pairs two different ways, so there are 120 11-cells. The 11-cell is 6 pentads' and 5 hexads' smallest joint compound, and the 120-cell is their largest, 120 pentads and 100 hexads.

The heptad{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''|pages=22–31}} building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|4}} edges and 9 {{radic|5}} edges), and 7 vertices.{{Sfn|Christie: Pentahemicosahedron|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;}} It is a compound of a 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]) and a 6-point (hexad [[W:hemi-icosahedron|hemi-icosahedron]]) intersecting in a 4-point (tetrahedron), that is, of three orthogonal 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]&lt;nowiki/&gt;s) intersecting in their common 4-point (tetrahedron). It is also the abstract cube contracted by removal of one vertex, which reduces one of the cube's two inscribed tetrahedra (demicubes) to a single remaining triangular face. As with all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Four &lt;small&gt;&lt;math&gt;(11-7)&lt;/math&gt;&lt;/small&gt; orthogonal 7-point (pentahemicosahedron) blocks can be joined at a common Petrie polygon (a skew hexagon which contains their orthogonal 6-edge paths) to construct a real 11-point ([[W:11-cell|11-cell]] semi-regular 4-polytope), which magically contains six &lt;small&gt;&lt;math&gt;(11-5)&lt;/math&gt;&lt;/small&gt; 5-point (1-pentad regular decahedra) as cells and five &lt;small&gt;&lt;math&gt;(11-6)&lt;/math&gt;&lt;/small&gt; 12-point (2-hexad semi-regular cuboctahedra) as cells. There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like [[W:Euclid's postulates|Euclid's postulates]], and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as honeycombs of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles. It possesses the complete symmetry of the polytope without any redundancy, because it contains one of each of the polytope's characteristic root elements.{{Efn|A [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be partitioned radially by its planes of symmetry (Coxeter's &quot;reflecting circles&quot;) into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated rotationally to generate its regular 4-polytope because it is the complete [[W:gene|gene]] for it, containing all of its elements and capturing all of its symmetry without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring that triggers the reflections.|name=Characteristic orthoscheme}} It is the gene for the polytope, which can be replicated to construct the polytope.

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its &lt;math&gt;A_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the regular compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;small&gt;&lt;math&gt;120\times 120 = 14400&lt;/math&gt;&lt;/small&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[16--cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its &lt;math&gt;B_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 384. The 120-cell is the convex hull of the regular compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 24-point (24-cell) is subdivided into 1152 instances of its [[24-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;F_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 1152. The 120-cell is the convex hull of the regular compound of 25 disjoint 24-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;25\times 1152 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has three equivalent origins in distinct root systems, though most are obviously constructed from one root, and not so obviously from the others. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything larger than the pentad can be constructed by compounding hexads, and we shall see that everything as large as a 24-cell has a construction from heptads which are a product of a pentad and a hexad. Surely every uniform polytope can be constructed by some function of expansions and contractions from any of three root systems, however indirect the recipe. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes, related by an equals sign between them, then give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics by [[W:Noether's theorem|Noether's theorem]].

=== Pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 octahedral cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell contains the 600 octahedral cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It also contains the 600 cubical cells of its 75 8-cells. It is a 600-cell in three ways, although the octahedra and cubes are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell and the 8-cell, respectively.|name=600 cells three ways}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (1-pentad 5-cell), out of the 8-point (2-hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (1-pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope that contains 5-point (1-pentad 5-cell) building blocks beside the largest one, it just isn't a ''regular'' 4-polytope. It is semi-regular with two kinds of cells, the 4-point (regular tetrahedron) and the 12-point (truncated cuboctahedron), which have just one kind of regular face between them, a 5-cell face. That semi-regular 4-polytope is the 11-cell.

We said earlier that the 11-cell had 11 disjoint 5-cells and 11 hemi-icosahedral cells in it, and that its construction from them might be hard to see, but now we are going to account numerically, at least, for how it can have 6 pentad cells plus 5 hexad cells magically adding to 11 of the same kind of elevenad cell. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, 11. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (5-cell regular decahedron) cells, and the 12 6-point (hexad) building blocks into 100 12-point (regular icosahedron) cells, and then magically adding one whole 5-point (5-cell regular decahedron) cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 12-point (regular icosahedron) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such (cell bundles of 6 + 5 = 11 cells), with each 11-cell bundle containing 11 of both kinds of cell. Very, very strange!{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden'': §II To Add is to Multiply|p=38|ps=; the sequence of regular polygons with harmonious proportions beginning {5}, {7}, {9}, {11} as a family of natural numbers; &quot;The numbers in this optimal family have remarkable properties [that allow] similar figures to be arranged easily or allow a figure to be dissected into a set of similar figures. This is the origin of the Greek idea of geometric progression — that ''multiplication can be accomplished instead by addition''.... For instance, it is well known that the square of the golden ratio is one more than itself (written above as 𝜙&lt;sup&gt;2&lt;/sup&gt; = 𝜙 + 1) and one less than 𝜙 is its reciprocal (1/𝜙 = 𝜙 - 1). The first relation equates multiplication with addition, while the second accomplishes division through subtraction. [These are pentagon {5} bisectional golden ratios, but] the heptagonal [{7} tri-sectional] ratios also behave in surprising ways — adding to multiply, subtracting to divide.... There are analogous constructions for [nonagon {9}] quadrisectional and [hendecagon {11}] pentasectional systems.&quot;}}

The 120-cell has three interchangeable cyclic expansion-contraction cycles, the pentad, hexad and heptad cyclic designs, rooted in the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems, respectively. Each of them constructs the whole sequence of regular 4-polytopes from a single building block. The key to all three cycles is that 5 + 6 = 11, and there is an 11-point quasi-regular 4-polytope in the sequence of convex 4-polytopes. It lies between the 8-point (16-cell) and the 16-point (8-cell), and is itself a &lt;math&gt;B_4&lt;/math&gt; polytope. Its role is to be a unified single building block common to all three cyclic expansion-contraction cycles. It is involved with the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems and expresses in one building block their relations to each other.

We have already seen how the hexad building cycle works. It is plain compounding of 4-orthoplexes at the beginning, increasing to densely intricate non-disjoint symmetry toward the end, making it difficult to see why it works once past the 24-cell. This is how the build works, in terms of sixes. The 16-cell is a 2-hexad (3 pairs of completely orthogonal great square planes). The 24-cell is 3 16-cells and a 4-hexad, 4 different ways: 16 non-disjoint hexads (great hexagon planes). The 120-point (600-cell) is 25 24-cells and a 20-hexad, 10 different ways: 200 non-disjoint hexads (great hexagon planes). The 600-point (120-cell) is 225 (9 times 25) 24-cells. ....

The pentad building cycle doesn't do plain compounding to regular 4-polytopes until the last one, the 120-cell. Fives don't even appear as pentagons until after the 24-cell. We have seen how there are 11 5-cells in the quasi-regular 11-cell, which occurs before the 8-cell, and also how everything can be compounded from irregular 5-cells (4-orthoschemes specialized for each 4-polytope). It is still not evident how a pentad cycle can work to build all the regular 4-polytopes from the regular 4-simplex. This is how is how the build works, in terms of fives. ....

This is how the build works, in terms of elevens. The 11-cell is 4 orthogonal heptads (intersecting in their skew hexagon Petrie polygon). The 24-cell is also 4 heptads (intersecting in 4 orthogonal digon axes, so each heptad contributes 6 unique points to a single fibration of 24 points). Heptads are quarter elevenads, so the 11-cell and 24-cell are both 1 elevenad. ....

See the tabular sequence of convex 4-polytopes in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Build with the blocks|§Build with the blocks]]'', below: table rows ''Inscribed, Pentad, Hexad, Heptad''.

=== 11 of them ===
There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of eleven 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum eleven of them. There is only a single Hopf fibration of the 11 great circles of an 11-cell. 

The 5-cell and the 11-cell are limit cases. The 5-cells are completely disjoint, but the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 10 other instances of itself.

== Cell rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer &amp; Segerman|2013|loc=''Puzzling the 120-cell''}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his section of the Wikipedia article on the 120-cell entitled ''[[120-cell#Visualization|§Visualization]]''.{{Sfn|Ruen &amp; Goucher et al. eds. 120-cell|2024|loc=Goucher's ''[[120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious place to look for the 11-cell's Hopf map is in Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other place it could be hiding. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The quasi-regular convex 11-cell|§The quasi-regular convex 11-cell]]'', and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen: triacontagon|2011|loc=''Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them except his illustrations. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Four orthogonal projections of the 120-cell to a central plane, selected from the table of 3 regular and 11 compound 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|100 icosahedra&lt;br&gt;120 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|5 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 icosahedra&lt;br&gt;6 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|120
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|4
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|50
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|7.2°
|align=center|36°
|align=center|6°
|align=center|7.2°
|}

....

Here we summarize our findings on the 11-cells so far, including in advance some results to be obtained in the next section two sections.

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section....

== The 15 major chords ring everything ==
The natural numbers are each a distinct flavor.

Each integer has its own distinct taste. The first six integers {0..5} are the six strongest, most familiar tastes in nature, instantly distinguished and utterly unique.

The seventh flavor {6} is the first ratio of two of the others. Thereafter follows in strange sequence all the ratios of all the flavors, each marriage of every two flavors, most of which are fractions &lt;math&gt;\{k/d\}&lt;/math&gt;, but some of which are integers &lt;math&gt;\{k\}&lt;/math&gt;.

Some of these ratios are successful flavors in their own right, as distinct and inimitable as the marriage of chocolate and oranges, but the marriage of many flavor pairs is mud. It is a rule of thumb among flavor specialists that nearly any taste can be successfully alloyed with either onions or chocolate, but the direct marriage of these two flavors is notably unsuccessful. Some flavors make famous pairings or infamous ones, and subtle new pairings are always to be discovered that work surprisingly well for some purpose, but the majority of flavor marriages are undistinguished compounds.

In polytope geometry, each chord of a polytope is both is a distinct 1-dimensional object, a chord of the sphere of a distinct length &lt;math&gt;l&lt;/math&gt;, and a distinct natural number &lt;math&gt;h&lt;/math&gt;, a unique flavor. If the polytope is regular, it is a noteworthy and successful flavor. The chord's length &lt;math&gt;l&lt;/math&gt; is a square root, related to the natural number &lt;math&gt;h = k/d&lt;/math&gt; and to a set of polytopes they both represent, by a formula discovered by Steinbach.{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden''; Figure 5. Optimal sections and proportions|p=37|ps=; the regular polygons {5}, {7}, {9} and {11} with their diagonals define respectively: {5} the golden bisection proportional to 𝜙; {7} an analogous trisection; {9} an analogous quadrasection; {11} an analogous pentasection.}} The chord length &lt;math&gt;l&lt;/math&gt; is related to the number of sides of the regular polygon formed by the chord (its numerator &lt;math&gt;k&lt;/math&gt;), and to the winding number or density of its regular skew Petrie polygon (its denominator &lt;math&gt;d&lt;/math&gt;).

The regular 4-polytopes do not contain an infinite variety of these chord flavors, but only 30 of them: only 15 unique pairs of 180° degree complementary chords, which combine to make every regular ''n''-dimensional thing found in nature.{{Efn|The 15 ''major chords'' suffice to construct all the regular ''n''-polytopes. Their 180° complements include 15 ''minor chords'' which occur only in the 120-cell, not in any smaller regular polytope.{{Efn|name=Major and minor chords}} Including the complement of the 180° {{radic|4}} digon {2} chord (the {0} chord {{radic|0}}), and the skew variant of the 90° {{radic|2}} square {4} chord (the {30/7} chord), there are 32 distinct natural numbers which are all the chords of all the regular polytopes in all dimensions.|name=32 chords}} All the regular polytopes (including the largest one on the cover of the box) can be built not only from a box containing a number of blocks of 3 shapes glued together by gravity, they can be also be built from a box containing a number of sticks of 15 lengths glued together by rubber joints.

[[File:15 major chords.png|500px|thumb|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the others occur only in the interior of 120-cell, not as edges of any of the regular 4-polytopes. There are 30 distinct 4-space chordal distances between vertices within the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to identify any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=Major and minor chords}} chords are properly labelled #1 — #15 in the order they fan out by size.{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''; Figure 3|ps=; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with a &quot;fan of chords&quot; diagram like the one here.|p=23}} They join vertex pairs which are 1 — 15 edges apart on a &lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon. A #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; chord forms &lt;small&gt;&lt;math&gt;f+1&lt;/math&gt;&lt;/small&gt; regular geodesic polygons which include a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; disjoint regular {{nowrap|&lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt;-gons}} (e.g. &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; a compound of 5 disjoint hexagons) plus a single [[W:Triacontagon#Triacontagram|regular skew {{nowrap|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram}}]] that wraps around the sphere &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times before closing its circle. Notice the maverick #11 chord, with its non-conforming origin point for its {{nowrap|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;}} polygram.]]

These 15 chords (natural numbers) include the integers 2, 3, 4, 5, 6, 10, 11, 15 and 30. The others are the fractions 5/2, 10/3, 13/7, 15/2, 15/4, and 15/7. Represented among the chord lengths (the square roots) measured in ''unit radii'' (not in {{radic|2}} radii, as metrics are often given elsewhere in this article) are the {{background color|paleturquoise|roots of the first 4 positive integers {{radic|1}}, {{radic|2}}, {{radic|3}} and {{radic|4}}}}, the {{background color|#FFCCCC|root of the very distinguished rational fraction {{radic|5/2}}}}, and the roots of various distinguished irrational fractions as Steinbach discovered: for example, 4 of them are {{background color|yellow|roots of golden numbers (a small integer ± 𝜙)}}. The 15 chords each form their regular polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, which is just &lt;small&gt;&lt;math&gt;\{k\}&lt;/math&gt;&lt;/small&gt; in 8 cases. Arranged in order #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; from shortest edge-length square root to longest, the #1 — #15 chords bridge vertex pairs which are &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; vertices apart on a [[W:Triacontagon#Petrie polygons|&lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon]]. Each chord makes a [[W:Triacontagon#Triacontagram|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; polygram]] of its edge length, a compound of the smaller &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. The 15 distinct polygons form a sequence you would not anticipate exactly unless you have Steinbach's formula:

{| style=&quot;background-color:transparent; white-space:nowrap;&quot;
|-
| {30/1} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|.073}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{30} 
| 120-cell edges
|-
| {30/2}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.191}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15}
|- style=&quot;background: yellow;&quot;|
| {30/3}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.382}} 
| &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi}&lt;/math&gt;&lt;/small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{10}''' 
| 600-cell edges
|-
| {30/4} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.573}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/2}
|- style=&quot;background: paleturquoise;&quot;|
| {30/5}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|1}}'''
| &lt;math&gt;1&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{6}''' 
| 24-cell, 8-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/6}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|1.382}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{5}'''
|- style=&quot;background: paleturquoise;&quot;|
| {30/7}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|2}}'''
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{4}''' 
| 16-cell edges
|- style=&quot;background: #FFCCCC;&quot;|
| {30/8}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.5}} 
| &lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;&lt;small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/4} 
| 5-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/9}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.618}} 
| &lt;math&gt;\phi&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{10/3}
|- style=&quot;background: paleturquoise;&quot;|
| {30/10}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|3}}''' 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{3}'''
|-
| {30/11}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.426}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{11/11}
|- style=&quot;background: yellow;&quot;|
| {30/12}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.618}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{5/2}
|-
| {30/13}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.810}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{13/7}
|-
| {30/14}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.928}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/7}
|- style=&quot;background: paleturquoise;&quot;|
| {30/15}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|4}}''' 
| &lt;math&gt;2&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{2}'''
|}

{| class=wikitable style=&quot;white-space:nowrap; float:right;&quot;
!colspan=6|Natural number &lt;small&gt;&lt;math&gt;h=\{k/d\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Fibers&lt;br&gt;&lt;small&gt;&lt;math&gt;f\{h\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Cells&lt;br&gt;&lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|4-tope&lt;br&gt;&lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Verts&lt;br&gt;&lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|[[W:Triacontagon#Triacontagram|30-gram]]&lt;br&gt;&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;
|-
!Chord
!Arc
!colspan=2|&lt;small&gt;&lt;math&gt;l/\sqrt{1}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;l/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;h&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#1
|15.5~°
|{{radic|0.073}}
|0.270
|{{radic|0.146}}
|&lt;small&gt;&lt;math&gt;30&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1\{30\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;5\{3,3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/1\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#2
|25.2~°
|{{radic|0.191}}
|0.437
|{{radic|0.382}}
|&lt;small&gt;&lt;math&gt;15&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;..\{3,3\}..&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/2\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#3
|36°
|{{radic|0.382}}
|0.618
|{{radic|0.764}}
|&lt;small&gt;&lt;math&gt;10&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;30\{3,3\}20&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/3\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#4
|44.5~°
|{{radic|0.573}}
|0.757
|{{radic|1.146}}
|&lt;small&gt;&lt;math&gt;15/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/4\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5
|60°
|{{radic|1}}
|1
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;6&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/5\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#6
|72°
|{{radic|1.382}}
|1.175
|{{radic|2.624}}
|&lt;small&gt;&lt;math&gt;5&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}} 
|&lt;small&gt;&lt;math&gt;10\{5,3\}12&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/6\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#7
|90°
|{{radic|2}}
|1.414
|{{radic|4}}
|&lt;small&gt;&lt;math&gt;4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7\{4\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;4\{4,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;9\{3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/7\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: #FFCCCC;&quot;|
|#8
|104.5~°
|{{radic|2.5}}
|1.581
|{{radic|5}}
|&lt;small&gt;&lt;math&gt;15/4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/4\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}} 
|&lt;small&gt;&lt;math&gt;15\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/8\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#9
|108°
|{{radic|2.618}}
|1.618
|{{radic|5.236}}
|&lt;small&gt;&lt;math&gt;10/3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10/3\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/9\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#10
|120°
|{{radic|3}}
|1.732
|{{radic|6}}
|&lt;small&gt;&lt;math&gt;3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;10\{3\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/10\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#11
|135.5~°
|{{radic|3.426}}
|1.851~
|{{radic|6.852}}
|&lt;small&gt;&lt;math&gt;11/11&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;11\{11/11\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;11\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#12
|144°
|{{radic|3.618}}
|1.902
|{{radic|7.235}}
|&lt;small&gt;&lt;math&gt;5/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2},3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/12\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#13
|154.8~°
|{{radic|3.810}}
|1.952
|{{radic|7.621}}
|&lt;small&gt;&lt;math&gt;13/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;13\{13/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;7\{5,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/13\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#14
|164.5~°
|{{radic|3.928}}
|1.982
|{{radic|7.857}}
|&lt;small&gt;&lt;math&gt;15/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;137\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/14\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{radic|8}}
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15\{2\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐𝜙&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;8\{3,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/15\}&lt;/math&gt;&lt;/small&gt;
|}

The polygons '''{10}''', '''{6}''', '''{5}''', '''{4}''', '''{3}''', and '''{2}''' are flat polygons that each lie on a great circle in a central plane of the 3-sphere. The polygons {30}, {15}, {15/2}, {15/4}, {10/3}, {11/11}, {5/2}, {13/7}, {15/7} and all the {30/''n''} polygons are skew polygrams spiraling through curved 3-space, each successive edge in a different central plane of the 3-sphere. Projected to the 1-sphere (a circle in the plane), they are [[W:Star polygon#Regular star polygon|regular star polygrams]], the simplest being the pentagram {5/2}. In 4-space they wind around the 3-sphere more than once ''without self-intersecting'', before they close their circuit. They lie on helical geodesic circles or ''isoclines'' of circumference greater than 2𝝅''r''.

Even the 6 integer chords which form flat polygons also form skew polygrams. Each of the 15 chords occurs as a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, a fiber bundle of Clifford parallel polygons that totals 30 vertices per bundle. Each chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; also forms a single skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram that winds through another, shorter chord's bundle of parallel polygons, intersecting them in succession multiple times each, and weaving that whole bundle of parallel fibers together as the cross-thread does on a loom. The polygram cross-thread has to wind multiple times through the whole bundle (&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times around the [[W:Triacontagon#Triacontagram|30-gram]] in which each edge connects every &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt;th vertex) to complete an enumeration of all 30 vertices in the bundle. For example, the #5 chord forms hexagon &lt;small&gt;&lt;math&gt;\{6\}&lt;/math&gt;&lt;/small&gt; flat planes in parallel bundles of 5, written &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt;, and also a skew &lt;small&gt;&lt;math&gt;\{15/2\}&lt;/math&gt;&lt;/small&gt;-gram of #5 edges which winds 2 times through the #3 chord's bundle of 3 decagon &lt;small&gt;&lt;math&gt;\{10\}&lt;/math&gt;&lt;/small&gt; flat planes. Every chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; forms both its &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram and its fiber bundle of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, and the edges of one chord's parallel polygons are also the edges of its skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram weaving through some shorter chord's bundle of parallel polygons. Each numbered chord is a natural number, a ratio of two integers and their polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, either flat or skew. Each chord number defines a &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; skew polygon that is a compound bundle of the &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. All these flat or skew polygons are regular polygons made of their numerator instances of a single chord #, whose length is that square root, Steinbach's function of the natural number.

The relationship between each bundle of flat Clifford parallel great circle polygons of a distinct length, and its corresponding bundle of skew polygrams of a longer length, is precisely that of a discrete simple rotation and its corresponding discrete isoclinic rotation. For example, the flat great circle hexagons of edge length {{radic|1}} lie in the invariant rotation planes of the 24-cell's characteristic 60° isoclinic rotation, in which the vertices move on skew hexagram rotation paths of edge length {{radic|3}}. The 24-cell contains 4 bundles of 4 Clifford parallel (disjoint) hexagon central planes each. The 24 vertices of the 24-cell in each bundle are linked by rotation in the 4 hexagon invariant planes in two different ways: simply and isoclinically. In a simple rotation the vertices of a single invariant hexagon exchange places on that great circle hexagon of edge {{radic|1}}. In an isoclinic rotation in which any one of these 4 disjoint hexagon central planes is invariant, all 4 of them are necessarily invariant planes in the rotation, because they rotate in parallel. In that isoclinic rotation the vertices of all 4 hexagon invariant planes (all 24 vertices of the 24-cell) exchange places concurrently, moving along 4 disjoint helical hexagrams of edge {{radic|3}} in parallel. It is a characteristic only of this distinct rotation in hexagon invariant planes that the helical vertex path or ''isocline'' is a hexagram of period 6, edge length {{radic|3}}, and circumference 720°. Rotations in other kinds of polygon invariant planes have different characteristic isoclines. The period of their isoclinic rotation must be ''the same or greater'' than the period of their simple rotation; in most cases (unlike this one) it is greater.

The 15 chords form not only distinct fiber bundles of polygons, but also distinct bundles of polyhedra: cell rings &lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt; and compound vertex figures &lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;. These combine to form their characteristic regular 4-polytope &lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Coxeter|1973|p=269|loc=Compounds|ps=; &quot;It is remarkable that the vertices of {5, 3, 3} include the vertices of all the other fifteen regular polytopes in four dimensions.&quot;}} All these elements of a regular 4-polytope constructed from one chord are themselves regular polytopes, and each of the 16 regular 4-polytopes is a function of one characteristic chord. If the regular 4-polytope is convex, its characteristic chord is its unit-radius edge. For a non-convex regular 4-polytope, the characteristic chord is its unit-radius ''mid-edge diameter'', the distance from the midpoint of each edge to the midpoint of the completely orthogonal edge. Each of the characteristic chords (each distinct natural number) has its expression as one of the 16 regular 4-polytopes and all the products of the distinct regular 1-polytopes, 2-polytopes and 3-polytopes that construct it, and the whole distinct construction from the natural number to the 4-polytope is an [[W:SO(4)|SO(4) symmetry group]] expression of that 4-polytope's characteristic discrete [[W:SO(4)#Isoclinic rotations|isoclinic rotation]], in the invariant central planes of its characteristic chord, by the characteristic chord arc.

...demonstrate this theory by proper enumeration of the chord lengths in the table...

...color-code the △ ☐ 𝜙 ✩ symbols ... chord colors in diagram should match them ... some of the diagram colors are wrong and must be changed, but otherwise diagram is pretty good ... i think in a colored diagram all chords of the same length should have the same size line dash, the polygons are not obvious enough and the 15 fanning-out chords are readily identified by their labels...

... add back the counts of chords in the box (very small font number under each table entry), and present these two images as the front and back a box of building sticks and rubber joints...What's in the other box...

...say the rational-root major chords {{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|5/2}} are the ''n''-edges: their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge...

...abbreviated text of the [[120-cell#Relationships among interior polytopes|''§Relationships among interior polytopes'']] section from the 120-cell article...to explain the color-coding

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius=1}}

See also the elements, properties and metrics of the sequence of regular 4-polytopes summarized in tabular form in the section ''[[#Build with the blocks|§Build with the blocks]]'', below.

== Alicia Boole Stott's original formulation of dimensional analogy ==
&lt;blockquote&gt;&quot;...a method by which bodies having a certain kind of semi-regularity may be derived from regular bodies in an Euclidean space of any number of dimensions...&quot;{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''; presents cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues|pp=12-45}}&lt;/blockquote&gt;

Alica Boole Stott's 1910 paper reads like an exceptionally well-written patent application. It is a comprehensive stepwise description of her original invention, a general method of dimensional analogy between regular and semiregular polytopes of the same edge length.

[[File:The Roots of Three.png|thumb|An illustration of Boole Stott's expansion of unit-edge triangle to hexagon captures the geometric basis in 3 of the &lt;math&gt;A_n&lt;/math&gt;root system.]]

The engineering-like articulation of her new principles is entirely discrete and group-theoretic. Boole Stott repeatedly refers to sets of polytope elements as ''groups''. All her transformations are discrete instances of one or another special case of her &quot;two inverse operations which may be called ''expansion'' and ''contraction''.&quot; Hers is an exploration of ''n''-dimensional space at its most basic: space defined by the discrete objects which fill it, themselves defined directly as groups of elements with a few operations on them, which transform them into each other. This is geometry which seems to precede Euclid's postulates, and does not depend on them. Instead it corresponds directly to the deeper group theory mathematics which underlies geometry, even though Boole Stott, like the ancients, did not have that mathematics.

Boole Stott enumerated her expansion and contraction operations as transformations between regular and semi-regular polytopes of a single edge length. In her 1910 paper she did not consider any polytopes of multiple edge lengths, or expansions of the subject polytope by a distance other than that exactly required to produce new edges of the original edge length, so of course she did not enumerate Moxness's non-uniform rhombicosidodecahedron (Hull #8). This may be why section 8&lt;sub&gt;3&lt;/sub&gt; is missing from her earlier drawings of the sections of {5,3,3}; she may not have had a construction for it.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The Boole Stott expansion (or contraction) operation is as precisely defined a transformation as Coxeter's reflection, rotation and translation operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions). Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;
where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;
where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;
For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} An instance of the expansion (or contraction) operation is an application of the general operation to a particular subject polytope, operating on all of its elements symmetrically at once. The general operation is restricted, such that one may only expand (or contract) the subject polytope by pushing all its edges, faces or cells directly away from (or toward) its center, without changing the size of the moved elements. This splits each vertex into several (or coalesces several vertices into one), and has corresponding effects on the other elements, producing new edges ''of the same size'' between pairs of separated vertices (or reducing some edges to single vertices). The distance all the elements are moved away from (or toward) the center must be such that the vertices separate producing new edges of only the original edge length (or come together reducing edges to single vertices). It is permitted to combine these three operations, e.g. by pushing the edges ''and'' the faces, or by pushing all three element sets, provided the combination results in a semi-regular polytope of only one edge length. The three operations are all commutative with one another, so combined operations can be considered either as a sequence of expansions (or contractions) linking several related polytopes, or as their concurrent movements, a single transformation of the subject polytope. Since there are 8 combinations of one to three operations there are 8 polytope variants (including the original regular polytope), and all are adjacent to one another by a single transformation. The variants of the original regular polytope are named for which element set(s) are moved. By enumerating all 8 variants of expansion (and contraction) for each of the six regular convex 4-polytopes, Boole Stott produced a table of regular and semi-regular 4-polytopes divided into 12 sets of 8: six expansion sets and six contraction sets.{{Sfn|Boole Stott|1910|loc=''Table of Polytopes in S&lt;sub&gt;4&lt;/sub&gt;''; note that Boole Stott's usage of ''S&lt;sub&gt;4&lt;/sub&gt;'' denotes a Euclidean 4-space ''E&lt;sub&gt;4&lt;/sub&gt;'', or a 3-sphere space embedded in it, ''not'' a 4-sphere space embedded in a Euclidean 5-space|p=30}} Each set includes just one of the six regular convex 4-polytopes, with its distinct unit-radius edge length, and the 7 possible semi-regular expansions (or contractions) of it, which all preserve that characteristic edge length. The polytopes in each set of 8 all have the same total number of elements as their original regular polytope, but each polytope in the set has a distinct combination of cell, face, and vertex figures. The 12 sets are not completely disjoint, and even within one set the 8 operations do not necessarily produce 8 unique polytopes. For example, the regular 5-cell (the 4-simplex polytope) has 5 cells, 10 faces, 10 edges, and 5 vertices (30 elements). Expanding the 5-cell produces 7 related semi-regular 4-polytopes of 30 elements each (two of which happen to be the same polytope). In each expansion, the 5 cells have expanded into semi-regular or regular polyhedra, the 10 faces and 10 edges ''may'' have expanded into prisms, and the 5 original vertices have expanded into regular or semi-regular cells.

== The regular 11-cell 4-polytope ==

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosadodecahedron). Each non-convex Jessen's icosahedron is a set of 6 reflex edges that are {{radic|5}} 5-cell edges, each set symmetrically arranged in 3 parallel pairs, not as far apart ({{radic|3}}) as they are long, in 3 orthogonal central planes. The 12-point (Jessen's sets of 6 reflex 5-cell edges) are inscribed in the 60-point with their endpoints as a vertex of a pentagon face. The 12 pentagon faces are joined to 5 other pentagon faces by the 5-cell edges; not to their nearest neighbor pentagons but to their next-to-nearest neighbor pentagons; and since there are 5 vertices to each pentagon face, there are 10 12-point (Jessen's sets of 6 reflex 5-cell edges). Each pentagon face is joined to another pentagon face by 5 reflex edges of five distinct Jessen's. The 5 reflex edges are Clifford parallel, the 5 edges of a pentagonal prism column that has a twist to it. The contraction of those 5 edges into 1, and those 10 vertices (two pentagons) into 2, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing all their edges as a single Jessen's. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (octahedron) by conflating its parallel pairs of edges, leaving the 11-cell's abstract 6-point (hexad cell). The hexad's situation is that it occurs in bundles of 5, disjoint but close together in the sense of adjacent, like the great circle rings in a Hopf fiber bundle. 

To summarize, the 12-point (2-hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell). We shall now be able to see how Fuller's perfect hexads combine with Todd's perfect pentads to form the perfect heptads of the 11-cell and 137-cell.  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=alias the hemi-cuboctahedron}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces. (The ''heptad'' arrives at the party, anonymously; they are our distinguished anti-celebrity, who most of the other guests have never even heard of.) The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. This 12-point (truncated tetrahedron) is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 100 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp: Truncated tetrahedron|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp: Truncated tetrahedron|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}} Of course Moxness has been there, seen that, already with his graphics software;{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} see his parameterized illustration of it exactly as it occurs in the 11-cell, below in his bestiary of A3, B3, and H3 Archimedian and Catalan polyhedra in the 120-cell.]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:Archimedean_and_Catalan_solid_hulls_with_their_Weyl_orbit_definitions.svg|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of the 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. In this catalog of Archimedean and Catalan polyhedra sighted in the 120-cell by Moxness's software,{{Sfn|Moxness: Archimedean and Catalan hulls|2023|loc=Hull #1 Archimedean Name A3 110 Truncated Tetrahedron A (upper left)}} it is the first entry (upper left), Moxness's Hull # = 1 with 12 vertices of 3D norm {{radic|11}}/2. On a {{radic|2}}-radius 3-sphere, 5 of these with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell. 20 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges are cells in the [[#The rings of the 11-cells|cell rings of the quasi-regular 11-cell]], which also has 12 regular 5-cells as cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

...

Both kinds of cells in the 11-cell have expressions in two different dimensions (on the 2-sphere and 3-sphere), because they are all centered on the center of the 3-sphere, unlike the cells of most 4-polytopes, which are centered on various off-center points within the 4-polytope's interior. 

One kind of 11-cell cell appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (semi-regular truncated cuboctahedron), a.k.a. the truncated tetrahedron. The other kind of cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now, except in the 120-cell. But this is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron, and a regular tetrahedron behind each face. The 11-cell is thus a honeycomb of mixed regular tetrahedra and truncated cuboctahedra (a.k.a. truncated tetrahedra) that meet at their shared 5-cell faces. 

Both the 5-cells (decahedra) and the truncated cuboctahedra (truncated tetrahedra) are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks in their 4-polytopes (off-center).{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} Therefore all the 11-cell's cells are concentric, which indeed makes a different kind of honeycomb, an inside-out cellular honeycomb with all the cells piled on top of each other overlapping in 4-space. They do not overlap in 3-sphere space however; their 3-volumes (but not their 4-contents) are disjoint. They make a proper 4D honeycomb like the cells of any 4-polytope, with volumetrically disjoint 3D cells that meet each other in the proper way at their shared 2D faces.

An 11-cell contains 6 completely disjoint 5-point (1-pentad 5-cell decahedron cells), and 5 volumetrically disjoint 12-point (2-hexad truncated cuboctahedron cells), 11 volumetrically disjoint cells altogether. Of course it is supposed to have 11 identical 11-point (elevenad hemi-icosahedral cells), and that is also true in its abstract sense, which is not the same sense as disjoint cellular containment: each pentad or hexad cell is inscribed in one of 11 larger 60-point (Moxness's rhombicosadodecahedron cells). The abstract pentads and hexads are only hemi-polyhedra, so the 11-cell's pentad cells are 12 (out of 120) disjoint instances of a pentad 5-point, and its hexad cells are 10 (out of 100) disjoint instances of a hexad 6-point, 1/10th of the 120-cell's cellular volume in both cases. The 11-cell's hemi-icosahedral cells are 11 (out of 120) non-disjoint instances of an elevenad 60-point. Each 11-cell shares each of its hexad cells with 9 other 11-cells, and each of its 11 vertices (out of 600) occurs in 9 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.|name=Tetrahedral vertex figures}}

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

...insert image thumb of 11-cell...

...Construct the 11-cell from the heptad....

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...Four concentric 4-polytopes may be mutually orthogonal. In any set of 4 mutually orthogonal regular 5-cells, one pair of faces in each orthogonal pair are completely orthogonal to each other. Thus three edges in one 5-cell in an orthogonal pair are opposite (completely orthogonal) to a face of the orthogonal 5-cell. .... A 5-cell has 5 vertices in its abstract hemi-icosahedral cells, but only 4 vertices in one concrete icosahedral cell; the 5th lies in an icosahedral cell of another 11-cell (not in another icosahedral cell of the same 11-cell).

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

....

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 11-cell ===

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon.... (wrong, 11-cell's Hopf map is the truncated icosahedron)

....

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk: Truncated icosahedron|2018}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

...

Since the rhombicosadodecahedron has 120 5-cell edges, the 120-cell has 120 completely disjoint 5-cells, and the rhombicosadodecahedron shares each vertex with 11 other rhombicosadodecahedra, how a rhombicosadodecahedron is bound to its neighbor 5-cells is constrained. A further constraint is that the abstract hemi-icosahedron has only 15 5-cell edges, so the 120 actual 5-cell edges of the rhombicosadodecahedron are conflated 8-to-1 in the hemi-icosahedron. To be conflated by contraction into a single edge, those 8 edges must lie parallel to each other in the 120-cell; either actually parallel in the same plane, or Clifford parallel in separate planes, as two edges in completely orthogonal planes may be parallel and orthogonal at the same time. Each 5-cell inscribed in the 120-cell contains 5 such tetrahedra occur in parallel pairs in 4 orthogonal planes (none of them completely orthogonal planes). 

.... 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
Each conflated vertex is the 10 vertices of two orthogonal regular 5-cells...
The 60-point contains 6 pairs of these orthogonal regular 5-cell cells...
...its 20 hexad cells are ..., a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere.

== The regular 137-cell 4-polytope ==

(description originally written for the 11-cell, which applies only to a fibration of 11 11-cells)

It is quasi-regular with two kinds of cells, the 12-point (regular icosahedron) and the 5-point (regular 5-cell), which have just one kind of regular face between them. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of regular facets (it is semi-regular with regular facets), it also means that its facets have expressions of two different dimensionalities because they are concentric in the 3-sphere. The 11-cell has two kinds of regular cells. One appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (regular icosahedron), but since it is embedded at the center of the 3-sphere (unlike the cells of most 4-polytopes) it also acts as an inscribed 4-polytope.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} The other cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now except in the 120-cell. This is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron. The 11-cell is a honeycomb of mixed icosahedron and half-icosahedron cells. Like the icosahedra, the decahedra are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks, off-center in their 4-polytopes. All the 11-cell's cells are concentric, which indeed makes an unusual honeycomb of cells, an inside-out honeycomb. It is a proper 4D honeycomb however, with volumetrically disjoint 3D cells that meet each other in the proper way at shared 2D faces. The embedded 12-point (regular icosahedron) cell is an icosahedral pyramid (as a 4-polytope), which is filled by 20 4-point (irregular tetrahedra) that meet at its center; the center point of the 11-cell acts as a 12th vertex, just as the center point of the 24-cell acts as a 25th vertex in some honeycombs. The 5-point (regular 5-cell) cell contains five 4-point (regular tetrahedra) which meet the 12-point (regular icosahedra)'s irregular tetrahedral cells face-to-face, so the 11-cell is quasi-regular in the usual sense of having two varieties of regular 3-polytope cells, but additionally both varieties of of its cells are bi-dimensional: another clue that the 11-cell has something to do with dimensional analogy

....

The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

....

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur to make an 11-cell. 11 of them occur in a 137-cell, all 120 of them occur in the 120-cell, and they occur nowhere else.

How can this regular 4-polytope possibly exist? By Schläfli's criterion {3, 3, ..} exists, but not as a finite polytope.{{Sfn|Coxeter|1973|loc=§7.7 Schläfli's criterion}} Coxeter found {3, 5, 3} as a finite hyperbolic honeycomb, but not in Euclidean 4-space.{{Sfn|Coxeter|1968|title=The Beauty of Geometry: Twelve Geometric Essays|loc=''Regular Honeycombs in Hyperbolic Space''|ps=; &quot;The honeycombs considered by the above authors [Schlegel, Schläfli, Klein, Sommerville] have finite cells and finite vertex figures. ... A further extension allows a cell or vertex figure to be a ''star''-polytope, so that the honeycomb covers the space several times. .... We shall find that there are four regular star-honeycombs in hyperbolic 4-space, as well as two infinite families of them in the hyperbolic plane.&quot;}} We find that {3, 3, ..} is a central expression of the &lt;math&gt;H_4&lt;/math&gt; root system, being the regular convex hull in 4 dimensions of 137 points, 96 pentads and 600 hexads. The minimal expression of &lt;math&gt;H_4&lt;/math&gt; is the regular convex hull {3, 3, 5} of 120 points, 600 tetrads and 75 hexads, and its maximal expression is the regular convex hull {5, 3, 3} of 600 points, 120 pentads and 675 hexads. {3, 3, ..} is a third regular &lt;math&gt;H_4&lt;/math&gt; polytope which fits between them. The properties and metrics of this regular sequence are presented in tabular form below in ''[[#Build with the blocks|§Build with the blocks]]''.

{|class=&quot;wikitable&quot;
!colspan=3|Regular &lt;math&gt;H_4&lt;/math&gt; polytopes
|-
|[[File:600-cell.gif|200px]]
|[[File:5-cell.gif|200px]]
|[[File:120-cell.gif|200px]]
|-
!120-point 600-cell
!137-point 137-cell
!600-point 120-cell
|}

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 137-cell ===
....

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=Apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|2022|loc=''[[Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]''}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie: On Fuller's use of language|2024|loc=''[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry'']]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center, 1948-1949|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with their mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the gender neutral shad can open their six beaks all the way, until they become flat squares and they becomes a cuboctahedron, or they can shut them all tight like a turtle retracting into their octahedron shell. The six mouths always move in unison. This is [[Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do their ''really'' odd trick -- where they flip their 6 jaws 90 degrees in their 6 faces and shut their 6 beaks on the opposite axis of their squares than the one they opened them on -- or whether they will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than their normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make their edge length exactly the same size as their radius, when they open their mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing their mouths in spherical synchrony, their 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron (the hexad), then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=''[[W:SO(4)|SO(4)]]''}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=Isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its orthogonal hexad's planes (wz, wx, wy).{{Efn|name=Six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of (expanded octahederon 16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh &amp; Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== Legendre's 12-point hexad binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a 12-point double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's perfectly realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, this 2-hexad expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre hexad) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction.{{Efn|name=Concentric polyhedra}} In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== ''n''-simplexes ===
....

The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== ''n''-orthoplexes ===
....
[[File:Great 5-cell √5 digons rectangle.png|thumb|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} ''regular'' great hexagons. These are the same 200 dodecagon central planes illustrated above, which also contain 6 120-cell edges and two opposing ''irregular'' great hexagons (truncated {{radic|6}} triangles). The 1200 {{radic|5}} 5-cell face triangles occupy 1200 other, non-central planes.]]

...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's Hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[16--cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[16--cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=Concentric polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[16--cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== ''n''-taliesins ===
The 4-simplex pentad building block combines with the 4-orthoplex hexad building block to make the 4-taliesin heptad building block. The heptad is the essential building block of the ''taliesin'' family of polytopes. The 11-cell and 57-cell hemi-polytopes and the 137-cell and 120-cell regular 4-polytopes are nuclear members of this trans-dimensional family. The 24-cell and 600-cell are extended members of it, because they also contain instances of the heptad building block.

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its spherical symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* [[W:Taliesin Myrddin Namkai-Meche|Taliesin Myrddin Namkai-Meche]], a young Reed College scholar who was stabbed to death by a crazy man in 2017 on a Portland, Oregon light rail train, when he rose to the nonviolent defense of two black teenagers who were being violently assaulted in racist and anti-Muslim language by a right-wing extremist.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in orthogonal [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].
* '''Taliesan polytope''', an (''n+1)''-polytope whose ''n''-polytope dimensional analogue inscribed concentrically in the ''n''-sphere forms a convex honeycomb of itself which is the (''n+1)''-polytope.{{Efn|name=Embedding point}}
* The taliesan polytope's honeycomb, a Hopf fibration of its great circles.
* The ''n''-dimensional family of dimensionally analogous polytopes related to a taliesan polytope by expansion and contraction operations.
* The characteristic [[w:Orthoscheme|orthoscheme]] of a teliesan polytope, an irregular simplex that is the complete gene for the entire expression of trans-dimensional symmetry relationships among the polytopes of the ''n''-dimensional family, as well as the complete gene for the teliesan polytope's rotational symmetry.{{Efn|name=Characteristic orthoscheme}}

=== ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== ''n''-equilaterals ===
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed at four different [[#n-leviathon|chord]] lengths from each other. In the {{radic|2}}-radius polytopes we have using as examples in this paper they are {{radic|2}}, {{radic|4}}, {{radic|6}} and {{radic|8}}, but in a ''unit-radius'' polytope they are: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}.

Each vertex is joined to 8 others by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}. The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== ''n''-pentagonals ===

[[File:600-cell.gif|thumb|The 120-point (600-cell), &lt;math&gt;H^4&lt;/math&gt;.]]
...

[[File:600-cell vertex geometry.png|thumb|Vertex geometry of the 600-cell, showing the 5 regular great circle polygons and the 8 vertex-to-vertex chord lengths with angles of arc. The golden ratio{{Efn|name=golden chords|group=}} governs the fractional roots of every other chord,{{Efn|name=fractional root chords}} and the radial golden triangles which meet at the center.|alt=|400x400px]]
The 120 vertices of the 600-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏&lt;sup&gt;−1&lt;/sup&gt;) beginning with a vertex; see column ''a''}} at eight different [[#n-leviathon|chord]] lengths from each other. These edges and chords of the 600-cell are simply the edges and chords of its five great circle polygons. In ascending order of length in a ''unit-radius'' polytope (not in a {{radic|2}} radius polytopes like our usual examples), they are {{radic|0.𝚫}}, {{radic|1}}, {{radic|1.𝚫}}, {{radic|2}}, {{radic|2.𝚽}}, {{radic|3}}, {{radic|3.𝚽}}, and {{radic|4}}.{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=Decimal fractions of square roots|group=}}

Notice that the four [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#''n-''equilaterals|hypercubic chords]] of the 24-cell ({{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|4}}) alternate with the four new chords of the 600-cell's additional great circles, the decagons and pentagons. The new ''golden chord'' lengths are necessarily square roots of fractions, but very special fractions related to the golden ratio{{Efn|1=The fractional-root ''golden chords'' are irrational fractions that are functions of {{radic|5}}. They exemplify that the [[W:golden ratio|golden ratio]] &lt;big&gt;𝜙&lt;/big&gt; {{=}} {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618 is a circle ratio related to &lt;big&gt;𝜋&lt;/big&gt;:&lt;br&gt;
: {{sfrac|𝜋|5}} = arccos (𝜙/2)
is one decagon edge, the 𝚽 = {{radic|0.𝚫}} = {{radic|0.382~}} ≈ 0.618 chord.
Reciprocally, in this function discovered by Robert Everest expressing &lt;big&gt;𝜙&lt;/big&gt; as a function of &lt;big&gt;𝜋&lt;/big&gt; and the numbers 1, 2, 3 and 5 of the Fibonacci series:&lt;br&gt;
: &lt;big&gt;𝜙&lt;/big&gt; = 1 – 2 cos ({{sfrac|3𝜋|5}})
{{sfrac|3𝜋|5}} is the arc length of the &lt;big&gt;𝜙&lt;/big&gt; = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618 chord.|name=golden chords|group=}} including the two [[W:golden section|golden section]]s of {{radic|5}}, as shown in the diagram.{{Efn|The 600-cell edges are decagon edges of length {{radic|0.𝚫}}, which is 𝚽, the ''smaller'' golden section of {{radic|5}}; the edges are in the inverse [[W:golden ratio|golden ratio]] {{sfrac|1|φ}} to the {{radic|1}} hexagon chords (the 24-cell edges).
The other fractional-root chords exhibit golden relationships as well. The chord of length {{radic|1.𝚫}} is a pentagon edge.
The next fractional-root chord is a decagon diagonal of length {{radic|2.𝚽}} which is &lt;big&gt;𝜙&lt;/big&gt;, the ''larger'' golden section of {{radic|5}}; it is in the golden ratio{{Efn|name=golden chords|group=}} to the {{radic|1}} chord (and the radius).{{Efn|Notice in the diagram how the &lt;big&gt;𝜙&lt;/big&gt; chord (the ''larger'' golden section) sums with the adjacent 𝚽 edge (the ''smaller'' golden section) to {{radic|5}}, as if together they were a {{radic|5}} chord bent to fit inside the {{radic|4}} diameter.}}
The last fractional-root chord is the pentagon diagonal of length {{radic|3.𝚽}}.
The [[W:Pentagon#Side length is given|diagonal of a regular pentagon]] is always in the golden ratio to its edge, and indeed &lt;big&gt;𝜙&lt;/big&gt;{{radic|1.𝚫}} is {{radic|3.𝚽}}.|name=fractional root chords|group=}}

....


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== ''n''-leviathon ===
...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of its &lt;math&gt;10^2&lt;/math&gt; disjoint instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction, before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]].{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}} .... It is only because she did not have the 11-cell to start with that she did not discover the regular 137-cell {3, 3, ..} by her method.

Between Alicia Boole Stott and her identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and her identification of symmetry groups with conservation laws (in her 1st theorem), we owe the deepest mathematics of geometry and physics originally to two pioneering women, colleagues of their contemporaries the greatest men of the academy, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just an [[W:abstract polytope|abstract 4-polytope]], not just a singleton convex 4-polytope, and not just a honeycomb.{{Sfn|Coxeter|1970|loc=''Twisted Honeycombs''}} The 11-point (11-cell) has a concrete ...regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}&lt;nowiki/&gt;} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, 3, 3} as the 137-point (137-cell).

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 11-cell, 137-cell and 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=''Clifford Spinors and Root System Induction: H4 and the Grand Antiprism''}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=Apology}}

== Build with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Duveneck|1978|loc=Carl Jung, quoted in ''Life on Two Levels''|p=ii|ps=.{{Sfn|Jung|1961|ps=: &quot;The best of truths is of no use unless it has become one's most personal inner experience. It is the duty of everyone who takes a solitary path to share with society what he finds on his journey of discovery.&quot;}}}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1954|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius={{radic|2}}}}

== Acknowledgements ==
...

== Notes ==
{{Regular convex 4-polytopes Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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{{Refend}}</text>
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      <text bytes="208667" sha1="mo8noubwy5wkvazoj34jaz22nlxpf4t" xml:space="preserve">{{align|center|David Brooks Christie}}
{{align|center|dc@samizdat.org}}
{{align|center|June 2024}}

&lt;blockquote&gt;[[W:Branko Grünbaum|Grünbaum]] and [[W:H.S.M. Coxeter|Coxeter]] independently discovered the [[W:11-cell|11-cell]], a regular 4-polytope with cells that are the [[W:hemi-icosahedron|hemi-icosahedron]], a hexad non-orientable polyhedron. The 11-cell is often described as an abstract 4-polytope, because its cells do not have a direct realization in Euclidean 3-space. However, the 11-cell has a concrete realization in Euclidean 4-space, inscribed in the [[120-cell|120-cell]], the largest regular convex 4-polytope, which contains inscribed instances of all the regular 4-polytopes. The 11-cell contains 11 hemi-icosahedra and 11 regular 5-cells. The 120-cell contains 120 dodecahedra, 120 regular 5-cells, 120 hemi-icosahedra and 120 11-cells. The 11-cell (singular) is the &lt;s&gt;120-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 5, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 11 [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] cells. Eleven 11-cells (plural) are the 137-cell, the &lt;s&gt;600-vertex regular 4-polytope {{nowrap|{&lt;small&gt;{{sfrac|5|2}}&lt;/small&gt;, 3, 3}&lt;nowiki/&gt;}}&lt;/s&gt; with 137 [[W:Rhombic triacontahedron|triacontahedron]] cells.&lt;/blockquote&gt;

== Introduction ==
[[W:Branko Grünbaum|Branko Grünbaum]] discovered the 11-cell around 1970,{{Sfn|Grünbaum|1976|loc=''Regularity of Graphs, Complexes and Designs''}} about a decade before [[W:H.S.M. Coxeter|H.S.M. Coxeter]] extracted hemi-icosahedral hexads from the permutations of eleven numbers, with observations on the perfection of Todd's cyclic pentads and other symmetries he had been studying.{{Sfn|Coxeter|1984|loc=''A Symmetrical Arrangement of Eleven Hemi-Icosahedra''}} Grünbaum started with the hexad, and the impetus for his discovery of the 11-cell was simply the impulse to build with them. Like a child building with blocks, he fit them together, three around each edge, until the arrangement closed up into a 3-sphere and surprise, ''eleven'' of them.

[[File:120-cell.gif|thumb|360px|The picture on the cover of the box.{{Sfn|Hise|2011|loc=File:120-cell.gif|ps=; &quot;Created by Jason Hise with Maya and Macromedia Fireworks. A 3D projection of a 120-cell performing a [[W:SO(4)#Geometry of 4D rotations|simple rotation]].&quot;}} The complex interior parts of the 120-cell, all its inscribed 11-cells, 600-cells, 24-cells, 8-cells, 16-cells and 5-cells, are completely invisible in this view, as none of their edges are rendered at all. The child must imagine them.]]
The 4-dimensional regular polytopes are the most wonderful set of child's building blocks. The simplest two 4-polytopes are the 5-point 4-[[W:Simplex|simplex]] (called the [[5-cell]], because it is built from 5 tetrahedra), and the 8-point 4-[[W:Orthoplex|orthoplex]] (called the [[16-cell]], because it is built from 16 tetrahedra). As building blocks they could not be more different. The 16-cell is the basic building block of everything 4-dimensional. Every other regular convex 4-polytope (''except'' the 5-cell) can be built as a compound of 16-cells, including first of all the [[w:Tesseract|16-point (8-cell) tesseract]], the 4-cubic, which is a compound of two 16-cells in [[W:Demihypercube|exact dimensional analogy]] to the way a cube is a compound of two tetrahedra. The regular 5-cell, on the other hand, is not found within any of the other regular convex 4-polytopes, except in the largest and most complex one, the [[120-cell|120-cell]], the biggest thing you can build from this set of building blocks (the picture on the cover of the box, which is built from everything in the box). However, we know from its [[W:Tetrahedral symmetry|tetrahedral symmetry]] group &lt;math&gt;A_4&lt;/math&gt; that the 5-cell has a fundamental relationship to all the other 4-polytopes, just not one as simple as compounding. The 5-cell is not immediately useful to children trying to learn to build with 4-dimensional building blocks, but the 16-cell is their very starting point, and the most frequently used tool in the box.

== 5-cells and hemi-icosahedra in the 11-cell ==
[[File:Symmetrical_5-set_Venn_diagram.svg|thumb|The 5-point (10-face) regular 5-cell (the regular 4-simplex). Grünbaum's rotationally symmetrical 5-set Venn diagram{{Sfn|Grünbaum|1975|loc=''Rotationally symmetrical 5-set Venn diagram'', Fig 1 (e)|ps=; partitions the individual elements of the 5-cell.}} is an illustration of the 5-cell labeling each of its &lt;math&gt;2^5&lt;/math&gt; elements.{{Sfn|Cmglee: Grunbaum's 5-point Venn Diagram|2019|ps=; each individual element of the 5-cell is labelled; image includes the Python code to render it, optimising for maximum area of the smallest regions.}}]]
[[File:Hemi-icosahedron.png|thumb|The 6-point (10 face) [[W:hemi-icosahedron|hemi-icosahedron]], an abstraction of the regular icosahedron, has half as many faces, edges and vertices. Each element of the abstract polyhedron represents two real elements found in different places in the 11-cell's realization.{{Sfn|Ruen: hemi-icosahedron|2007}}]]
The most apparent relationship between the pentad 5-cell and the hexad hemi-icosahedron is that they both have 10 triangular faces. When we see a facet congruence between a 4-polytope and a 3-polytope we anticipate a dimensional analogy. Another clue that the hemi-icosahedron has something to do with dimensional analogy comes from its realization in the 6-point 5-simplex. Another hexad is the 6-point 3-orthoplex; thus the hemi-icosahedron is related by dimensional analogy to the 4-simplex from above and to the 4-orthoplex from below, which are ''only'' related to each other indirectly by dimensional analogies, having no chord congruences in 4-space. The elevenad has only been at the party 5 minutes, and it is already inter-dimensionally ''involved'' with the two earliest arrivals, 5-cell and 16-cell, who are famously stand-offish with each other. Interesting!

The cell of the 11-cell is an abstract hexad hemi-icosahedron with 5 central planes, most handsomely illustrated by Séquin.{{Sfn|Séquin &amp; Hamlin|2007|loc=Figure 2. 57-Cell: (a) vertex figure|ps=; The 6-point (hemi-isosahedron) is the vertex figure of the 11-cell's dual 4-polytope the 57-point ([[W:57-cell|57-cell]]). Séquin &amp; Hamlin have a lovely colored illustration of the hemi-icosahedron in their paper on the 57-cell, revealing rings of polytopes nestled in its interior, subdivided into triangular faces by 5 central planes of its icosahedral symmetry. Their illustration cannot be directly included here, because it has not been uploaded to [[W:Wikimedia Commons|Wikimedia Commons]] under an open-source copyright license, but you can view it online by clicking through this citation to their paper, which is available on the web.}} The 11 hemi-icosahedral cells have 10 triangle faces each; the 5-cell's 5 tetrahedral cells have 10 faces and 10 edges all together. If 11-cell faces are 5-cell faces, then 3 of each 5-cell's 5 vertices are a hemi-icosahedron face, and its other 2 vertices must be some 11-cell edge lying opposite the face. Coxeter determined that the 11-cell does indeed have an edge opposite each face, but it does not belong to the same hemi-icosahedral cell as its opposing face. He found that the 10 edges opposite each hemi-icosahedron's faces are the 10 edges of a single 5-cell, which does not share any vertices, edges or faces with the hemi-icosahedron. Of course, in the 11-point 11-cell there is just one such 5-point 5-cell that is completely disjoint from each 6-point hemi-icosahedron.{{Sfn|Coxeter|1984|p=110|loc=§6. The Petrie polygon [of the 11-cell]|ps=; &quot;We may reasonably call this edge and face ''opposites''. It is easy to find the face opposite to a given edge by looking at the faces to which a given edge belongs. ... Conversely, given a face, we can find the opposite edge by seeing which vertices belong to neither of the hemi-icosahedra which share that face. The ten edges opposite to the ten faces of one hemi-icosahedron are the edges of the complementary &lt;math&gt;a_4&lt;/math&gt; [4-simplex], that is, the joins of all pairs of the five vertices [of the 11-cell] not belonging to the given hemi-icosahedron.&quot;}}

There are 11 disjoint 5-cell 4-polytopes inscribed in each 11-cell, which also contains 11 hemi-icosahedral cells, 55 faces, 55 edges and 11 vertices. The real 11-cell is more complex and harder to see than the abstract 11-cell representing it, because the real hemi-icosahedron is also more complex than the abstract hemi-icosahedron. Seeing the real 11-cell will be easier once we have identified the real hemi-icosahedron, and exactly where the 11-cell's real elements reside in the 4-polytopes with which they intermingle.

The 5-cell has 10 triangular faces, and the 11-cell has 10 triangular faces in each of its hemi-icosahedral cells, but that is not how their faces correspond. Each hemi-icosahedron's 10 faces come from 10 disjoint 5-cells, and there is just one 5-cell with which it does not share a face.{{Efn|As Coxeter observes (in the previous citation), that unrepresented 5-point (5-cell) is the other 5 vertices of the 11-point (11-cell) that are not vertices of this 6-point (hemi-icosahedron), the hemi-icosahedron's disjoint complement.}} Each 5-cell has 10 faces from 10 distinct hemi-icosahedra, and there is just one hemi-icosahedron with which it does not share a face.

In the abstract 11-cell each face represents two conflated icosahedron faces, two actual faces in different places, so the 11-cell's 55 faces represent 110 actual faces: the faces of 11 ''disjoint'' 5-cells. Each hemi-icosahedron vertex represents conflated icosahedral vertices: multiple actual vertices separated by a small distance which has been reduced to a point at the coarse scale of the abstraction. Seemingly adjacent hemi-icosahedron faces do not actually meet at an edge; there is a polygon separating them, which has been abstracted to an edge. The 5-cells do not touch each other; they are all completely disjoint.

In the 5-cell each face bonds two tetrahedral cells together, and in the 11-cell each face bonds two pairs of hemi-icosahedral cells together, because each 11-cell face represents two actual faces in two different planes completely orthogonal to each other. Each duplex 11-cell face bonds to two 5-cells in different places, without binding them together (they are disjoint). One actual 5-cell face meets one half of a duplex 11-cell face, so 110 5-cell faces are bonded to 55 duplex 11-cell faces. The 11-cell's 11 vertices include all 55 distinct vertices of the 11 disjoint 5-cells, so they must be conflations of 5 vertices. Therefore for any of this to be possible, the 11-cell must not be alone; multiple 11-cells must be sharing vertices.

[[File:120-Cell showing the individual 8 concentric hulls and in combination.svg|thumb|400px|right|
Orthogonal projections of the 120-cell by Moxness{{Sfn|Moxness: 8 concentric hulls|2022|loc=Hull #8 (lower right)|ps=; &quot;Orthogonal projection of the 120-cell using any 3 of these Cartesian coordinate dimensions forms an outer hull of a Chamfered dodecahedron of Norm=√8. Hulls 1, 2, &amp; 7 are each overlapping pairs of Dodecahedrons. Hull 3 is a pair of Icosidodecahedrons. Hulls 4 &amp; 5 are each pairs of Truncated icosahedrons. Hulls 6 &amp; 8 are pairs of Rhombicosidodecahedrons.&quot;}} using 3 of its 4 Cartesian coordinate dimensions to render 8 polyhedral hulls which are 3D sections through distinct hyperplanes. Hull #8 with 60 vertices (lower right) is the real polyhedral cell of the 11-cell that the abstract hemi-icosahedron represents.]]
We shall see in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Compounds in the 120-cell|§Compounds in the 120-cell]]'' that the 11-cell is not in fact alone, but first let us see if we can find an existing illustration of the actual hemi-icosahedron as it is realized in the 120-cell. Moxness developed software which uses Hamilton's [[w:Quaternion|quaternion]]&lt;nowiki/&gt;s to render the polyhedra which are found in the interior of ''n''-dimensional polytopes.{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} [[w:William_Rowan_Hamilton|Hamilton]] was the first wise child to discover a 4-dimensional building block, [[w:History_of_quaternions#Hamilton's_discovery|in his flash of genius on Broom bridge]] in 1843, though he didn't think of his quaternion formula {{math|1=''i''&lt;sup&gt;2&lt;/sup&gt; = ''j''&lt;sup&gt;2&lt;/sup&gt; = ''k''&lt;sup&gt;2&lt;/sup&gt; = ''ijk'' = −1}} as the 16-point (8-cell) 4-polytope, or that he'd discovered the 4-cubic and the tetrad (w, x, y, z) Cartesian [[w:Euclidean_geometry#19th_century|coordinates of Euclidean 4-space]]. Moxness built his software out of Hamilton's quaternions, as quite a lot of graphics software is built, because quaternions make rotations and projections in 3D or 4D space as simple as matrix multiplications. The quaternions are 4-cubic building blocks analogous to the 3-cubic blocks everyone built with as a child (only they fit together even better, because they are [[w:8-cell#Radial_equilateral_symmetry|radially equilateral]] like the cuboctahedron and the 24-cell, but we digress). Moxness used his software to render illustrations of some of the polyhedra inside the 120-cell, which he published. Notice his &quot;Hull # = 8 with 60 vertices&quot;, lower right. It is a real icosahedron that the abstract hemi-icosahedron represents, as it occurs in the 120-cell. Moxness's 60-point (Hull #8) is a concrete realization of the 6-point (hemi-icosahedron) in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;, embedded in Euclidean 4-space &lt;math&gt;R^4&lt;/math&gt;. Its little pentagon faces are 120-cell faces. Its larger triangle faces happen to be [[600-cell]] faces, separated from each other by rectangles.

[[File:Irregular great hexagons of the 120-cell radius √2.png|thumb|Every 6 edges of the 120-cell that lie on a great circle join with 5-cell edges to form two opposing irregular great hexagons (truncated triangles). The 120-cell contains 1200 of its own edges and 1200 5-cell edges in 200 central planes. The 5-cell ''faces'' do not lie in central planes.]]
The edges of still larger 5-cell face triangles are there too, but they are invisible chords below the surface of Moxness's 60-point polyhedron. To see them, notice that {6} 120-cell edges (little pentagon edges) lie on a great circle, alternating with {6} rectangle diagonals. Also lying on this {12} irregular great dodecagon are {6} 5-cell edges, joining every other 120-cell edge as an invisible chord running under the 120-cell edge between them, and forming two opposing {6} irregular great hexagons (truncated triangles) of alternating 120-cell edges and 5-cell edges. The great dodecagon is a great circle of the hemi-icosahedral cell, and also a great circle of the 120-cell, because unlike the cells of most 4-polytopes, the 11-cell's cells are embedded in 4-space at the exact center of the 4-polytope, making them inscribed 4-polytopes themselves.{{Efn|name=Embedding point}} There are 25 great dodecagons and 120 5-cell edges in Moxness's 60-point (Hull #8), and 200 great dodecagons and 1200 5-cell edges in the 120-cell. But the 5-cell ''faces'' do not lie in these central planes; each of the 3 edges of a 5-cell face lies in a different hyperplane, that is, in a different 60-point (Hull #8) polyhedron altogether.

[[File:Nonuniform_rhombicosidodecahedron_as_rectified_rhombic_triacontahedron_max.png|thumb|Moxness's 60-point (Hull #8) is a nonuniform [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] similar to the one from the catalog shown here,{{Sfn|Piesk: Rhombicosidodecahedron|2018}} but a slightly shallower truncation with smaller red pentagons (120-cell faces) and narrower rhombs. Rhombicosidodecahedra are made by truncating the [[W:Rhombic triacontahedron|rhombic triacontahedron]], which is the 30-sided polyhedron with only one kind of face, the dual of the 30-point icosidodecahedron. The 120-cell contains 120 of these 11-cell cells.]]
Moxness's 60-point (Hull #8) is a nonuniform form of an Archimedean solid, the 60-point [[W:Rhombicosidodecahedron|rhombicosidodecahedron]] from [[W:Johannes Kepler|Kepler's]] 1619 [[W:Harmonices Mundi|''Harmonices Mundi'']], which has the same 120 edges, 20 triangular faces and 12 pentagon faces, but with 30 squares between them instead of 30 rectangles. Without the squares ''or'' the rectangles it would be the 30-point ([[W:icosidodecahedron|icosidodecahedron]]), which has the same relationship to Moxness's 60-point (Hull #8) that the 6-point (hemi-icosahedron) does: they are both abstractions of it by conflation of its 60 points, 2-into-1 (icosidodecahedron) and 10-into-1 (hemi-icosahedron), in what [[w:Alicia_Boole_Stott|Alicia Boole Stott]] named a ''contraction'' operation.{{Efn|The 5-point (regular 5-cell) can be another abstraction of Moxness's 60-point (Hull #8), 12-vertices-into-1, but it is not strictly the result of a contraction operation as described by Boole Stott, since she did not apply her expansion and contraction operations to uniform polytopes with more than one edge length.}} In fact Moxness was not the first to find a rhombicosidodecahedron in the 120-cell; Alicia Boole Stott was. It is the 6th section of {5,3,3} beginning with a cell, which Boole Stott identified as the ''e&lt;sub&gt;2&lt;/sub&gt; expansion'' of the icosahedron (or equivalently of its dual polyhedron the dodecahedron).{{Sfn|Boole Stott|1910|loc=§Examples of the e&lt;sub&gt;2&lt;/sub&gt; expansion|p=7}} But that rhombicosidodecahedron identified by Boole Stott is not Moxness's Hull #8: rather, it is the semi-regular Archimedean solid, with a single edge length and square faces. Moxness's Hull #8 with its two distinct edge lengths and rectangular faces is Coxeter's 8th section of {5,3,3} beginning with a cell, which is missing from the sections illustrated by Boole Stott.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The 30-point (icosidodecahedron) is the quasi-regular product of 5-point (pentad) and 6-point (hexad), and we are back to Coxeter's origins of the 11-cell, and the two child's building blocks, one so useless the 5-point (5-cell), and the other so useful the 8-point (16-cell) which has four orthogonal 6-point (octahedral hexad) central sections, and can be compounded into everything larger. Some children building with it notice that just like its 6-point (octahedral hexad) orthogonal central section, the 30-point (icosidodecahedron) is the orthogonal central section of the 120-point (600-cell). It is less often noticed that Moxness's 60-point (rhombicosadodecahedron) is a non-orthogonal (pentad rather than tetrad) central section of the 600-point (120-cell), intersecting one-tenth of its 600 vertices and one-tenth of its 120 regular 5-cells. The 120-cell contains 120 of Moxness's 60-point (11-cell cells). They are non-disjoint, with 12 60-points sharing each vertex.

We have come far enough with our pentad building blocks, usually so useless to children less wise than Todd or Coxeter, to see that Moxness's 60-point (Hull #8) is the real hemi-icosahedron cell, with 120 5-cell edges in it as chords lurking under its surface. The rhombicosadodecahedron has enough of those 5-cell edges in its interior to make 20 disjoint tetrahedral cells which belong to 20 distinct 5-cells.

The 600-point (120-cell) may be constructed as a semi-regular honeycomb of two kinds of cells, the 120 non-disjoint 60-point (Moxness's rhombicosadodecahedra) and their neighbors the 120 disjoint 5-point (5-cells), cell-bonded together by shared tetrahedral cells of edge-length {{radic|5}}. Each rhombicosadodecahedron is bonded to 20 islanded 5-cells, sharing with each a 5-cell tetrahedron (4 of its interior 5-cell faces, and 6 of its interior 5-cell edges). The rhombicosadodecahedron is also bonded directly to 10 other rhombicosadodecahedra (since each 11-cell has 11 rhombicosadodecahedron cells), again not by sharing one of its exterior faces, but this time by sharing an interior face, a {{radic|6}} triangle which lies in a central plane.{{Efn|The {{radic|6}} chord is another hidden chord lurking beneath the 60-point rhombicosadodecahedron's surface, like the {{radic|5}} 5-cell edge. Both these hidden chords lie on the same great circle with the 120-cell edges (the pentagon face edges). Three {{radic|6}} chords make a triangle which lies in a central plane. This {{radic|6}} great triangle must not be confused with the {{radic|5}} face triangle of the 5-cell, which does not lie in a central plane.}} The 120 rhombicosadodecahedra are also bonded directly together at ''all'' their exterior faces: at their pentagon faces (which are 120-cell faces of dodecahedral cells); at their small triangular faces (which are 600-cell faces of small tetrahedral cells, 10 of which are inscribed in each dodecahedral cell); and at their rectangular faces (which are faces of 10-point [[w:Pentagonal_prism|pentagonal prism]] cells, where the pentagonal prism is a column inscribed in a dodecahedral cell, joining its opposite pentagon faces). The 120 rhombicosadodecahedra are bonded directly to each other at their own exterior faces; no dodecahedra, small tetrahedra, or pentagonal prisms lie between them, and those objects are not cells of this honeycomb; we mention them here only to help locate the rhombicosadodecahedra within the 120-cell. The only objects which fill holes between the rhombicosadodecahedra are the 120 islanded 5-cells.{{Efn|This honeycomb of the 120-cell resembles the [[600-cell#Icosahedra|honeycomb of the 600-cell consisting of 24 face-bonded regular icosahedra]] with 24 clusters of 5 tetrahedra filling the holes between them. In the 120-cell there are 5 times as many of everything as in the 600-cell, so there are 120 (rhombicosadodecahedron) icosahedral cells with 600 tetrahedra filling the holes between them.}}

It is possible for rhombicosadodecahedra to meet each other at more than one kind of exterior face because they are not meeting there as cells of the same 11-cell; they belong to different 11-cells. The 11 rhombicosadodecahedra in the same 11-cell do not meet at their own exterior faces, they meet at their interior {{radic|6}} triangle faces. As a cellular 4-polytope, the 11-cell is inside-out. Its cells meet each other at interior planes, sharing interior {{radic|6}} triangle faces, instead of in the usual way that 4-polytopes' cells share an exterior face. It is a deeper kind of intertwining to be sure, but the 11 rhombicosadodecahedra in the same 11-cell are still volumetrically disjoint cells, and the 11-cell is a perfectly real 4-polytope with real cells. It is a semi-regular 4-polytope, because it has two different kinds of cells: 11 rhombicosadodecahedra and 11 5-cells. The 11 5-cells are islanded (not bound to each other), and the 11 rhombicosadodecahedra are bound both to 5-cells (at {{radic|5}} 5-cell faces) and to each other (at {{radic|6}} central triangle faces). In the section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#What's in the box|§What's in the box]]'', below, we will look closely at the relation between the {{radic|5}} chord and the {{radic|6}} chord, which is the relation between the pentad and the hexad, and at the general phenomenon of hemi-polytopes which meet each other at a central plane. Then in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Cell rings of the 11-cells|§Cell rings of the 11-cells]]'' we will look at the two-cell cellular structure of the 11-cell in more detail.

Considered as a single complex of overlapping cells rather than as distinct honeycombs of volumetrically disjoint cells, the 120-cell contains 60-point (rhombicosadodecahedral cells), islanded 5-point (5-cells), 20-point (dodecahedral cells), and 10-point ([[w:Pentagonal_prism|pentagonal prism]] cells). The 60-points meet each other in pairs at their pentagon faces, meet regular dodecahedra at those pentagon faces also, meet each other in quads at their 600-cell faces where they share an interior 4-point (600-cell tetrahedral cell) and its four faces (just as they share a larger interior 5-cell tetrahedron and its faces), and meet each other and the 10-point (pentagonal prisms) at their rectangle faces. We might try to imagine such a complex with its 60-point facets removed, leaving only the dodecahedra and the pentagonal prisms, so the 120-cell becomes an open geodesic space frame of joints and struts without its facets, with 120 dodecahedral joints (an exploded 120-cell with its 120 dodecahedra moved apart) joined by 720 pentagonal prism struts (stubby pentagonal columns, whose pentagon ends are the pentagon faces of the joints). But this vision is an abstraction, because the 120-cell is also a honeycomb in which the 120 dodecahedra are directly in contact, and meet each other at their faces. A 120-cell is always both these honeycombs at one and the same time, just as it is a compound of 5 disjoint 600-cells two entirely different ways. One cannot remove the 60-points, except in imagination by neglecting to see them. They are always there, and there is never any space between adjacent dodecahedra. It is only that in this honeycomb of 120 regular dodecahedra, some pairs of dodecahedra are not adjacent, but separated by a pentagonal prism strut. That is, they are separated by another dodecahedron which lies between them, and the pentagonal prism is a column inscribed in that dodecahedron, joining its opposite pentagon faces. Each pentagonal prism is a dodecahedron minus (a truncated dodecahedron), and each rhombicosadodecahedron is a dodecahedron plus (augmented with parts truncated from 30 neighboring truncated dodecahedra pentagonal prisms). There are 120 of each of these 3 kinds of dodecahedron. 

Moxness's 60-point (rhombicosadodecahedron cell) is a central object of the 120-cell, and contains many wonders. The centers of its 12 little pentagons are the vertices of the regular icosahedron, which frames ''all'' the regular polyhedra by expansion and contraction operations, as Alicia Boole Stott discovered before 1910,{{Sfn|Polo-Blanco: ''Theory and history of geometric models of Alicia Boole Stott''|2007|loc=§5.3.2 1910 paper on semi-regular polytopes|pp=152-158|ps=; summarizes Boole Stott's method and results from {{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}}, including her tables of expansion-contraction dimensional analogies and a few of her illustrations.}} and those wise young friends Coxeter &amp; Petrie, building together with polyhedral blocks, rediscovered before 1938.{{Efn|&quot;Just as a tetrahedron can be inscribed in a cube, so a cube can be inscribed in a dodecahedron. By reciprocation, this leads to an octahedron circumscribed about an icosahedron. In fact, each of the twelve vertices of the icosahedron divides an edge of the octahedron according to the &quot;[[W:Golden section|golden section]]&quot;. Given the icosahedron, the circumscribed octahedron can be chosen in five ways, giving a [[W:Compound of five octahedra|compound of five octahedra]], which comes under our definition of [[W:Stellated icosahedron|stellated icosahedron]]. (The reciprocal compound, of five cubes whose vertices belong to a dodecahedron, is a stellated [[W:Triacontahedron|triacontahedron]].) Another stellated icosahedron can at once be deduced, by stellating each octahedron into a [[W:Stella octangula|stella octangula]], thus forming a [[W:Compound of ten tetrahedra|compound of ten tetrahedra]]. Further, we can choose one tetrahedron from each stella octangula, so as to derive a [[W:Compound of five tetrahedra|compound of five tetrahedra]], which still has all the rotation symmetry of the icosahedron (i.e. the [[W:Icosahedral group|icosahedral group]]), although it has lost the reflections. By reflecting this figure in any plane of symmetry of the icosahedron, we obtain the complementary set of five tetrahedra. These two sets of five tetrahedra are enantiomorphous, i.e. not directly congruent, but related like a pair of shoes. [Such] a figure which possesses no plane of symmetry (so that it is enantiomorphous to its mirror-image) is said to be ''[[W:Chiral|chiral]]''.&quot;{{Sfn|Coxeter &amp; Petrie|1938|p=4|loc=''[[W:The Fifty-Nine Icosahedra|The Fifty-Nine Icosahedra]]''}}|name=Concentric polyhedra}} Before we move on from the 60-point (hemi-icosahedral cell) of the 11-cell for now, it is important to notice one more thing about the hidden {{radic|5}} chords lurking below its surface. The 12 little pentagon faces are spanned by sets of 6 {{radic|5}} chords (5-cell edges). The 6 chords of each set are disjoint (they don't touch or form 5-cell faces), but they are symmetrically arranged in 3 parallel pairs, {{radic|3}} apart, which lie in 3 orthogonal central planes. It turns out they are the 6 reflex edges of a 12-point non-convex polyhedron called the [[w:Jessen's_icosahedron|Jessen's icosahedron]], inscribed in the 60-point cell. But here we find ourselves far out in the 3-sphere system, almost to the [[W:Borromean_rings|Borromean rings]] of the giant 600-cell. We shall have to go back and orient ourselves at the origin again and work our way outwards, before eventually in ''[[#The perfection of Fuller's cyclic design|§The perfection of Fuller's cyclic design]]'' we meet that rare child Bucky Fuller's orthogonal 12-point (tensegrity icosahedron), an in-folded cuboctahedron, the unique pyritohedral fish swimming deep in the 3-sphere ocean.

== Compounds in the 120-cell ==

=== 120 of them ===
The largest regular convex 4-polytope is the 120-cell, the convex hull of a [[120-cell#Relationships among interior polytopes|regular compound of 120 disjoint regular 5-cells]], with 600 vertices, 1200 {{radic|5}} [[120-cell#Chords|chords]] and 1200 {{radic|5}} interior triangles (in a {{radic|2}} radius polytope). The 120-cell contains hemi-icosahedra and 5-cells in equal number (11 of each in each 11-cell), and it has exactly 120 5-cells, so there are 120 distinct hemi-icosahedra inscribed in the 120-cell. Because the 11-cells share their 11 hemi-icosahedra and their 11 5-cells in the same permutation, there are also 120 distinct 11-cells inscribed in the 120-cell. The 120-cell has 120 dodecahedral cells, 120 5-cells, 120 hemi-icosahedra, and 120 11-cells.

=== How many building blocks, how many ways ===
[[120-cell#Geometry|The 120-cell is the maximally complex regular 4-polytope]], which contains inscribed instances of every kind of 1-, 2-, 3-, and 4-polytope in existence, up to the heptagon {7}. For example, it is a regular compound of 5 inscribed disjoint instances of the 600-cell (two different ways). Children with their 16-cell building blocks will soon learn to protect their sanity by thinking of its compounds by their other names, as a number of symmetrically distributed identical vertices, rather than as a number of identical cells (especially since the cell type is not the same in all the compounds). They are the 8-point (16-cell), the 16-point (8-cell), the 24-point (24-cell), the 120-point (600-cell), and the 600-point (120-cell).

Spoiler alert: the 8-point compounds by 2 in the 16-point, and by 3 in the 24-point. The 16-point compounds in the 24-point by 3 ''non-disjoint instances'' of itself which share pairs of vertices. (We can think of non-disjoint instances as overlapping instances, except that disjoint instances overlap in space too, they just don't have overlapping vertex sets.) The 24-point compounds by 5 disjoint instances of itself in the 120-point, and the 120-point compounds by 5 disjoint instances of itself in the 600-point. So far, our child is happily building and satisfied with his castle. Then things get hairy.

The 24-point also compounds by &lt;math&gt;5^2&lt;/math&gt; non-disjoint instances in the 120-point; it compounds into 5 disjoint instances of itself, 10 (not 5) different ways. The resulting 120-point, magically, contains 25 24-points, not just 5 (or 10). This means that 15 disjoint 8-point building blocks will construct a 120-point (3 times 10 different ways), which then magically contains 75 8-points. 

The 600-point is 5 disjoint 120-points, just 2 different ways (not 5 or 10 ways). So it is 10 non-disjoint 120-points. This means the 8-point building block compounds by 3 times &lt;math&gt;5^2&lt;/math&gt; (75) disjoint instances of itself into the 600-point, which then magically contains &lt;math&gt;3^2&lt;/math&gt; times  &lt;math&gt;5^2&lt;/math&gt; (225) instances of the 24-point, and  &lt;math&gt;3^3&lt;/math&gt; times &lt;math&gt;5^2&lt;/math&gt; (675) instances of the original 8-point. 

They will be rare wise children who figure all this out for themselves, and even wiser who can see ''why'' it is so. [[W:Pieter Hendrik Schoute|Schoute]] was the first to see that the 120-point (600-cell) is a compound of 5 24-point (24-cell)s ''10 different ways'', and after he saw it a hundred years lapsed until Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne proved his result, and showed why.{{Sfn|Denney, Hooker, Johnson, Robinson, Butler &amp; Claiborne|2020|loc=''The geometry of H4 polytopes''|ps=; This hexad of scholars from New Orleans, Louisiana extracted the truth from the permutations of the 600-cell's elements as perspicaciously as Coxeter did from the permutations of the 11-cell.}}

=== What's in the box ===
The picture on the back of the box of building blocks, showing its contents:
{|class=&quot;wikitable&quot;
!pentad
!hexad 
!heptad
|-
|[[File:4-simplex_t0.svg|120px]]
|[[File:3-cube t2.svg|120px]]
|[[File:6-simplex_t0.svg|120px]]
|-
|[[File:Pentahemidemicube.png|120px]]
|[[File:Tetrahemihexahedron_rotation.gif|120px]]
|[[File:Pentahemicosahedron.png|120px]]
|-
!120
!100
!120
|}
That's everything in the box. It contains all the 4-dimensional parts needed to build the 120-cell (the picture on the cover of the box), and everything else that's invisible in that picture which fits inside the 120-cell: all the astonishing regular 4-polytopes. The pentad building block is half a pair of orthogonal 5-point (regular 5-cells), the hexad building block is half an 8-point (16-cell), and the heptad building block is a quarter of an 11-point (11-cell). Everything can be built from these blocks by embedding them at the center of the 3-sphere, inscribing them as 4-polytopes to make compounds as with the 16-cell building block, rather than by assembling them face-to-face as 3-polytopes to make honeycombs in which they are cells or vertex figures.{{Efn|name=Embedding point}}

The pentad building block is the 5-point '''pentahemidemicube''', an abstract polyhedron with 10 equilateral triangle faces, 10 {{radic|5}} edges, and 5 vertices.{{Sfn|Christie: Pentahemidemicube|2024|loc=File:Pentahemidemicube.png|ps=; &quot;The 5-point ''pentahemidemicube'' is an abstract polyhedron with 10 equilateral triangle faces, 10 edges, and 5 vertices. It has a real presentation as the regular 5-cell.&quot;}} It has a real presentation as the 5-point (regular 5-cell 4-polytope), a regular tetrahedron expanded by addition of a fifth vertex equidistant from all the others in a fourth dimension. The pentad block is a tetrahedral pyramid (a 5-cell) inscribed in a cube: a tetrahedron inscribed in a cube (a demicube) expanded by addition of a fifth vertex, one of the cube's other 4 vertices. Each pentad (5-cell) building block lies in the 120-cell orthogonal to three other pentad (5-cell) building blocks. They are four completely disjoint building blocks that don't touch, but their corresponding vertices on opposite sides of the 3-sphere are connected by {{radic|6}} edges. Each regular 5-cell in the 120-cell is a pentad (pentahemidemicube) building block, in one of 30 orthogonal sets of 4. There are 120 pentad building blocks in the box.

Every vertex of the 16-cell, and therefore every vertex of the 8-cell, 24-cell, 600-cell and 120-cell, is surrounded by a 6-point (octahedral vertex figure). The hexad building block is that 6-point octahedron embedded at the center of the 4-polytope instead of at a vertex, that is, inscribed in the 3-sphere as a 4-polytope, the 6-point '''[[w:Tetrahemihexahedron|tetrahemihexahedron]]''' or '''hemi-cuboctahedron'''.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=File:Tetrahemihexahedron rotation.gif|ps=; &quot;The tetrahemihexahedron or hemicuboctahedron is a uniform star polyhedron, indexed as U&lt;sub&gt;4&lt;/sub&gt;. It has 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices.&quot;}} The hexads have 4 red triangle 5-cell faces, bounded by 12 {{radic|5}} edges. Their 3 yellow square faces lie in central planes, because the building block is an abstract 4-polytope. The hexad is an irregular abstract polyhedron that also has 6 {{radic|6}} edges, which are the diagonals of its yellow square faces. There are 100 hexad building blocks in the box.

Two orthogonal 5-point (pentad) building blocks intersect in a 4-point (inscribed tetrahedron) to form a 6-point (hexad) building block. Two orthogonal 6-point (hexad) building blocks intersect in a 4-point (central square) to form an 8-point (16-cell) building block, the most usable building block, from which everything larger can be built by compounding.

The 11-point (11-cell) is the Cartesian product of a pentad building block and a hexad building block that are completely orthogonal to each other. There are 120 pentads to pair with a completely orthogonal hexad, and although there are two chiral ways of pairing completely orthogonal objects (left-handed or right-handed) they only result in 120 pairs two different ways, so there are 120 11-cells. The 11-cell is 6 pentads' and 5 hexads' smallest joint compound, and the 120-cell is their largest, 120 pentads and 100 hexads.

The heptad{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''|pages=22–31}} building block is the 7-point '''pentahemicosahedron''', an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges (9 {{radic|4}} edges and 9 {{radic|5}} edges), and 7 vertices.{{Sfn|Christie: Pentahemicosahedron|2024|loc=File:Pentahemicosahedron.png|ps=; &quot;The 7-point ''pentahemicosahedron'' is an abstract polyhedron with 8 faces (3 squares and 5 triangles), 18 edges, and 7 vertices.&quot;}} It is a compound of a 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]) and a 6-point (hexad [[W:hemi-icosahedron|hemi-icosahedron]]) intersecting in a 4-point (tetrahedron), that is, of three orthogonal 5-point (pentad [[w:Hemi-cube_(geometry)|hemi-demicube]]&lt;nowiki/&gt;s) intersecting in their common 4-point (tetrahedron). It is also the abstract cube contracted by removal of one vertex, which reduces one of the cube's two inscribed tetrahedra (demicubes) to a single remaining triangular face. As with all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other. Four &lt;small&gt;&lt;math&gt;(11-7)&lt;/math&gt;&lt;/small&gt; orthogonal 7-point (pentahemicosahedron) blocks can be joined at a common Petrie polygon (a skew hexagon which contains their orthogonal 6-edge paths) to construct a real 11-point ([[W:11-cell|11-cell]] semi-regular 4-polytope), which magically contains six &lt;small&gt;&lt;math&gt;(11-5)&lt;/math&gt;&lt;/small&gt; 5-point (1-pentad regular decahedra) as cells and five &lt;small&gt;&lt;math&gt;(11-6)&lt;/math&gt;&lt;/small&gt; 12-point (2-hexad semi-regular cuboctahedra) as cells. There are 120 heptad building blocks in the box.

=== Building the building blocks themselves ===
We didn't make the 5-point (5-cell) or the 12-point (16-cell) building blocks out of anything, we just accepted them a priori like [[W:Euclid's postulates|Euclid's postulates]], and started to build with them. But it turns out that while they are the smallest ''regular'' 4-polytopes, they are not indivisible, and can be made as honeycombs of identical smaller ''irregular'' 4-polytopes. They are not a priori miracles; like everything else fundamental in nature, including Euclid's postulates, at root they are an expression of a distinct symmetry group.

Every regular convex ''n''-polytope can be subdivided into instances of its characteristic ''n''-[[W:Orthoscheme|orthoscheme]] that meet at its center. An ''n''-orthoscheme (not an ''n''-orthoplex!) is an ''irregular'' ''n''-simplex with faces that are various right triangles instead of congruent equilateral triangles. It possesses the complete symmetry of the polytope without any redundancy, because it contains one of each of the polytope's characteristic root elements.{{Efn|A [[W:Schläfli orthoscheme|Schläfli orthoscheme]] is a [[W:Chiral|chiral]] irregular [[W:Simplex|simplex]] with [[W:Right triangle|right triangle]] faces that is characteristic of some polytope if it will exactly fill that polytope with the reflections of itself in its own [[W:facet (geometry)|facet]]s (its ''mirror walls''). Every regular polytope can be partitioned radially by its planes of symmetry (Coxeter's &quot;reflecting circles&quot;) into instances of its [[W:Orthoscheme#Characteristic simplex of the general regular polytope|characteristic orthoscheme]] surrounding its center. The characteristic orthoscheme can be replicated rotationally to generate its regular 4-polytope because it is the complete [[W:gene|gene]] for it, containing all of its elements and capturing all of its symmetry without any redundancy. It has the shape described by the same [[W:Coxeter-Dynkin diagram|Coxeter-Dynkin diagram]] as the regular polytope without the ''generating point'' ring that triggers the reflections.|name=Characteristic orthoscheme}} It is the gene for the polytope, which can be replicated to construct the polytope.

The regular 4-simplex (5-cell) is subdivided into 120 instances of its [[5-cell#Orthoschemes|characteristic 4-orthoscheme]] (an irregular 5-cell) by all of its &lt;math&gt;A_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 120. The 120-cell is the convex hull of the regular compound of 120 disjoint regular 5-cells, so it can be subdivided into &lt;small&gt;&lt;math&gt;120\times 120 = 14400&lt;/math&gt;&lt;/small&gt; of these 4-orthoschemes, so that is its symmetry order. The 5-cell is both 1∕120th of the 120-cell, and 1∕120th of itself.

The regular 4-orthoplex (16-cell) is subdivided into 384 instances of its [[16--cell#Tetrahedral constructions|characteristic 4-orthoscheme]] (another irregular 5-cell) by all of its &lt;math&gt;B_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 384. The 120-cell is the convex hull of the regular compound of 75 disjoint 16-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;75\times 384 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

The regular 24-point (24-cell) is subdivided into 1152 instances of its [[24-cell#Characteristic orthoscheme|characteristic 4-orthoscheme]] (yet another irregular 5-cell) by all of its &lt;math&gt;F_4&lt;/math&gt; planes of symmetry at once intersecting at its center, so its symmetry is of order 1152. The 120-cell is the convex hull of the regular compound of 25 disjoint 24-cells (which have 2-fold reflective symmetry), so its symmetry is of order &lt;small&gt;&lt;math&gt;25\times 1152 / 2 = 14400&lt;/math&gt;&lt;/small&gt;.

This is as much group theory as we need to practice to see that every uniform polytope has three equivalent origins in distinct root systems, though most are obviously constructed from one root, and not so obviously from the others. Every polytope can be constructed from its characteristic pentad, including the hexad and heptad. Everything larger than the pentad can be constructed by compounding hexads, and we shall see that everything as large as a 24-cell has a construction from heptads which are a product of a pentad and a hexad. Surely every uniform polytope can be constructed by some function of expansions and contractions from any of three root systems, however indirect the recipe. These construction formulae of &lt;math&gt;A^n&lt;/math&gt;, &lt;math&gt;B^n&lt;/math&gt; and &lt;math&gt;C^n&lt;/math&gt; orthoschemes, related by an equals sign between them, then give us a uniform set of conservation laws for each uniform ''n''-polytope, which we may call its physics by [[W:Noether's theorem|Noether's theorem]].

=== Pentads and hexads together ===
The most numerically strange compounding is how five 24-point (24-cells) compound magically into ''twenty-five'' 24-point (24-cells), because 25 is exactly 1 more than 24. We are used to things compounding by multiplication, and multiplication never adds one! On the other hand, when things compound by squaring rather than by multiplying, sometimes (by magical coincidence) they may add one.  &lt;math&gt;5^2&lt;/math&gt; is one of those times. In the 600-cell, compounding five 24-cells by squaring them has the magical effect of turning every one of the 24 octahedral cells into a whole 24-cell (24 cells), and then magically adding one whole 24-cell (24 cells) to the bundle (24 * 25 = 600 cells).{{Efn|The 600-cell contains the 600 octahedral cells of its 25 24-cells, as well as its own 600 tetrahedral cells. It also contains the 600 cubical cells of its 75 8-cells. It is a 600-cell in three ways, although the octahedra and cubes are not ''cells of the 600-cell'', since they are not volumetrically disjoint in the 600-cell; they are ''cells'' only of the 24-cell and the 8-cell, respectively.|name=600 cells three ways}} Very strange!

We made compounds of all the 4-polytopes except the smallest, the 5-point (1-pentad 5-cell), out of the 8-point (2-hexad 16-cell) building blocks, but we didn't make any 4-polytopes out of the 5-point (1-pentad 5-cell) building blocks except the largest, the 600-point (120-cell). But it turns out there is another 4-polytope that contains 5-point (1-pentad 5-cell) building blocks beside the largest one, it just isn't a ''regular'' 4-polytope. It is semi-regular with two kinds of cells, the 4-point (regular tetrahedron) and the 12-point (truncated cuboctahedron), which have just one kind of regular face between them, a 5-cell face. That semi-regular 4-polytope is the 11-cell.

We said earlier that the 11-cell had 11 disjoint 5-cells and 11 hemi-icosahedral cells in it, and that its construction from them might be hard to see, but now we are going to account numerically, at least, for how it can have 6 pentad cells plus 5 hexad cells magically adding to 11 of the same kind of elevenad cell. Compounding the 5-point (pentad) and the 6-point (hexad) by squaring them both and taking their ''difference'' has the magical effect of yielding their sum, 11. In the 120-cell, compounding 10 5-point (pentad) building blocks and 12 6-point (hexad) building blocks by squaring them (which means multiplying 10 by 12, ''rectangling'' them) has the magical effect of turning the 10 5-point (pentad) building blocks into 120 5-point (5-cell regular decahedron) cells, and the 12 6-point (hexad) building blocks into 100 12-point (regular icosahedron) cells, and then magically adding one whole 5-point (5-cell regular decahedron) cell to the original 10 (in each resulting cell bundle), and magically ''subtracting'' one whole 12-point (regular icosahedron) building block from the original 12 (in each resulting cell bundle), magically yielding 11 such (cell bundles of 6 + 5 = 11 cells), with each 11-cell bundle containing 11 of both kinds of cell. Very, very strange!{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden'': §II To Add is to Multiply|p=38|ps=; the sequence of regular polygons with harmonious proportions beginning {5}, {7}, {9}, {11} as a family of natural numbers; &quot;The numbers in this optimal family have remarkable properties [that allow] similar figures to be arranged easily or allow a figure to be dissected into a set of similar figures. This is the origin of the Greek idea of geometric progression — that ''multiplication can be accomplished instead by addition''.... For instance, it is well known that the square of the golden ratio is one more than itself (written above as 𝜙&lt;sup&gt;2&lt;/sup&gt; = 𝜙 + 1) and one less than 𝜙 is its reciprocal (1/𝜙 = 𝜙 - 1). The first relation equates multiplication with addition, while the second accomplishes division through subtraction. [These are pentagon {5} bisectional golden ratios, but] the heptagonal [{7} tri-sectional] ratios also behave in surprising ways — adding to multiply, subtracting to divide.... There are analogous constructions for [nonagon {9}] quadrisectional and [hendecagon {11}] pentasectional systems.&quot;}}

The 120-cell has three interchangeable cyclic expansion-contraction cycles, the pentad, hexad and heptad cyclic designs, rooted in the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems, respectively. Each of them constructs the whole sequence of regular 4-polytopes from a single building block. The key to all three cycles is that 5 + 6 = 11, and there is an 11-point quasi-regular 4-polytope in the sequence of convex 4-polytopes. It lies between the 8-point (16-cell) and the 16-point (8-cell), and is itself a &lt;math&gt;B_4&lt;/math&gt; polytope. Its role is to be a unified single building block common to all three cyclic expansion-contraction cycles. It is involved with the &lt;math&gt;A_4&lt;/math&gt;, &lt;math&gt;B_4&lt;/math&gt; and &lt;math&gt;H_4&lt;/math&gt; root systems and expresses in one building block their relations to each other.

We have already seen how the hexad building cycle works. It is plain compounding of 4-orthoplexes at the beginning, increasing to densely intricate non-disjoint symmetry toward the end, making it difficult to see why it works once past the 24-cell. This is how the build works, in terms of sixes. The 16-cell is a 2-hexad (3 pairs of completely orthogonal great square planes). The 24-cell is 3 16-cells and a 4-hexad, 4 different ways: 16 non-disjoint hexads (great hexagon planes). The 120-point (600-cell) is 25 24-cells and a 20-hexad, 10 different ways: 200 non-disjoint hexads (great hexagon planes). The 600-point (120-cell) is 225 (9 times 25) 24-cells. ....

The pentad building cycle doesn't do plain compounding to regular 4-polytopes until the last one, the 120-cell. Fives don't even appear as pentagons until after the 24-cell. We have seen how there are 11 5-cells in the quasi-regular 11-cell, which occurs before the 8-cell, and also how everything can be compounded from irregular 5-cells (4-orthoschemes specialized for each 4-polytope). It is still not evident how a pentad cycle can work to build all the regular 4-polytopes from the regular 4-simplex. This is how is how the build works, in terms of fives. ....

This is how the build works, in terms of elevens. The 11-cell is 4 orthogonal heptads (intersecting in their skew hexagon Petrie polygon). The 24-cell is also 4 heptads (intersecting in 4 orthogonal digon axes, so each heptad contributes 6 unique points to a single fibration of 24 points). Heptads are quarter elevenads, so the 11-cell and 24-cell are both 1 elevenad. ....

See the tabular sequence of convex 4-polytopes in ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#Build with the blocks|§Build with the blocks]]'', below: table rows ''Inscribed, Pentad, Hexad, Heptad''.

=== 11 of them ===
There are 120 distinct 11-cells inscribed in the 120-cell, in 11 fibrations of eleven 11-cells each, but unlike the 120 completely disjoint 5-cells, none of the 11-cells are disjoint. 11-cells are always plural, with at minimum eleven of them. There is only a single Hopf fibration of the 11 great circles of an 11-cell. 

The 5-cell and the 11-cell are limit cases. The 5-cells are completely disjoint, but the other convex 4-polytopes, including the regular 16-cell, 8-cell, 24-cell and 600-cell, occur inscribed in the 120-cell in some number of disjoint instances, and also in a larger number of distinct instances. For example, the 120-cell contains 75 disjoint 16-cells and 675 distinct 16-cells. A fibration of 11-cells contains 0 disjoint 11-cells and 11 distinct 11-cells. The 11-point (11-cell) is abstract only in the sense that no disjoint instances of it exist. It exists only in compound objects, always in the presence of 10 other instances of itself.

== Cell rings of the 11-cells ==
Most wise children who succeed in building the 120-cell (the picture on the cover of the box of building blocks) do so their first time by assembling it like a round jig-saw puzzle, out of 12 rings of dodecahedral building blocks.{{Sfn|Schleimer &amp; Segerman|2013|loc=''Puzzling the 120-cell''}} Each ring starts out as a stack of 10 dodecahedra piled face-to-pentagon-face, which must then be bent into a circle in the fourth dimension, bringing its top and bottom pentagons together. The entire 120-cell can then be assembled from exactly 12 of these rings, such that every dodecahedron is in contact with 12 other dodecahedra at its 12 pentagon faces, if and only if the wise child learns to link the 12 rings together, so that each passes through all the others. This bundle of 12 interlinked rings of 10 dodecahedra is the 120-cell.

This building process is very clearly described by Goucher in his section of the Wikipedia article on the 120-cell entitled ''[[120-cell#Visualization|§Visualization]]''.{{Sfn|Ruen &amp; Goucher et al. eds. 120-cell|2024|loc=Goucher's ''[[120-cell#Visualization|§Visualization]]'' describes the decomposition of the 120-cell into rings two different ways; his subsection ''[[120-cell#Intertwining rings|§Intertwining rings]]'' describes the discrete Hopf fibration of 12 linked rings of 10 dodecahedra}} Goucher was not the first to describe the dodecahedral Hopf fibration of the 120-cell, as the numerous previously published papers describing it referenced in the Wikipedia article make clear, but his is one of the clearest and most multi-faceted geometric descriptions of how to build it.

The [[W:Hopf fibration|Hopf fibration]] is the general dimensional analogy between the 2-sphere and 3-sphere, which enables us to make a distinct Hopf map describing each special case dimensional analogy between a discrete 3-polytope and 4-polytope that we discover. A Hopf map describes a fiber bundle of linked disjoint great circles on the [[W:Clifford torus|Clifford torus]] completely filling the 3-sphere, where each great circle is conflated to a single point on the dimensionally analogous 2-sphere map. Each map of a discrete fibration is an abstract 3-polytope of the real 4-polytope, mapping a particular set of its rings. The Hopf fibration has been studied and described by many mathematicians, some of whom like Goucher have also described it geometrically in succinct English so the rest of us can visualize it.{{Sfn|Dorst|2019|loc=§1. Villarceau Circles|p=44|ps=; &quot;In mathematics, the path that the (1, 1) knot on the torus traces is also known as a [[W:Villarceau circle|Villarceau circle]].
Villarceau circles are usually introduced as two intersecting circles that are the cross-section of a torus by a well-chosen plane cutting it.
Picking one such circle and rotating it around the torus axis, the resulting family of circles can be used to rule the torus.
By nesting tori smartly, the collection of all such circles then form a [[W:Hopf fibration|Hopf fibration]].... we prefer to consider the Villarceau circle as the (1, 1) torus knot rather than as a planar cut.&quot;}}

A dimensional analogy is a finding that some real ''n''-polytope is an abstraction of some other real polytope in a space of higher ''n'', where the ''n''-polytope is embedded in some distinct place in that higher-dimensional space. A Hopf map is a general example, an abstract 3-polytope embedded in 4-space as a distinct 4-polytope on the 3-sphere. The crucial point is that it is the object in 4-space which is the real object, and the object in 3-space which is only an abstract object (a lossy description of the real object), not the other way around.{{Sfn|Huxley|1937|loc=''Ends and Means: An inquiry into the nature of ideals and into the methods employed for their realization''|ps=; Huxley observed that directed operations determine their objects, not the other way around, because their direction matters; he concluded that since the means ''determine'' the ends,{{Sfn|Sandperl|1974|loc=letter of Saturday, April 3, 1971|pp=13-14|ps=; ''The means determine the ends.''}} they can't justify them.}}

To a topologist a Hopf map is not necessarily an actual 3-dimensional object within the 4-dimensional object it is a map of, any more than any map is necessarily an object in the domain of the thing it maps, but in the case of the Hopf maps of the characteristic rotations of the regular convex 4-polytopes, each Hopf map is in fact a 3-polytope found somewhere within the 4-polytope. Thus for example the 12-point (regular icosahedron) is the Hopf map of the bundle of 12 interlinked rings of 10 dodecahedra just described. Each axial great circle of a cell ring (each fiber in the fiber bundle) is conflated to one distinct point on the 12-point (regular icosahedron) map.

In like fashion, we can look within the 120-cell for the Hopf map of the 11-cell's rings. Such a map would tell us how the eleven cells relate to each other, revealing the cells' external structure in complement to the way we found out the internal structure of each 60-point (rhombicosadodecahedron) cell, above. The obvious place to look for the 11-cell's Hopf map is in Moxness's 60-point (rhombicosadodecahedron the hemi-icosahedral cell) itself; there really is no other place it could be hiding. So here we return to our inspection of Moxness's rhombicosadodecahedron as an augmented dodecahedron, one of three kinds of dodecahedra that occur as 120 identical cells in rings within the 120-cell. We have seen that the regular dodecahedron forms 12 interlinked rings of 10 dodecahedral cells, and we anticipate that the rhombicosadodecahedron must form interlinked rings in a Hopf fibration as well, but we have not yet counted the rhombicosadodecahedral rings or measured their circumference. As possibly a distinct Hopf fibration with its own distinct map, it need not also have 12 rings of 10 cells.  

Let us compare the way regular dodecahedra and rhombicosadodecahedra form rings. The essential difference is that the dodecahedra meet face-to-face like good 4-polytope cells, but the rhombicosadodecahedra only meet edge-to-edge. Grünbaum found that they meet three around an edge. It almost looks at first as if there might be a pentagonal prism between two adjacent rhombicosadodecahedra, but no, the {{radic|5}} edge they meet at is a chord that runs between every other little pentagon face, under the little pentagon face between them, while the two pentagonal prisms connected to the middle pentagon form an arc over the top of the {{radic|5}} chord. Pentagonal prisms fill both 3-dimensional concavities on either side of the meeting-edge, but they do not separate the two rhombicosadodecahedra entirely to make them completely disjoint.  

The Hopf map for the pentagonal prisms is easier to figure out, because they do meet a neighboring cell at a common face. They meet a regular dodecahedron cell, so again we might at first suspect that there is a regular dodecahedron between them, just as there certainly is one of them between two non-adjacent regular dodecahedra. But again no, two pentagonal prisms meet the same face of a regular dodecahedron, not its opposite faces, so they meet each other at that common face, which is also common to two regular dodecahedra, of course. But are they the same two dodecahedra that these two pentagonal prisms are cored out of? Well yes, they must be, since the pentagonal column is orthogonal to the pentagon face it meets at each end. So the pentagonal prisms also form rings of 10 cells, which are just like the usual rings of 10 dodecahedra, albeit with smaller cells (columnar, not round) with only 10 vertices instead of 20. The pentagonal prisms must form the same discrete Hopf fibration of 12 rings of 10 cells as the regular dodecahedra do, not a distinct fibration with a different map. This is possible even though each pentagonal prism cell touches only 10 vertices, not 20, because the central rule of a fibration is that the great circles of its rings (which are axial to its rings of cells) collectively must visit every vertex of the polytope exactly once. As Goucher describes, in the 120-cell the 12 great decagons axial to its dodecahedral cells (also axial to its pentagonal prism cells) run between opposing dodecahedron vertices (and so diagonally through the volume of the pentagonal prism, not along one of the column's edges). These cell diameters are actually 600-cell edges (which we glimpsed earlier as the edges of the triangular faces of the rhombicosadodecahedron). In this fashion, the transit of a dodecahedral cell enumerates exactly the same two vertices as the transit of its inscribed pentagonal prism column does. Both honeycombs are the same discrete fibration of 12 great decagon rings.

We return to the more puzzling rhombicosadodecahedron rings. How many rhombicosadodecahedral cells does it take to make a ring of these augmented dodecahedra? Ordinary cells joined at their edges would need fewer cells to make a ring than the same cells joined at their faces, because the edge radius is greater than the face radius. But these are not ordinary cells, because they have concave reflex edges. They are joined at those concave edges, whose radius is different than the convex edge radius, but neither radius is the same as the dodecahedron's radius. We shall take a much closer look at the internal geometry of the hemi-icosahedral cell in the next section ''[[User:Dc.samizdat/A symmetrical arrangement of eleven 11-cells#The quasi-regular convex 11-cell|§The quasi-regular convex 11-cell]]'', and determine all its radii and other properties, but for present purposes all we need to know is the width of the cell between opposing {{radic|5}} edges, the chord of the great circle polygon, so we can see what polygon it is and how many cells it takes to make a cell ring on its great circle axis. But even that is tricky, because in 4 dimensions there are two such widths. There is the distance between two opposing parallel {{radic|5}} edges in the same central plane, which we will discover in the next section is {{radic|3}}, an angular distance of 60°. That would give us a ring of 6 rhombicosadodecahedral cells, if each cell is bound to its neighbors on either side by those two opposing edges which lie in the same plane. But there is another possibility, because in 4 dimensions there is another way in which two edges can be opposing, when they lie in two different, completely orthogonal central planes on opposite sides of the 3-sphere.

In this regard, it is significant that a hemi-icosahedral cell is a central section of the 120-cell, with the same center and radius. The dodecahedron, like the cells of all the regular 4-polytopes, is an off-center section which lives on one side of the 3-sphere. But the 11-cell has its unusual hemi-icosahedral cell, a rhombicosadodecahedron embedded in 4-space at the center of the 120-cell like an inscribed 4-polytope. It reaches all the way around the 3-sphere, even though it is only a hyperdisk rather than a hypersphere. This central symmetry is why the 6-point (hemi-icosahedron) is not only the cell of the 11-cell, but also a Hopf map of its rings: it is an abstract 6-point (3-polytope) hyperdisk map of the real 11-point (4-polytope) hypersphere's rings. This will be useful, because a Hopf map is useful in reasoning about its 4-polytope's rings, but the central significance of the 60-point being a central hyperdisk is that it contains completely orthogonal planes. Therefore there is another way that three rhombicosadodecahedra in the 120-cell may be edge-bonded in line by a pair of opposing {{radic|5}} edges belonging to the middle rhombicosadodecahedron: that pair of edges may be opposing edges in completely orthogonal planes 180° apart on opposite sides of the 3-sphere, rather than two nearby edges {{radic|3}} apart in the same plane on the same side of the 3-sphere.

We will see exactly how this works in the next section, but for the present, note that completely orthogonal {{radic|5}} edges are {{radic|6}} apart, an angular distance of 120°. Unfortunately that is not enough information to tell us the number of cells in the hemi-icosahedral cell ring. 120° might appear to imply that the hemi-icosahedral ring could be closed with just three cells, but [[W:SO(4)|rotations in 4-dimensional Euclidean space]] are not as straightforward as that, and orthogonal great circle rings sometimes wind more than once around the 3-sphere before closing. Obviously a ring whose cells spanned 180° would have to do that, if it had more than 2 cells per ring. The fundamental situation is actually worse than that: in 4-space not all circles have circumference &lt;math&gt;2\pi r&lt;/math&gt;. There is not just one other way that two {{radic|5}} edges can be opposing edges and lie in the same great circle ring, but potentially many other ways. Not just two completely orthogonal edges, exactly opposite each other on the 3-sphere, but any pair of {{radic|5}} edges in the rhombicosadodecahedron that are [[w:Clifford_parallel|Clifford parallel]] will be opposing edges on some great circle, but possibly on a strange circle that has a circumference greater than &lt;math&gt;2\pi r&lt;/math&gt;. It is entirely possible that the 11-cell's strange rings lie on such a strange circle.

Is there some way to figure out the circumference of the hemi-icosahedral cell ring, before we know all the details of its rotational geometry? We could guess, and then look for existing illustrations of that circular symmetry. What would be the best guess? It is perfectly obvious what our first guess should be.

If the 120-cell has any rings of 11 cells, it possesses an 11-fold 2-dimensional symmetry. Does such a symmetry exist in the 120-cell? A cell ring symmetry would be a polygonal symmetry in central planes. The 120-cell's central planes contain only certain distinct regular and irregular polygons. The Petrie polygon of the 120-cell is the regular skew 30-gon, so that is the largest possible central polygon that could be found in the 120-cell, and the least-common-denominator of all its [[120-cell#Chords|chords]]. Any polygonal symmetry found in the 120-cell must be present in the 30-gon.

Ruen illustrated many of the Wikipedia articles on polytopes, including articles on various families of polygons. The copiously illustrated [[W:Triacontagon|Wikipedia article on the triacontagon (30-gon)]] describes in detail not only the properties and symmetries of the regular 30-gon itself, but also its compounds and stars, a family of [[W:Triacontagon#Triacontagram|triacontagram]]s.

&lt;blockquote&gt;A triacontagram is a 30-sided [[W:star polygon|star polygon]] (though the word is extremely rare). There are 3 regular forms given by [[W:Schläfli symbol|Schläfli symbol]]s {30/7}, {30/11}, and {30/13}, and 11 compound star figures with the same [[W:vertex configuration|vertex configuration]].{{Sfn|Ruen: triacontagon|2011|loc=''Triacontagon''|ps=; This Wikipedia article is one of a large series edited by Ruen. They are encyclopedia articles which contain only textbook knowledge; Ruen did not contribute any original research to them except his illustrations. They are an invaluable resource, because they collect in one place illustrations and properties of most families of polytopes. As a series of articles, they are remarkably comprehensive.}}&lt;/blockquote&gt;

In this single Wikipedia article we find illustrations of every discrete fibration of central polygons that occurs in the 120-cell, which contains fibrations of all the central polygons of every regular convex 4-polytope. These illustrations are indispensable for visualizing the relationships among objects in the interior of the 120-cell. Here are four of them, illustrating respectively: the 5-fold pentad symmetry of the 120 5-points in the 120-cell, the 6-fold hexagonal symmetry of the 225 24-points in the 120-cell,{{Sfn|Sadoc|2001|pp=576-577|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the six-fold screw axis}} the 10-fold pentagonal symmetry of the 10 120-points in the 120-cell,{{Sfn|Sadoc|2001|p=576|loc=§2.4 Discretising the fibration for the {3, 3, 5} polytope: the ten-fold screw axis}} and the 11-fold heptad symmetry{{Sfn|Sadoc|2001|loc=§2.5 The 30/11 symmetry: an example of other kind of symmetries|pp=577-578}} of the 120 11-points in the 120-cell.

{| class=&quot;wikitable&quot; width=&quot;450&quot;
!colspan=5|Four orthogonal projections of the 120-cell to a central plane, selected from the table of 3 regular and 11 compound 30-grams
|-
!style=&quot;white-space: nowrap;&quot;|Polygon
!Compound {30/4}=2{15/2}
!Compound {30/5}=5{6}
!Compound {30/10}=10{3}
!Regular star {30/11}
|- 
!style=&quot;white-space: nowrap;&quot;|Inscribed 4-polytopes
|align=center|120 disjoint regular 5-cells
|align=center|225 (25 disjoint) 24-cells
|align=center|10 (5 disjoint) 600-cells
|align=center|120 (11 disjoint) 11-cells
|-
!style=&quot;white-space: nowrap;&quot;|Edge length ({{radic|2}} radius)
|align=center|{{radic|5}}
|align=center|{{radic|2}}
|align=center|{{radic|6}}
|align=center|{{radic|5}}
|-
!align=center style=&quot;white-space: nowrap;&quot;|Edge arc
|align=center|104.5~°
|align=center|60°
|align=center|120°
|align=center|135.5~°
|-
!style=&quot;white-space: nowrap;&quot;|Interior angle
|align=center|132°
|align=center|120°
|align=center|60°
|align=center|48°
|-
!style=&quot;white-space: nowrap;&quot;|Triacontagram
|[[File:Regular_star_figure_2(15,2).svg|200px]]
|[[File:Regular_star_figure_5(6,1).svg|200px]]
|[[File:Regular_star_figure_10(3,1).svg|200px]]
|[[File:Regular_star_polygon_30-11.svg|200px]]
|- 
!style=&quot;white-space: nowrap;&quot;|Ring polyhedra in 120-cell
|align=center|600 tetrahedra
|align=center|600 octahedra
|align=center|120 dodecahedra
|align=center|100 icosahedra&lt;br&gt;120 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cells per ring
|align=center|5 tetrahedra
|align=center|6 octahedra
|align=center|10 dodecahedra
|align=center|5 icosahedra&lt;br&gt;6 decahedra
|- 
!style=&quot;white-space: nowrap;&quot;|Cell rings per fibration
|align=center|120
|align=center|20
|align=center|12
|align=center|60
|- 
!style=&quot;white-space: nowrap;&quot;|Number of fibrations
|align=center|3
|align=center|20
|align=center|12
|align=center|4
|-
!style=&quot;white-space: nowrap;&quot;|Ring-axial great polygons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}&lt;/math&gt; digons
|align=center style=&quot;white-space: nowrap;&quot;|400 &lt;math&gt;\sqrt{2}&lt;/math&gt; hexagons
|align=center style=&quot;white-space: nowrap;&quot;|720 &lt;math&gt;\sqrt{2}(\phi-1)&lt;/math&gt; decagons
|align=center style=&quot;white-space: nowrap;&quot;|1200 &lt;math&gt;\sqrt{5}\curlywedge(2-\phi)&lt;/math&gt; hexagons
|- 
!style=&quot;white-space: nowrap;&quot;|Coplanar great polygons
|align=center|6 digons
|align=center|2 hexagons
|align=center|1 decagon
|align=center|6 digons
|- 
!style=&quot;white-space: nowrap;&quot;|Vertices per fiber
|align=center|12
|align=center|12
|align=center|10
|align=center|12
|- 
!style=&quot;white-space: nowrap;&quot;|Fibers per fibration
|align=center|50
|align=center|10
|align=center|60
|align=center|50
|- 
!style=&quot;white-space: nowrap;&quot;|Fiber separation
|align=center|7.2°
|align=center|36°
|align=center|6°
|align=center|7.2°
|}

....

Here we summarize our findings on the 11-cells so far, including in advance some results to be obtained in the next section two sections.

...finish and organize the following section, pushing some of it into subsequent sections; then reduce it to a short summary of findings to be put here, to conclude this section....

== The 15 major chords ring everything ==
The natural numbers are each a distinct flavor.

Each integer has its own distinct taste. The first six integers {0..5} are the six strongest, most familiar tastes in nature, instantly distinguished and utterly unique.

The seventh flavor {6} is the first ratio of two of the others. Thereafter follows in strange sequence all the ratios of all the flavors, each marriage of every two flavors, most of which are fractions &lt;math&gt;\{k/d\}&lt;/math&gt;, but some of which are integers &lt;math&gt;\{k\}&lt;/math&gt;.

Some of these ratios are successful flavors in their own right, as distinct and inimitable as the marriage of chocolate and oranges, but the marriage of many flavor pairs is mud. It is a rule of thumb among flavor specialists that nearly any taste can be successfully alloyed with either onions or chocolate, but the direct marriage of these two flavors is notably unsuccessful. Some flavors make famous pairings or infamous ones, and subtle new pairings are always to be discovered that work surprisingly well for some purpose, but the majority of flavor marriages are undistinguished compounds.

In polytope geometry, each chord of a polytope is both is a distinct 1-dimensional object, a chord of the sphere of a distinct length &lt;math&gt;l&lt;/math&gt;, and a distinct natural number &lt;math&gt;h&lt;/math&gt;, a unique flavor. If the polytope is regular, it is a noteworthy and successful flavor. The chord's length &lt;math&gt;l&lt;/math&gt; is a square root, related to the natural number &lt;math&gt;h = k/d&lt;/math&gt; and to a set of polytopes they both represent, by a formula discovered by Steinbach.{{Sfn|Steinbach|2000|loc=''Sections Beyond Golden''; Figure 5. Optimal sections and proportions|p=37|ps=; the regular polygons {5}, {7}, {9} and {11} with their diagonals define respectively: {5} the golden bisection proportional to 𝜙; {7} an analogous trisection; {9} an analogous quadrasection; {11} an analogous pentasection.}} The chord length &lt;math&gt;l&lt;/math&gt; is related to the number of sides of the regular polygon formed by the chord (its numerator &lt;math&gt;k&lt;/math&gt;), and to the winding number or density of its regular skew Petrie polygon (its denominator &lt;math&gt;d&lt;/math&gt;).

The regular 4-polytopes do not contain an infinite variety of these chord flavors, but only 30 of them: only 15 unique pairs of 180° degree complementary chords, which combine to make every regular ''n''-dimensional thing found in nature.{{Efn|The 15 ''major chords'' suffice to construct all the regular ''n''-polytopes. Their 180° complements include 15 ''minor chords'' which occur only in the 120-cell, not in any smaller regular polytope.{{Efn|name=Major and minor chords}} Including the complement of the 180° {{radic|4}} digon {2} chord (the {0} chord {{radic|0}}), and the skew variant of the 90° {{radic|2}} square {4} chord (the {30/7} chord), there are 32 distinct natural numbers which are all the chords of all the regular polytopes in all dimensions.|name=32 chords}} All the regular polytopes (including the largest one on the cover of the box) can be built not only from a box containing a number of blocks of 3 shapes glued together by gravity, they can be also be built from a box containing a number of sticks of 15 lengths glued together by rubber joints.

[[File:15 major chords.png|500px|thumb|The major{{Efn|The 120-cell itself contains more chords than the 15 chords numbered #1 - #15, but the others occur only in the interior of 120-cell, not as edges of any of the regular 4-polytopes. There are 30 distinct 4-space chordal distances between vertices within the 120-cell (15 pairs of 180° complements), including #15 the 180° diameter (and its complement the 0° chord). In this article, we do not have occasion to identify any of the 15 unnumbered ''minor chords'' by their arc-angles, e.g. 41.4~° which, with length {{radic|0.5}}, falls between the #3 and #4 chords.|name=Major and minor chords}} chords are properly labelled #1 — #15 in the order they fan out by size.{{Sfn|Steinbach|1997|loc=''Golden fields: A case for the Heptagon''; Figure 3|ps=; Steinbach derived a formula relating the diagonals and edge lengths of successive regular polygons, and illustrated it with a &quot;fan of chords&quot; diagram like the one here.|p=23}} They join vertex pairs which are 1 — 15 edges apart on a &lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon. A #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; chord forms &lt;small&gt;&lt;math&gt;f+1&lt;/math&gt;&lt;/small&gt; regular geodesic polygons which include a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; disjoint regular {{nowrap|&lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt;-gons}} (e.g. &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; a compound of 5 disjoint hexagons) plus a single [[W:Triacontagon#Triacontagram|regular skew {{nowrap|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram}}]] that wraps around the sphere &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times before closing its circle. Notice the maverick #11 chord, with its non-conforming origin point for its {{nowrap|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;}} polygram.]]

These 15 chords (natural numbers) include the integers 2, 3, 4, 5, 6, 10, 11, 15 and 30. The others are the fractions 5/2, 10/3, 13/7, 15/2, 15/4, and 15/7. Represented among the chord lengths (the square roots) measured in ''unit radii'' (not in {{radic|2}} radii, as metrics are often given elsewhere in this article) are the {{background color|paleturquoise|roots of the first 4 positive integers {{radic|1}}, {{radic|2}}, {{radic|3}} and {{radic|4}}}}, the {{background color|#FFCCCC|root of the very distinguished rational fraction {{radic|5/2}}}}, and the roots of various distinguished irrational fractions as Steinbach discovered: for example, 4 of them are {{background color|yellow|roots of golden numbers (a small integer ± 𝜙)}}. The 15 chords each form their regular polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, which is just &lt;small&gt;&lt;math&gt;\{k\}&lt;/math&gt;&lt;/small&gt; in 8 cases. Arranged in order #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; from shortest edge-length square root to longest, the #1 — #15 chords bridge vertex pairs which are &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; vertices apart on a [[W:Triacontagon#Petrie polygons|&lt;small&gt;&lt;math&gt;\{30\}&lt;/math&gt;&lt;/small&gt; Petrie polygon]]. Each chord makes a [[W:Triacontagon#Triacontagram|&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; polygram]] of its edge length, a compound of the smaller &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. The 15 distinct polygons form a sequence you would not anticipate exactly unless you have Steinbach's formula:

{| style=&quot;background-color:transparent; white-space:nowrap;&quot;
|-
| {30/1} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|.073}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{30} 
| 120-cell edges
|-
| {30/2}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.191}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15}
|- style=&quot;background: yellow;&quot;|
| {30/3}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.382}} 
| &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi}&lt;/math&gt;&lt;/small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{10}''' 
| 600-cell edges
|-
| {30/4} 
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|0.573}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/2}
|- style=&quot;background: paleturquoise;&quot;|
| {30/5}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|1}}'''
| &lt;math&gt;1&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{6}''' 
| 24-cell, 8-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/6}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|1.382}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{5}'''
|- style=&quot;background: paleturquoise;&quot;|
| {30/7}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|2}}'''
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{4}''' 
| 16-cell edges
|- style=&quot;background: #FFCCCC;&quot;|
| {30/8}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.5}} 
| &lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;small&gt;&lt;math&gt;\sqrt{\frac{5}{2}}&lt;/math&gt;&lt;/small&gt;&lt;/small&gt;&lt;/small&gt;&lt;small&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/4} 
| 5-cell edges
|- style=&quot;background: yellow;&quot;|
| {30/9}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|2.618}} 
| &lt;math&gt;\phi&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|{10/3}
|- style=&quot;background: paleturquoise;&quot;|
| {30/10}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|3}}''' 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{3}'''
|-
| {30/11}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.426}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{11/11}
|- style=&quot;background: yellow;&quot;|
| {30/12}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.618}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{5/2}
|-
| {30/13}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.810}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{13/7}
|-
| {30/14}
| style=&quot;padding:0px 10px 0px 10px;&quot;|{{radic|3.928}} 
|
| style=&quot;padding:0px 10px 0px 10px;&quot;|{15/7}
|- style=&quot;background: paleturquoise;&quot;|
| {30/15}
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{{radic|4}}''' 
| &lt;math&gt;2&lt;/math&gt;
| style=&quot;padding:0px 10px 0px 10px;&quot;|'''{2}'''
|}

{| class=wikitable style=&quot;white-space:nowrap; float:right;&quot;
!colspan=6|Natural number &lt;small&gt;&lt;math&gt;h=\{k/d\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Fibers&lt;br&gt;&lt;small&gt;&lt;math&gt;f\{h\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Cells&lt;br&gt;&lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|4-tope&lt;br&gt;&lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|Verts&lt;br&gt;&lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;
!rowspan=2|[[W:Triacontagon#Triacontagram|30-gram]]&lt;br&gt;&lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;
|-
!Chord
!Arc
!colspan=2|&lt;small&gt;&lt;math&gt;l/\sqrt{1}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;l/\sqrt{2}&lt;/math&gt;&lt;/small&gt;
!&lt;small&gt;&lt;math&gt;h&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#1
|15.5~°
|{{radic|0.073}}
|0.270
|{{radic|0.146}}
|&lt;small&gt;&lt;math&gt;30&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;1\{30\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;5\{3,3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/1\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#2
|25.2~°
|{{radic|0.191}}
|0.437
|{{radic|0.382}}
|&lt;small&gt;&lt;math&gt;15&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;..\{3,3\}..&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/2\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#3
|36°
|{{radic|0.382}}
|0.618
|{{radic|0.764}}
|&lt;small&gt;&lt;math&gt;10&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;30\{3,3\}20&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/3\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#4
|44.5~°
|{{radic|0.573}}
|0.757
|{{radic|1.146}}
|&lt;small&gt;&lt;math&gt;15/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/4\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#5
|60°
|{{radic|1}}
|1
|{{radic|2}}
|&lt;small&gt;&lt;math&gt;6&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/5\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#6
|72°
|{{radic|1.382}}
|1.175
|{{radic|2.624}}
|&lt;small&gt;&lt;math&gt;5&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}} 
|&lt;small&gt;&lt;math&gt;10\{5,3\}12&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/6\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#7
|90°
|{{radic|2}}
|1.414
|{{radic|4}}
|&lt;small&gt;&lt;math&gt;4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;7\{4\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐}}
|&lt;small&gt;&lt;math&gt;4\{4,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{4,3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;9\{3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/7\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: #FFCCCC;&quot;|
|#8
|104.5~°
|{{radic|2.5}}
|1.581
|{{radic|5}}
|&lt;small&gt;&lt;math&gt;15/4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/4\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}} 
|&lt;small&gt;&lt;math&gt;15\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/8\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#9
|108°
|{{radic|2.618}}
|1.618
|{{radic|5.236}}
|&lt;small&gt;&lt;math&gt;10/3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;3\{10/3\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/9\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#10
|120°
|{{radic|3}}
|1.732
|{{radic|6}}
|&lt;small&gt;&lt;math&gt;3&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;10\{3\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;6\{3,4\}4&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;4\{4,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/10\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#11
|135.5~°
|{{radic|3.426}}
|1.851~
|{{radic|6.852}}
|&lt;small&gt;&lt;math&gt;11/11&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;11\{11/11\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;11\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/11\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: yellow;&quot;|
|#12
|144°
|{{radic|3.618}}
|1.902
|{{radic|7.235}}
|&lt;small&gt;&lt;math&gt;5/2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;6\{5/2\}&lt;/math&gt;&lt;/small&gt; {{align|right|𝜙{{spaces|1|thin}}}}
|&lt;small&gt;&lt;math&gt;10\{5,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,\tfrac{5}{2},3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},5\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/12\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#13
|154.8~°
|{{radic|3.810}}
|1.952
|{{radic|7.621}}
|&lt;small&gt;&lt;math&gt;13/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;13\{13/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|✩}}
|&lt;small&gt;&lt;math&gt;7\{5,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{5,3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,\tfrac{5}{2}\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/13\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: seashell;&quot;|
|#14
|164.5~°
|{{radic|3.928}}
|1.982
|{{radic|7.857}}
|&lt;small&gt;&lt;math&gt;15/7&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15/7\}&lt;/math&gt;&lt;/small&gt; {{align|right|&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;137\{\tfrac{5}{2},3\}1&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{\tfrac{5}{2},3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/14\}&lt;/math&gt;&lt;/small&gt;
|- style=&quot;background: paleturquoise;&quot;|
|#15
|180°
|{{radic|4}}
|2
|{{radic|8}}
|&lt;small&gt;&lt;math&gt;2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;15\{2\}&lt;/math&gt;&lt;/small&gt; {{align|right|☐𝜙&lt;small&gt;△&lt;/small&gt;}}
|&lt;small&gt;&lt;math&gt;8\{3,3\}2&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{3,3,4\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;2\{15\}&lt;/math&gt;&lt;/small&gt;
|&lt;small&gt;&lt;math&gt;\{30/15\}&lt;/math&gt;&lt;/small&gt;
|}

The polygons '''{10}''', '''{6}''', '''{5}''', '''{4}''', '''{3}''', and '''{2}''' are flat polygons that each lie on a great circle in a central plane of the 3-sphere. The polygons {30}, {15}, {15/2}, {15/4}, {10/3}, {11/11}, {5/2}, {13/7}, {15/7} and all the {30/''n''} polygons are skew polygrams spiraling through curved 3-space, each successive edge in a different central plane of the 3-sphere. Projected to the 1-sphere (a circle in the plane), they are [[W:Star polygon#Regular star polygon|regular star polygrams]], the simplest being the pentagram {5/2}. In 4-space they wind around the 3-sphere more than once ''without self-intersecting'', before they close their circuit. They lie on helical geodesic circles or ''isoclines'' of circumference greater than 2𝝅''r''.

Even the 6 integer chords which form flat polygons also form skew polygrams. Each of the 15 chords occurs as a compound of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, a fiber bundle of Clifford parallel polygons that totals 30 vertices per bundle. Each chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; also forms a single skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram that winds through another, shorter chord's bundle of parallel polygons, intersecting them in succession multiple times each, and weaving that whole bundle of parallel fibers together as the cross-thread does on a loom. The polygram cross-thread has to wind multiple times through the whole bundle (&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; times around the [[W:Triacontagon#Triacontagram|30-gram]] in which each edge connects every &lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt;th vertex) to complete an enumeration of all 30 vertices in the bundle. For example, the #5 chord forms hexagon &lt;small&gt;&lt;math&gt;\{6\}&lt;/math&gt;&lt;/small&gt; flat planes in parallel bundles of 5, written &lt;small&gt;&lt;math&gt;5\{6\}&lt;/math&gt;&lt;/small&gt;, and also a skew &lt;small&gt;&lt;math&gt;\{15/2\}&lt;/math&gt;&lt;/small&gt;-gram of #5 edges which winds 2 times through the #3 chord's bundle of 3 decagon &lt;small&gt;&lt;math&gt;\{10\}&lt;/math&gt;&lt;/small&gt; flat planes. Every chord #&lt;small&gt;&lt;math&gt;n&lt;/math&gt;&lt;/small&gt; forms both its &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram and its fiber bundle of &lt;small&gt;&lt;math&gt;f&lt;/math&gt;&lt;/small&gt; parallel &lt;small&gt;&lt;math&gt;\{h\}&lt;/math&gt;&lt;/small&gt; polygons, and the edges of one chord's parallel polygons are also the edges of its skew &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt;-gram weaving through some shorter chord's bundle of parallel polygons. Each numbered chord is a natural number, a ratio of two integers and their polygon &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt;, either flat or skew. Each chord number defines a &lt;small&gt;&lt;math&gt;\{30/n\}&lt;/math&gt;&lt;/small&gt; skew polygon that is a compound bundle of the &lt;small&gt;&lt;math&gt;\{k/d\}&lt;/math&gt;&lt;/small&gt; polygon. All these flat or skew polygons are regular polygons made of their numerator instances of a single chord #, whose length is that square root, Steinbach's function of the natural number.

The relationship between each bundle of flat Clifford parallel great circle polygons of a distinct length, and its corresponding bundle of skew polygrams of a longer length, is precisely that of a discrete simple rotation and its corresponding discrete isoclinic rotation. For example, the flat great circle hexagons of edge length {{radic|1}} lie in the invariant rotation planes of the 24-cell's characteristic 60° isoclinic rotation, in which the vertices move on skew hexagram rotation paths of edge length {{radic|3}}. The 24-cell contains 4 bundles of 4 Clifford parallel (disjoint) hexagon central planes each. The 24 vertices of the 24-cell in each bundle are linked by rotation in the 4 hexagon invariant planes in two different ways: simply and isoclinically. In a simple rotation the vertices of a single invariant hexagon exchange places on that great circle hexagon of edge {{radic|1}}. In an isoclinic rotation in which any one of these 4 disjoint hexagon central planes is invariant, all 4 of them are necessarily invariant planes in the rotation, because they rotate in parallel. In that isoclinic rotation the vertices of all 4 hexagon invariant planes (all 24 vertices of the 24-cell) exchange places concurrently, moving along 4 disjoint helical hexagrams of edge {{radic|3}} in parallel. It is a characteristic only of this distinct rotation in hexagon invariant planes that the helical vertex path or ''isocline'' is a hexagram of period 6, edge length {{radic|3}}, and circumference 720°. Rotations in other kinds of polygon invariant planes have different characteristic isoclines. The period of their isoclinic rotation must be ''the same or greater'' than the period of their simple rotation; in most cases (unlike this one) it is greater.

The 15 chords form not only distinct fiber bundles of polygons, but also distinct bundles of polyhedra: cell rings &lt;small&gt;&lt;math&gt;c\{p,q\}&lt;/math&gt;&lt;/small&gt; and compound vertex figures &lt;small&gt;&lt;math&gt;v\{q,r\}&lt;/math&gt;&lt;/small&gt;. These combine to form their characteristic regular 4-polytope &lt;small&gt;&lt;math&gt;\{p,q,r\}&lt;/math&gt;&lt;/small&gt;.{{Sfn|Coxeter|1973|p=269|loc=Compounds|ps=; &quot;It is remarkable that the vertices of {5, 3, 3} include the vertices of all the other fifteen regular polytopes in four dimensions.&quot;}} All these elements of a regular 4-polytope constructed from one chord are themselves regular polytopes, and each of the 16 regular 4-polytopes is a function of one characteristic chord. If the regular 4-polytope is convex, its characteristic chord is its unit-radius edge. For a non-convex regular 4-polytope, the characteristic chord is its unit-radius ''mid-edge diameter'', the distance from the midpoint of each edge to the midpoint of the completely orthogonal edge. Each of the characteristic chords (each distinct natural number) has its expression as one of the 16 regular 4-polytopes and all the products of the distinct regular 1-polytopes, 2-polytopes and 3-polytopes that construct it, and the whole distinct construction from the natural number to the 4-polytope is an [[W:SO(4)|SO(4) symmetry group]] expression of that 4-polytope's characteristic discrete [[W:SO(4)#Isoclinic rotations|isoclinic rotation]], in the invariant central planes of its characteristic chord, by the characteristic chord arc.

...demonstrate this theory by proper enumeration of the chord lengths in the table...

...color-code the △ ☐ 𝜙 ✩ symbols ... chord colors in diagram should match them ... some of the diagram colors are wrong and must be changed, but otherwise diagram is pretty good ... i think in a colored diagram all chords of the same length should have the same size line dash, the polygons are not obvious enough and the 15 fanning-out chords are readily identified by their labels...

... add back the counts of chords in the box (very small font number under each table entry), and present these two images as the front and back a box of building sticks and rubber joints...What's in the other box...

...say the rational-root major chords {{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|5/2}} are the ''n''-edges: their lengths are the ratio of an ''n''-simplex edge to an ''n''-orthoplex edge...

...abbreviated text of the [[120-cell#Relationships among interior polytopes|''§Relationships among interior polytopes'']] section from the 120-cell article...to explain the color-coding

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius=1}}

See also the elements, properties and metrics of the sequence of regular 4-polytopes summarized in tabular form in the section ''[[#Build with the blocks|§Build with the blocks]]'', below.

== Alicia Boole Stott's original formulation of dimensional analogy ==
&lt;blockquote&gt;&quot;...a method by which bodies having a certain kind of semi-regularity may be derived from regular bodies in an Euclidean space of any number of dimensions...&quot;{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''; presents cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues|pp=12-45}}&lt;/blockquote&gt;

Alica Boole Stott's 1910 paper reads like an exceptionally well-written patent application. It is a comprehensive stepwise description of her original invention, a general method of dimensional analogy between regular and semiregular polytopes of the same edge length.

[[File:The Roots of Three.png|thumb|An illustration of Boole Stott's expansion of unit-edge triangle to hexagon captures the geometric basis in 3 of the &lt;math&gt;A_n&lt;/math&gt;root system.]]

The engineering-like articulation of her new principles is entirely discrete and group-theoretic. Boole Stott repeatedly refers to sets of polytope elements as ''groups''. All her transformations are discrete instances of one or another special case of her &quot;two inverse operations which may be called ''expansion'' and ''contraction''.&quot; Hers is an exploration of ''n''-dimensional space at its most basic: space defined by the discrete objects which fill it, themselves defined directly as groups of elements with a few operations on them, which transform them into each other. This is geometry which seems to precede Euclid's postulates, and does not depend on them. Instead it corresponds directly to the deeper group theory mathematics which underlies geometry, even though Boole Stott, like the ancients, did not have that mathematics.

Boole Stott enumerated her expansion and contraction operations as transformations between regular and semi-regular polytopes of a single edge length. In her 1910 paper she did not consider any polytopes of multiple edge lengths, or expansions of the subject polytope by a distance other than that exactly required to produce new edges of the original edge length, so of course she did not enumerate Moxness's non-uniform rhombicosidodecahedron (Hull #8). This may be why section 8&lt;sub&gt;3&lt;/sub&gt; is missing from her earlier drawings of the sections of {5,3,3}; she may not have had a construction for it.{{Sfn|Coxeter|1973|p=258-259|loc=§13.9 Sections and Projections: Historical remarks|ps=; &quot;Alicia Boole Stott (1860-1940) ... also constructed the sections i&lt;sub&gt;3&lt;/sub&gt; of {5, 3, 3}, exhibiting the nets in her Plate V. “Diagrams VIII-XIV” refer to the sections 1&lt;sub&gt;3&lt;/sub&gt;-7&lt;sub&gt;3&lt;/sub&gt;; but 8&lt;sub&gt;3&lt;/sub&gt; is missing. Incidentally, Diagram XIII (our 6&lt;sub&gt;3&lt;/sub&gt;) is a rhombicosidodecahedron, the Archimedean solid.&quot;}}

The Boole Stott expansion (or contraction) operation is as precisely defined a transformation as Coxeter's reflection, rotation and translation operations.{{Efn|Let Q denote a rotation, R a reflection, T a translation, and let Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T denote a product of several such transformations, all commutative with one another. Then RT is a glide-reflection (in two or three dimensions), QR is a rotary-reflection, QT is a screw-displacement, and Q&lt;sup&gt;2&lt;/sup&gt; is a double rotation (in four dimensions). Every orthogonal transformation is expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt;&lt;br&gt;
where 2''q'' + ''r'' ≤ ''n'', the number of dimensions. Transformations involving a translation are expressible as
{{indent|12}}Q&lt;sup&gt;''q''&lt;/sup&gt; R&lt;sup&gt;''r''&lt;/sup&gt; T&lt;br&gt;
where 2''q'' + ''r'' + 1 ≤ ''n''.&lt;br&gt;
For ''n'' {{=}} 4 in particular, every displacement is either a double rotation Q&lt;sup&gt;2&lt;/sup&gt;, or a screw-displacement QT (where the rotation component Q is a simple rotation). Every enantiomorphous transformation in 4-space (reversing chirality) is a QRT.{{Sfn|Coxeter|1973|pp=217-218|loc=§12.2 Congruent transformations}}|name=transformations}} An instance of the expansion (or contraction) operation is an application of the general operation to a particular subject polytope, operating on all of its elements symmetrically at once. The general operation is restricted, such that one may only expand (or contract) the subject polytope by pushing all its edges, faces or cells directly away from (or toward) its center, without changing the size of the moved elements. This splits each vertex into several (or coalesces several vertices into one), and has corresponding effects on the other elements, producing new edges ''of the same size'' between pairs of separated vertices (or reducing some edges to single vertices). The distance all the elements are moved away from (or toward) the center must be such that the vertices separate producing new edges of only the original edge length (or come together reducing edges to single vertices). It is permitted to combine these three operations, e.g. by pushing the edges ''and'' the faces, or by pushing all three element sets, provided the combination results in a semi-regular polytope of only one edge length. The three operations are all commutative with one another, so combined operations can be considered either as a sequence of expansions (or contractions) linking several related polytopes, or as their concurrent movements, a single transformation of the subject polytope. Since there are 8 combinations of one to three operations there are 8 polytope variants (including the original regular polytope), and all are adjacent to one another by a single transformation. The variants of the original regular polytope are named for which element set(s) are moved. By enumerating all 8 variants of expansion (and contraction) for each of the six regular convex 4-polytopes, Boole Stott produced a table of regular and semi-regular 4-polytopes divided into 12 sets of 8: six expansion sets and six contraction sets.{{Sfn|Boole Stott|1910|loc=''Table of Polytopes in S&lt;sub&gt;4&lt;/sub&gt;''; note that Boole Stott's usage of ''S&lt;sub&gt;4&lt;/sub&gt;'' denotes a Euclidean 4-space ''E&lt;sub&gt;4&lt;/sub&gt;'', or a 3-sphere space embedded in it, ''not'' a 4-sphere space embedded in a Euclidean 5-space|p=30}} Each set includes just one of the six regular convex 4-polytopes, with its distinct unit-radius edge length, and the 7 possible semi-regular expansions (or contractions) of it, which all preserve that characteristic edge length. The polytopes in each set of 8 all have the same total number of elements as their original regular polytope, but each polytope in the set has a distinct combination of cell, face, and vertex figures. The 12 sets are not completely disjoint, and even within one set the 8 operations do not necessarily produce 8 unique polytopes. For example, the regular 5-cell (the 4-simplex polytope) has 5 cells, 10 faces, 10 edges, and 5 vertices (30 elements). Expanding the 5-cell produces 7 related semi-regular 4-polytopes of 30 elements each (two of which happen to be the same polytope). In each expansion, the 5 cells have expanded into semi-regular or regular polyhedra, the 10 faces and 10 edges ''may'' have expanded into prisms, and the 5 original vertices have expanded into regular or semi-regular cells.

== The regular 11-cell 4-polytope ==

Long ago it seems now and far [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#The 5-cell and the hemi-icosahedron in the 11-cell|above]], we noticed 12-point (Jessen's icosahedra) spanning Moxness's 60-point (rhombicosadodecahedron). Each non-convex Jessen's icosahedron is a set of 6 reflex edges that are {{radic|5}} 5-cell edges, each set symmetrically arranged in 3 parallel pairs, not as far apart ({{radic|3}}) as they are long, in 3 orthogonal central planes. The 12-point (Jessen's sets of 6 reflex 5-cell edges) are inscribed in the 60-point with their endpoints as a vertex of a pentagon face. The 12 pentagon faces are joined to 5 other pentagon faces by the 5-cell edges; not to their nearest neighbor pentagons but to their next-to-nearest neighbor pentagons; and since there are 5 vertices to each pentagon face, there are 10 12-point (Jessen's sets of 6 reflex 5-cell edges). Each pentagon face is joined to another pentagon face by 5 reflex edges of five distinct Jessen's. The 5 reflex edges are Clifford parallel, the 5 edges of a pentagonal prism column that has a twist to it. The contraction of those 5 edges into 1, and those 10 vertices (two pentagons) into 2, by abstraction to a less detailed map, loses the distinction that the 5 close-together vertices are actually separate points, and that the 5 Jessen's are actually separate objects, superimposing all their edges as a single Jessen's. The further abstraction of identifying both ends of each reflex edge contracts the remaining 12-point (Jessen's icosahedron) into a 6-point (octahedron) by conflating its parallel pairs of edges, leaving the 11-cell's abstract 6-point (hexad cell). The hexad's situation is that it occurs in bundles of 5, disjoint but close together in the sense of adjacent, like the great circle rings in a Hopf fiber bundle. 

To summarize, the 12-point (2-hexad) is situated in pairs as a 24-point (24-cell), in bundles of 8 as a 96-edge 24-point (not the same 24-cell), and in bundles of 5 as a 60-point (hemi-icosahedral cell of an 11-cell). We shall now be able to see how Fuller's perfect hexads combine with Todd's perfect pentads to form the perfect heptads of the 11-cell and 137-cell.  

...you may finally be in a postion now to reveal how 12-points combine across bundles (of 2, 5, or 8 hexads) into bundles of 12 ''pentads''... but probably not yet to show how they combine into ''bundles of 11''...  

[[File:Tetrahemihexahedron_rotation.gif|thumb|The [[W:Hemi-cuboctahedron|hemi-cuboctahedron]] is an abstract polyhedron with a real presentation as a uniform star octahedron with 7 faces (4 triangles and 3 squares), 12 edges, and 6 vertices, the [[W:Tetrahemihexahedron|tetrahemihexahedron]]. Like all hemi-polyhedra, some of its faces pass through the center of the polyhedron, where they intersect each other.{{Sfn|Ruen: Tetrahemihexahedron rotation|2019|loc=alias the hemi-cuboctahedron}}]]
We have seen how the compound of 5 12-point (Jessen's icosahedra) that is Moxness's 60-point (rhombiscosadodecahedron) can be abstracted, by conflating its vertices 10-into-1 into the 6-point (hemi-icosahedron). Each of those 5 12-point (Jessen's icosahedra) can also be abstracted individually, by conflating its vertices 2-into-1 into the 6-point ([[W:Tetrahemihexahedron|hemi-cuboctahedron]]), which has 4 triangular faces and 3 square faces. (The ''heptad'' arrives at the party, anonymously; they are our distinguished anti-celebrity, who most of the other guests have never even heard of.) The prefix ''hemi'' means that the polyhedron's faces form a group with half as many members as the regular polyhedron, oriented in the same direction as the regular polyhedron's faces. The three square faces of the hemi-cuboctahedron, like the three facial orientations of the cube, are mutually perpendicular. The &quot;half-as-many&quot; characteristic also means that hemi faces must pass through the center of the polyhedron, where they all intersect each other. The inverted nature of the 5 Jessen's inscribed in the hemi-icosahedron, which have their 8 equilateral triangle faces actually located in central planes, is a distinguishing property of all hemi-polyhedra. Visually in the hemi-cuboctahedron, each square is divided into four yellow right triangles, with two visible from each side. The 4 red triangles are 5-cell faces, from 4 disjoint 5-cells. Thus their edges are {{radic|5}} reflex edges of 3 distinct Jessen's icosahedra, from 3 distinct rhombiscosadodecahedra. The hemi-cuboctahedron has ''only'' these {{radic|5}} edges. They do not actually meet at the corners of the red triangles, since in the 120-cell the hemi-cuboctahedron's 6 vertices are actually pairs of distinct Jessen's vertices, separated a {{radic|6}} edge.

[[File:Truncatedtetrahedron.gif|thumb|4 triangles opposite 4 hexagons. Two of these 12-point ([[W:Truncated tetrahedron|truncated cuboctahedra]]) fit together hexagon-to-hexagon so that 2 triangles are opposite each other. This 12-point (truncated tetrahedron) is an expansion of the 6-point (hemi-cuboctahedron). It occurs as the 12-point (inverted Jessen's icosahedron vertex figure) of the 120-cell. The quasi-regular 11-point (11-cell) has 6 of this hexad cell and 5 pentad cells. In a {{radic|2}}-radius 120-cell, 100 of these with {{radic|5}} triangle faces and {{radic|6}} hexagon long edges contribute their {{radic|5}} faces to the 120 11-cells.{{Sfn|Cyp: Truncated tetrahedron|2005|loc=File:Truncatedtetrahedron.gif}}{{Sfn|Cyp: Truncated tetrahedron|2005|loc=[[Wikipedia:User:Cyp/Poly.pov]]|ps=; source code for the [[W:POV-ray|Persistence of Vision ray-tracing engine]] that generates this image and others.}} Of course Moxness has been there, seen that, already with his graphics software;{{Sfn|Moxness: Quaternion graphics software|2023|ps= ; describes the theory and implementation of quaternion-based polytope graphics software.}} see his parameterized illustration of it exactly as it occurs in the 11-cell, below in his bestiary of A3, B3, and H3 Archimedian and Catalan polyhedra in the 120-cell.]]
We can expand the abstract 6-point (hemi-cuboctahedron) into the actual 12-point cuboctahedral vertex polyhedron it represents in the 120-cell, by moving those pairs of conflated vertices {{radic|6}} apart. The result is the proper right-side-out orthogonal projection of the inside-out 12-point (Jessen's icosahedral vertex figure). It has all the same-size elements as the 12-point (Jessen's) shown above, particularly its {{radic|5}} and {{radic|6}} edges. Its faces are 4 of the 12-point (cuboctahedron)'s 8 triangular faces, plus the cuboctahedron's 4 hexagonal planes; it is a 12-point (truncated cuboctahedron) and also a 12-point (truncated tetrahedron).

[[File:Archimedean_and_Catalan_solid_hulls_with_their_Weyl_orbit_definitions.svg|thumb|The 60-point hemi-icosahedral cell of the 11-cell is a compound of 5 of the 12-point ([[W:Truncated tetrahedron|truncated cuboctahedron]]), inversely isomorphic to the 5 12-point Jessen's icosahedra that span the cell. In this catalog of Archimedean and Catalan polyhedra sighted in the 120-cell by Moxness's software,{{Sfn|Moxness: Archimedean and Catalan hulls|2023|loc=Hull #1 Archimedean Name A3 110 Truncated Tetrahedron A (upper left)}} it is the first entry (upper left), Moxness's Hull # = 1 with 12 vertices of 3D norm {{radic|11}}/2. On a {{radic|2}}-radius 3-sphere, 5 of these with {{radic|2}}∕𝝓 triangle edges and {{radic|2}} hexagon long edges compound to a 60-point (rhombiscosadodecahedron) cell. 20 of them with {{radic|5}} triangle edges and {{radic|6}} hexagon long edges are cells in the [[#The rings of the 11-cells|cell rings of the quasi-regular 11-cell]], which also has 12 regular 5-cells as cells.]]
Vertex figures may be extracted in various sizes, depending on how deeply one truncates the vertex. The truncated cuboctahedron with {{radic|5}} and {{radic|6}} edges is the full-sized 12-point vertex figure (the deepest 12-point truncation) of the 120-cell vertex, with 4 triangular 5-cell faces. A shallower truncation gives a smaller 12-point (truncated cuboctahedron) with {{radic|2}}∕𝝓 and {{radic|2}} edges, and 5 of these smaller objects compound to make Moxness's  60-point (rhombiscosadodecahedron) with its 20 {{radic|2}}∕𝝓 triangular faces (600-cell faces), the real hemi-icosahedral cell. 

The 11-cell's {{radic|5}} faces (the 5-cell faces) are ''not'' faces of its rhombiscosadodecahedron cells, which have only smaller triangle faces. The 60-point (hemi-icosahedral) cells contain only individual {{radic|5}} edges, not whole {{radic|5}} faces; the {{radic|5}} faces have an edge in 3 different rhombiscosadodecahedra. The rhombiscosadodecahedron's small triangle faces, like the big {{radic|5}} faces which lie between three cells, are found as triangular faces of a 12-point (truncated cuboctahedron), but there are two different sizes of the 12-point (truncated cuboctahedron). The 120-cell has 5 small 12-point (truncated cuboctahedra) vertex figures around each of its 600 vertices, and 300 big 12-point (truncated cuboctahedra) around its own center bearing the 1200 faces of its 120 regular 5-cells.

...

Both kinds of cells in the 11-cell have expressions in two different dimensions (on the 2-sphere and 3-sphere), because they are all centered on the center of the 3-sphere, unlike the cells of most 4-polytopes, which are centered on various off-center points within the 4-polytope's interior. 

One kind of 11-cell cell appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (semi-regular truncated cuboctahedron), a.k.a. the truncated tetrahedron. The other kind of cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now, except in the 120-cell. But this is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron, and a regular tetrahedron behind each face. The 11-cell is thus a honeycomb of mixed regular tetrahedra and truncated cuboctahedra (a.k.a. truncated tetrahedra) that meet at their shared 5-cell faces. 

Both the 5-cells (decahedra) and the truncated cuboctahedra (truncated tetrahedra) are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks in their 4-polytopes (off-center).{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} Therefore all the 11-cell's cells are concentric, which indeed makes a different kind of honeycomb, an inside-out cellular honeycomb with all the cells piled on top of each other overlapping in 4-space. They do not overlap in 3-sphere space however; their 3-volumes (but not their 4-contents) are disjoint. They make a proper 4D honeycomb like the cells of any 4-polytope, with volumetrically disjoint 3D cells that meet each other in the proper way at their shared 2D faces.

An 11-cell contains 6 completely disjoint 5-point (1-pentad 5-cell decahedron cells), and 5 volumetrically disjoint 12-point (2-hexad truncated cuboctahedron cells), 11 volumetrically disjoint cells altogether. Of course it is supposed to have 11 identical 11-point (elevenad hemi-icosahedral cells), and that is also true in its abstract sense, which is not the same sense as disjoint cellular containment: each pentad or hexad cell is inscribed in one of 11 larger 60-point (Moxness's rhombicosadodecahedron cells). The abstract pentads and hexads are only hemi-polyhedra, so the 11-cell's pentad cells are 12 (out of 120) disjoint instances of a pentad 5-point, and its hexad cells are 10 (out of 100) disjoint instances of a hexad 6-point, 1/10th of the 120-cell's cellular volume in both cases. The 11-cell's hemi-icosahedral cells are 11 (out of 120) non-disjoint instances of an elevenad 60-point. Each 11-cell shares each of its hexad cells with 9 other 11-cells, and each of its 11 vertices (out of 600) occurs in 9 hexad cells.{{Efn| Notice the tetrahedral analogy between 3 16-point (8-cell 4-cubics) in the 24-point (24-cell) sharing pairs of vertices, and 200 12-point (truncated cuboctahedra) in the 600-point (120-cell) sharing quads of vertices. 8-cell, 11-cell and 5-cell (the 11-cell's other cell) all have tetrahedral vertex figures, and 75 16-point (8-cell 4-cubics) share quads of vertices in the 120-cell, whose 1200 own edges form 600 tetrahedral vertex figures.|name=Tetrahedral vertex figures}}

Physically, the 11-cell's hexad cells combine in pairs, hexagon-to-hexagon, to form a female-to-female quick-connect garden hose connector in an 11 cell hose ring. The triangle faces of the pentad cells are the male garden hose quick-connectors. Four orthogonal rings intersect in each 12-point (hexad female connector). Five non-orthogonal rings intersect in each 5-point (pentad male connector). Thus the physical 11-cell.

The 4-space realization of the abstract 6-point (hemi-icosahedron) cell is Moxness's 60-point (rhombicosadodecahedron), a central section of the 120-cell. Each abstract vertex is identified as a 10-point (great decagon) in the 60-point (rhombicosadodecahedron), which has 5 disjoint 12-point (Jessen's icosahedra) inscribed in it, each of which can itself be abstracted as a 6-point (hemi-icosahedron) with each abstract vertex identified as a 2-point ({{radic|5}} 5-cell edge) in the Jessen's icosahedron. There are 11 interlinked rings of 11 rhombicosadodecahedron cells each, forming a single dense fibration. The axial great circles of the fibration (its actual fibers) are 200 irregular great dodecagons. 

...insert image thumb of 11-cell...

...Construct the 11-cell from the heptad....

The big 12-point (truncated cuboctahedra) are related to the small 12-point (truncated cuboctahedra) by a contraction operation, not that such a contraction ever occurs in a rigid 120-cell. They are concentric like nested Russian dolls, and could be transformed into each other by symmetrical expansion or contraction of all their edges at once. This is almost the same thing the shad does when rotating his 4 pairs of triangular faces toward each other and away again as if they were inscribed in an invisible rifled cylinder barrel, except that in 3 dimensions the triangular face edges don't change size, only the radii do, and in 4 dimensions the edges of both size vary. That too is properly dimensionally analogous, because rotation in 4-space is a double rotation, spinning in two completely orthogonal tumbler cylinders at once, like two completely orthogonal keys turned simultaneously in a baffling 4-dimensional tumbler lock. That is not just a poetic visual metaphor; it is actually what happens when rotating objects in 4-space twist equally in all 4 dimensions at once to move the short distance between adjacent Clifford parallel subspaces, in an isoclinic rotation. The contraction operation is Fuller's jitterbug, but different embedded in 4-space than it is in 3-space: precisely the dimensionally analogous difference between [[w:SO(3)|SO(3)]] single rotations and [[w:SO(4)|SO(4)]] double rotations. And of course the difference that the 120-cell isn't actually flexing, it just embodies the expansion-contraction relationship between two concentric truncated cuboctahedra, whose 4 remaining triangle faces are congruent with 4 triangle faces of the same flexible cuboctahedron, at different stages of the cuboctahedron's kinematic transformation cycle. 

...

...Four concentric 4-polytopes may be mutually orthogonal. In any set of 4 mutually orthogonal regular 5-cells, one pair of faces in each orthogonal pair are completely orthogonal to each other. Thus three edges in one 5-cell in an orthogonal pair are opposite (completely orthogonal) to a face of the orthogonal 5-cell. .... A 5-cell has 5 vertices in its abstract hemi-icosahedral cells, but only 4 vertices in one concrete icosahedral cell; the 5th lies in an icosahedral cell of another 11-cell (not in another icosahedral cell of the same 11-cell).

...don't meet at a vertex, but we can pick out 6 disjoint 5-cell edges among these 20 edges that are 3 completely orthogonal pairs of edges....

....

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 11-cell ===

Besides being the cell of the 11-cell, Moxness's 60-point (rhombicosadodecahedron) is also the Hopf map of the 11-cells (plural), that is, of the 120-cell as a fibration of 120 11-cells. Each little pentagon edge lifts to a 120-cell pentagon.... (wrong, 11-cell's Hopf map is the truncated icosahedron)

....

[[File:Polyhedron_truncated_20_from_yellow_max.png|thumb|A 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), an Archimedean solid, one of 13 convex isogonal nonprismatic solids whose 32 faces are two or more types of regular polygons. It is the only one of these shapes that does not contain triangles or squares, and it is the [[W:Goldberg polyhedron|Goldberg polyhedron]] GP&lt;sub&gt;V&lt;/sub&gt;(1,1). Its geometry is associated with [[W:Ball (association football)|footballs]] (soccer balls), [[W:Geodesic dome|geodesic dome]]s, and the fullerene [[W:Buckminsterfullerene|C&lt;sub&gt;60&lt;/sub&gt;]] (&quot;buckyball&quot;) molecule. With a uniform degree of truncation it has regular hexagon faces, but it also occurs with irregular hexagon faces with {{radic|5}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|6}} edges in a {{radic|2}} radius polyhedron.{{Sfn|Piesk: Truncated icosahedron|2018}} Moxness's &quot;Overall Hull&quot; [[#The 5-cell and the hemi-icosahedron in the 11-cell|(above)]] is a truncated icosahedron.]]
The 11-point (11-cell) is an abstraction of yet another abstraction, the 60-point (32-cell). The 60-point is less abstract than the 11-point but still an abstraction, because its 60 vertices are 10-to-1 conflations such that there is only one of it in the 600-point (120-cell). It represents all 10 600-cells on top of each other, if you like, but the 10 60-points do not correspond to the 10 600-cells. The 120 11-cells are precisely a web binding together all 10 600-cells, a hologram of the whole 120-cell which comes into existence only when there is a compound of 5 disjoint 600-cells (5 sets of 120 vertices that are 120 sets of 5 vertices in the configuration of a regular 5-cell). No part of the hologram is contained by any object smaller than the whole 120-cell. However, the hologram has an analogous 60-point (32-face 3-polytope) Hopf map that is very real: the 60-point (truncated icosahedron) makes a helpful 3-dimensional picture of the 600-point (120) with its 60-point (32-cell rhombicosadodecahedron) cells. Both 60-points have 12 pentad facets and 20 hexad facets, which are polygons in the 3-polytope and polyhedral cells in the 4-polytope. The actual situation is more complex, however. In the 60-point (32-cell) the 60-point (truncated icosahedron)'s 5-point (pentagon) faces are 5-point (regular 5-cell) 4-polytopes, and its 6-point (hexagon) faces are 12-point (truncated cuboctahedra) 3-polytopes. The 5-points are not ordinary 4-polytope cells (3-polytopes), they are 4-polytopes themselves.

[[File:4-simplex_t0.svg|thumb|The projection (above) of the 60-point (32-cell) has pentagon faces which are 5-point (regular 5-cells), each of which is five 4-point (regular tetrahedra). The pentagons should have their diagonal edges drawn, like this, to represent the 5-point (4-simplex).]]
This illustration would be better representation of the 32-cell if we added the other 5 edges to the 5-points, so they looked like the pentagonal projection of the 5-cell. Then we could see this object as a conventional 4-polytope with only 3-polytope cells: the 60-point (80-cell) with 60 4-point (tetrahedra) and 20 12-point (truncated cuboctahedra). But that would be an abstraction of its true nature, because the 5 tetrahedra of each pentad cell lie in different hyperplanes and actually form a 4-polytope. But the hexad cells are single cells.

The 32-cell has 11 interlinked rings of 11 cells each, forming a single dense fibration. Its rings of 11 cells contain 5 of the 5-point cell and 6 of the 12-point cell, which is where 11 comes from. The 5-point cells have 10 triangle faces, and the 12-point cells are irregular with 4 triangle and 4 hexagon faces. 5-points bond to 12-points at triangle faces, and 12-points bond to other 12-points at hexagon faces, but the 5-points never bond to other 5-points at a triangle face: there is a shell of 10 12-point cells surrounding each 5-point cell; they are completely disjoint, like separate islands. The 12-points, on the other hand, are all connected to each other in a tetrahedral lattice: each has 4 hexagonal faces like doors which lead into a neighboring 12-point, as well as 4 triangular faces like doors into a neighboring 5-point cell. You can walk through any door and continue in as straight a line as you can through 10 cells and find yourself back in the cell where you started, in eleven steps. (You can go straight through a 12-point room between a triangular door on one side and the hexagonal door opposite it, but on entering a 5-point room you are staring staring straight at an edge opposite, not  door, so you have to make a choice to bend left or right.) The abstraction that the 11-cell has only 11 cells in total comes from pretending that it's the same sequence of 11 cells you loop through, whichever door you start through, but that isn't so; there are 11 distinct paths through a total of 32 distinct cells, but they aren't furnished rooms so it's pretty hard to tell them apart (except by counting your steps) so it's easier to think of them as just the 11 places that are between 1 and 11 steps away from here, even through there are actually 32 places. 

...

Since the rhombicosadodecahedron has 120 5-cell edges, the 120-cell has 120 completely disjoint 5-cells, and the rhombicosadodecahedron shares each vertex with 11 other rhombicosadodecahedra, how a rhombicosadodecahedron is bound to its neighbor 5-cells is constrained. A further constraint is that the abstract hemi-icosahedron has only 15 5-cell edges, so the 120 actual 5-cell edges of the rhombicosadodecahedron are conflated 8-to-1 in the hemi-icosahedron. To be conflated by contraction into a single edge, those 8 edges must lie parallel to each other in the 120-cell; either actually parallel in the same plane, or Clifford parallel in separate planes, as two edges in completely orthogonal planes may be parallel and orthogonal at the same time. Each 5-cell inscribed in the 120-cell contains 5 such tetrahedra occur in parallel pairs in 4 orthogonal planes (none of them completely orthogonal planes). 

.... 

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
Each conflated vertex is the 10 vertices of two orthogonal regular 5-cells...
The 60-point contains 6 pairs of these orthogonal regular 5-cell cells...
...its 20 hexad cells are ..., a section of the 120 cell...

The axial great circles of the 11 rings (the actual fibers of the fibration) are not 11-gons, they are irregular hexagons made of alternating triangle and hexagon edges. The triangle edges are much larger than the hexagon edges; in fact, the triangle edge is the largest edge found in any regular 4-polytope, and the hexagon edge is the smallest. The hexagons are just the triangles with their vertices separated a little bit, and replaced by a little edge, so their edges don't quite meet at a vertex any more, they are slightly apart. The 12-point rooms are  

The regular 5-cell has only digon central planes containing a 2-point ({{radic|5}} 5-cell edge). Each great circle fiber has 6 disjoint great digons inscribed in it, which are edges in 6 disjoint 5-cells. 

The 12-point (Legendre polytope) occurs in the 120-cell as a 12-point vertex figure of two different sizes (truncation depths), an inverted 12-point (Jessen's icosahedron), the 12-point (truncated cuboctahedron). Each 12-point (truncated cuboctahedron) has a distinct 6-point (hemi-polyhedron) inscribed in it, which is the 12-point with pairs of its vertices identified. Each of the larger 12-point (truncated cuboctahedra) has 4 orthogonal 5-point (5-cell) faces, from 2 opposing pairs of orthogonal 5-cells. There are 5 such concentric 12-point (truncated cuboctahedra) around each vertex of the 120-cell. 12 of them, found at 12 120-cell vertices which form an icosahedron inscribed in the 3-sphere, combine to form an icosahedral compound with 12 such 5-cell faces, concentric with the center of the 120-cell, by contributing one face each to the compound. The 4 {{radic|5}} faces in each truncated cuboctahedron, and the 12 {{radic|5}} faces in the icosahedral compound they form, are disjoint from one another with {{radic|6}} long edges of truncated cuboctahedra connecting them. The 12 disjoint {{radic|5}} triangle faces of the compound belong to 12 disjoint regular 5-cells. The icosahedral compound they form is a 60-point ([[W:Truncated icosahedron|truncated icosahedron]]), with 12 {{radic|5}} triangle faces and 20 {{radic|6}} &lt;math&gt;\curlywedge&lt;/math&gt; {{radic|5}} hexagon faces. It lies isomorphic to 11 of the 120 60-point (rhombicosadodecahedron) cells, such that each {{radic|5}} pentagon face is located under a smaller rhombicosadodecahedron pentagon face (a 120-cell face), and there are 120 of these truncated icosahedra as there are 120 rhombicosadodecahedral cells.  The 11 rings of 11 rhombicosadodecahedron cells are 11 distributions of 11 rhombicosadodecahedra {{radic|5}} equidistant from each other (each {{radic|5}} distant from the other 10) on the 3-sphere.

== The regular 137-cell 4-polytope ==

(description originally written for the 11-cell, which applies only to a fibration of 11 11-cells)

It is quasi-regular with two kinds of cells, the 12-point (regular icosahedron) and the 5-point (regular 5-cell), which have just one kind of regular face between them. That quasi-regular 4-polytope is the 11-cell.

In 4 dimensions quasi-regular means not only that a 4-polytope has two kinds of regular facets (it is semi-regular with regular facets), it also means that its facets have expressions of two different dimensionalities because they are concentric in the 3-sphere. The 11-cell has two kinds of regular cells. One appears to be a 3-polytope building block (as is usual for cells of 4-polytopes), the 12-point (regular icosahedron), but since it is embedded at the center of the 3-sphere (unlike the cells of most 4-polytopes) it also acts as an inscribed 4-polytope.{{Efn|Which of three possible roles a 3-polytope embedded in 4-space plays depends on the point at which it is embedded. A 3-polytope inscribed in a 4-polytope concentric to their common center acts as an inscribed 4-polytope, even though it resembles a polyhedron. A 3-polytope concentric to an off-center point in the 4-polytope's interior acts as a polyhedral cell. A 3-polytope concentric to a point on the surface of a 4-polytope acts as a vertex figure. However, all 3-polytopes embedded in 4-space in any role express themselves both ways, as a spherical 3-polytope and as a flat 4-polytope. For example, any vertex figure can be viewed both as its spherical 3-polytope, and as a 4-pyramid with its flat 3-polytope as the base.|name=Embedding point}} The other cell appears to be a 4-polytope building block, the 5-point (regular 5-cell), which is almost unprecedented: 4-polytopes don't usually have other 4-polytopes as their cells, and we have seen the regular 5-cell nowhere else until now except in the 120-cell. This is less incongruous than it appears, since the regular 5-cell 4-polytope is also the regular decahedron 3-polytope, with half as many triangle faces as a regular icosahedron. The 11-cell is a honeycomb of mixed icosahedron and half-icosahedron cells. Like the icosahedra, the decahedra are inscribed as cells at the center of the 3-sphere, in the usual way 4-polytopes are inscribed as building blocks in other 4-polytopes, rather than in the way cells are usually embedded as building blocks, off-center in their 4-polytopes. All the 11-cell's cells are concentric, which indeed makes an unusual honeycomb of cells, an inside-out honeycomb. It is a proper 4D honeycomb however, with volumetrically disjoint 3D cells that meet each other in the proper way at shared 2D faces. The embedded 12-point (regular icosahedron) cell is an icosahedral pyramid (as a 4-polytope), which is filled by 20 4-point (irregular tetrahedra) that meet at its center; the center point of the 11-cell acts as a 12th vertex, just as the center point of the 24-cell acts as a 25th vertex in some honeycombs. The 5-point (regular 5-cell) cell contains five 4-point (regular tetrahedra) which meet the 12-point (regular icosahedra)'s irregular tetrahedral cells face-to-face, so the 11-cell is quasi-regular in the usual sense of having two varieties of regular 3-polytope cells, but additionally both varieties of of its cells are bi-dimensional: another clue that the 11-cell has something to do with dimensional analogy

....

The 11-cells (collectively) are a single fibration of the 120-cell, an isoclinic rotation in 60 pairs of completely orthogonal invariant planes of two different kinds. One invariant plane in each pair is a 5-point (great pentagon pentad), and its completely orthogonal plane is a 12-point (irregular great dodecagon hexad). Each rotating pair of invariant planes generates an 11-point (11-cell elevenad building block). The 60 pairs of planes can rotate through along either of two chiral pathways (left-handed and right-handed), so there are two rotations for each pair of planes...

....

The contraction of this 60-point (60-{{radic|5}}-edge truncated isocahedron), a compound of 12 5-point (regular 5-cells), into a 60-point (60-{{radic|5}}-chord rhombicosadodecahedron), a compound of 5 12-point (Legendre polytopes), occurs in such a way that the two 60-points have 55 {{radic|5}} (5-cell edges) in common which do not move in the contraction; they are the edges of an 11-cell. The 11-cell's 55 faces are those framed by these edges. .... There are 11 distinct ways this contraction relationship can occur, and all 11 of them occur in each 11-cell. 

The 11-cell is a compound of 11 distinct 60-point (rhombicosadodecahedra), and of 11 distinct 5-cells, and of  55 distinct 12-point (Legendre polytopes). Each 12-point is diminished by excluding one vertex, and inscribed at the 11 corresponding vertices of the 11-cell. The 11 vertices are each a unique conflation of 5 distinct 120-cell vertices (a pentad), the realization of which is a unique 120-cell pentagon face. There are 120 distinct ways this diminishment and compounding can occur to make an 11-cell. 11 of them occur in a 137-cell, all 120 of them occur in the 120-cell, and they occur nowhere else.

How can this regular 4-polytope possibly exist? By Schläfli's criterion {3, 3, ..} exists, but not as a finite polytope.{{Sfn|Coxeter|1973|loc=§7.7 Schläfli's criterion}} Coxeter found {3, 5, 3} as a finite hyperbolic honeycomb, but not in Euclidean 4-space.{{Sfn|Coxeter|1968|title=The Beauty of Geometry: Twelve Geometric Essays|loc=''Regular Honeycombs in Hyperbolic Space''|ps=; &quot;The honeycombs considered by the above authors [Schlegel, Schläfli, Klein, Sommerville] have finite cells and finite vertex figures. ... A further extension allows a cell or vertex figure to be a ''star''-polytope, so that the honeycomb covers the space several times. .... We shall find that there are four regular star-honeycombs in hyperbolic 4-space, as well as two infinite families of them in the hyperbolic plane.&quot;}} We find that {3, 3, ..} is a central expression of the &lt;math&gt;H_4&lt;/math&gt; root system, being the regular convex hull in 4 dimensions of 137 points, 96 pentads and 600 hexads. The minimal expression of &lt;math&gt;H_4&lt;/math&gt; is the regular convex hull {3, 3, 5} of 120 points, 600 tetrads and 75 hexads, and its maximal expression is the regular convex hull {5, 3, 3} of 600 points, 120 pentads and 675 hexads. {3, 3, ..} is a third regular &lt;math&gt;H_4&lt;/math&gt; polytope which fits between them. The properties and metrics of this regular sequence are presented in tabular form below in ''[[#Build with the blocks|§Build with the blocks]]''.

{|class=&quot;wikitable&quot;
!colspan=3|Regular &lt;math&gt;H_4&lt;/math&gt; polytopes
|-
|[[File:600-cell.gif|200px]]
|[[File:5-cell.gif|200px]]
|[[File:120-cell.gif|200px]]
|-
!120-point 600-cell
!137-point 137-cell
!600-point 120-cell
|}

=== Coordinates ===
....

=== As a configuration ===
....

=== Hopf map of the 137-cell ===
....

== The perfection of Fuller's cyclic design ==
[[File:Jessen's unit-inscribed-cube dimensions.png|thumb|400px|Jessen's icosahedron on the 2-sphere of diameter {{radic|5}} has an inscribed unit-cube. It has 4 cubic axes (not shown) through the equilateral face centers (the inscribed cube's vertices), 6 non-orthogonal {{radic|5}} long diameter axes, and 3 orthogonal parallel pairs of {{radic|4}} reflex edges, {{radic|1}} apart.]]

This section is not an historical digression,{{Efn|No doubt this entire essay is too discursive, and mathematically educated writers reach their findings more directly. I have told my story this way, still in a less halting and circuitous manner than it came to me, because it is important to me to show how I came by my understanding of these objects, and I am not a mathematician. I have been a child building with blocks, and my only guides have been the wiser children who built with the blocks before me, and told me how they did it; that, and my own nearly physical experience building with them, and imagination. I am at pains to show how that can be done, even by as mathematically illiterate a child as I am.|name=Apology}} but a deep dive to the heart of the matter, like Coxeter on Todd's perfect pentads. In this case the heart is found in the [[Kinematics of the cuboctahedron|kinematics of the cuboctahedron]],{{Sfn|Christie|2022|loc=''[[Kinematics of the cuboctahedron|Kinematics of the cuboctahedron]]''}} first described by [[W:Buckminster Fuller|Buckminster Fuller]].{{Sfn|Christie: On Fuller's use of language|2024|loc=''[[W:User:Dc.samizdat#Bucky Fuller and the languages of geometry|Bucky Fuller and the languages of geometry'']]}}

After inventing the rigid geodesic dome, Fuller studied a family of domes which have no continuous compression skeleton, but only disjoint rigid beams joined by tension cables. Fuller called these envelopes ''tension integrity structures'', because they possess independent tension and compression elements, but no elements which do both. One of the simplest [[w:Tensegrity|tensegrity]] structures is the [[w:Tensegrity#Tensegrity_icosahedra|tensegrity icosahedron]], first described by [[W:Kenneth Snelson|Kenneth Snelson]], a master student of Fuller's.{{Efn|Fuller failed to credit [[W:Kenneth Snelson|Snelson]] for the first ascent of the tensegrity icosahedron, a sad lapse for a great educator, as if Coxeter had not gracefully acknowleged Grünbaum. Snelson taught it to Fuller, his teacher, at a Black Mountain College summer session&lt;ref&gt;{{Citation|year=1949|title=R. Buckminster Fuller|publisher=Museum and Arts Center, 1948-1949|place=Black Mountain College|url=https://www.blackmountaincollege.org/buckminster-fuller}}&lt;/ref&gt; where Fuller taught the geodesic domes he had invented, and the nascent principles of tension integrity geodesics he was exploring. It would have burnished Fuller's own reputation to gratefully acknowledge his exceptionally quick student's discovery. No doubt Fuller was about to discover the tensegrity icosahedron himself, but Snelson saw it first.&lt;ref&gt;{{Citation|last=Snelson|first=Kenneth|author-link=W:Kenneth Snelson|publisher=Stanford University|title=Bucky Conversations: Conversations on the Life and Work of an Enigmatic Genius|year=2003|url=https://searchworks.stanford.edu/view/mf245gr4637|postscript=; Ken Snelson, at a symposium on Fuller's legacy, acknowledged that Fuller led him up to the tensegrity icosahedron. Snelson said that he then conceived it on his own, built the first physical model, and presented it to Fuller.}}&lt;/ref&gt;|name=Snelson and Fuller}}

A tensegrity icosahedron is an icosahedral geodesic dome whose 6 orthogonal reflex compression struts float gently in space, linked only by 24 tension cables which frame equilateral faces of the icosahedron, the whole 2-sphere expanding and contracting symmetrically with ''infinitesimal mobility'', a spring-like symmetrical motion leveraged from whatever tiny amount of elasticity remains in the steel struts and cables.

The polyhedron that is the basis for this flexible structure is the Jessen's icosahedron, that we found 5 of in Moxness's 60-point (rhombicosadodecahedron), called by [[w:Adrien_Douady|Douady]] the ''six-beaked shaddock'' because it resembles the fish whose normal affect is with their mouth 90° open, but a squarish shadfish with mouths on all six sides. At the limits, the gender neutral shad can open their six beaks all the way, until they become flat squares and they becomes a cuboctahedron, or they can shut them all tight like a turtle retracting into their octahedron shell. The six mouths always move in unison. This is [[Kinematics of the cuboctahedron#Jitterbug transformations|Fuller's ''jitterbug'' transformation]] of the 12-point (vector equilibrium'')'', his name for the unstable [[Kinematics of the cuboctahedron|kinematically flexing cuboctahedron]]. Fuller found that its always-symmetric transformation through 4 distinct forms of the same 12-vertex polyhedron was a closed cycle with two equilibrium points, one stable and the other unstable. The shad's normal 90° open visage is the stable point, the shape the [[Kinematics_of_the_cuboctahedron#Elastic-edge transformation|elastic tensegrity icosahedron]] rests in and strives to return to. The widest-open square-faced cuboctahedron is the unstable inflection point, where the shad gets to decide non-deterministically (that is, without being compelled one way or the other) whether or not to do their ''really'' odd trick -- where they flip their 6 jaws 90 degrees in their 6 faces and shut their 6 beaks on the opposite axis of their squares than the one they opened them on -- or whether they will just shut them all the same way again. Interestingly, the regular icosahedron is one of the shad's guises too, just slightly more gaping than their normal visage. Fuller made a meal of the shad, finding all the insightful things to say about the kinematics of the only fish who can make their edge length exactly the same size as their radius, when they open their mouths all the way. Fuller built physical models of the 12-point (vector equilibrium), and even gave demonstrations to audiences of the flexible shad, opening and closing their mouths in spherical synchrony, their 4 pairs of opposing equilateral triangles spiraling toward and away from each other in parallel, always opposed like the two triangles inscribed in a hexagon, counter-rotating like dual [[W:Propellor|tri-propellors]] as they dance toward each other until their edges meet in an octahedron (the hexad), then backing away again (still rotating in the same directions), overlaid with Fuller's own deep commentary in physical language anyone can understand. Bucky flew the shad through the inflection points in its [[W:Spinor|spinor]] orbit, explaining its [[W:Möbius_loop|Möbius loop]] with vivid apt similes like trimming a submarine's ballast tanks, stalling an airplane at apogee, and nature's abhorrence of the unstable equilibrium point.{{Sfn|Fuller|1975|ps=; In this film Fuller carefully folds a model of the cuboctahedron made of rigid struts with flexible joints through the entire transformation cycle; he also shows how a rigid regular icosahedron can be rotated inside an inscribing &quot;vector edge cube&quot; (a cube with an octahedron inscribed in it), keeping the 12 vertices on the surface of the cube (and on the edges of the octahedron inscribed in the cube) at all times.}}

The Jessen's unique element set is its 6 long reflex edges, which occur in 3 parallel opposing pairs. Each pair lies in its own central plane, and the 3 central planes are the orthogonal central planes of the octahedron, the orthonormal (x,y), (y,z), and (x,z) planes of a Cartesian basis frame. The 6 reflex edges are all disjoint from one another, but each pair of them forms a merely conceptual great rectangle with the pair of invisible exterior chords that lies in the same central plane. These three great rectangles are storied elements in topology, the [[w:Borromean_rings|Borromean rings]]. They are three chain links that pass through each other and would not be separated even if all the other cables in the tensegrity icosahedron were cut; it would fall flat but not apart, provided of course that it had those 6 invisible exterior chords as still uncut cables.

[[File:Hemi-icosahedron √2 radius dimensions.png|thumb|400px|The hemi-icosahedral cell of the 11-cell is a compound of 5 of this Jessen's icosahedron embedded in &lt;math&gt;R^4&lt;/math&gt;, shown here on a {{radic|2}}-radius 3-sphere with {{radic|5}} reflex edges. It has an inscribed 8-point ({{radic|3}} cubic). Beware: these are not the dimensions of the 3-polytope Jessen's in Euclidean space &lt;math&gt;R^3&lt;/math&gt;, they are the metrics of the Jessen's embedded in spherical 3-space &lt;math&gt;S^3&lt;/math&gt;. Even more misleading, this is a not a normal orthogonal projection of that object, it is the object inverted. For example, the chord labeled {{radic|3}} is actually the diagonal of a unit cube, not its edge.]]
We have illustrated the 2-sphere Jessen's with {{radic|5}} diameter, and the 3-sphere Jessen's with {{radic|2}} radius, because that is how their metrics correspond. (Not coincidentally, the edge length of the unit-radius 4-simplex is {{radic|5}} ∕ {{radic|2}}.) But in the Jessen's this correspondence is curiously cyclic; in the embedding into 4-space the characteristic root factors seem to have moved around. In particular, the {{radic|5}} chord has moved to the former {{radic|4}} chord, because which axes of symmetry are the long diameters has changed.

We might have expected to find the 6-point (hemi-icosahedron)'s 5-cell triangular faces identified with the Jessen's 8 equilateral triangle faces somehow, but they are not the same size, so that is not the way the two polytopes are identified. The {{radic|5}} reflex edges of the 12-point (Jessen's) are the 6-point (hemi-icosahedron) edges. An 11-cell face (a 5-cell face) has its three {{radic|5}} edges in three different Jessen's icosahedra in three different hemi-icosahedra, and like most faces, it does not lie in a central plane. Astonishingly however, the Jessen's 8 equilateral {{radic|6}} faces ''are'' great triangles lying in central planes, in this 4-space embedding of the Jessen's polyhedron. It is very hard to look at our illustration of the Jessen's embedded in the 120-cell (where it occurs 600 times), and visualize that its 8 regular ''faces'' lie in 8 different ''central'' planes, but they do. In its 3-sphere embedding, the Jessen's is inside-out, and the {{radic|6}} triangles are not really cell faces at all, but central polygons of the cell. Opposing {{radic|6}} triangles lie in completely orthogonal central planes, where they are inscribed in great hexagons (but not in the same great hexagon, in two orthogonal great hexagons). The opposing [[W:Clifford parallel|Clifford parallel]] planes are parallel ''and'' orthogonal to each other, dimensionally analogous to the opposing edges of a tetrahedron.{{Efn|Clifford parallel planes, whether orthogonal or not, are ''isoclinic'', which means that a double rotation in two completely orthogonal invariant planes by the same angle will bring them together. The [[W:SO(4)|special orthogonal group SO(4)]]{{Sfn|Mebius|2015|loc=''[[W:SO(4)|SO(4)]]''}} of [[W:SO(4)#Isoclinic rotations|isoclinic rotations in 4-dimensional space]] is what generates the regular convex 4-polytopes and relates them to each other. Fortunately, we do not quite need to explicate SO(4) in this essay in order to make sense of the 11-cells, since at root they are an expression of a different symmetry which is not a rotation, the pyritohedral symmetry of discrete dimensional analogies.|name=Isoclinic}} The two planes do intersect, but only at one point, the center of the 3-sphere.{{Efn|name=Six orthogonal planes of the Cartesian basis}} Our illustration, though accurate, is profoundly misleading because it is an inverted projection, not the usual orthogonal projection. The actual 11-cell faces are {{radic|5}} faces which do not appear in this illustration at all. The Jessen's (the building block we find 5 of in the rhombiscosadodecahedral cell of the 11-cell) is a section centered on a vertex of the 120-cell, so it is not itself a cell but a vertex figure, like the 120 icosahedra of the 600-cell. That is why we find 600 Jessen's, of course. The inside-out Jessen's is the central apex and radii of a cell without the cell's faces, and now we should be able to illustrate the entire vertex cell properly by drawing it with both its faces and its radii, in their proper right-side-out relation to the single 120-cell vertex at its center. 

Before we do, consider where the 12 vertices of that 12-point (vertex figure) lie in the 120-cell. We shall figure out exactly what kind of right-side-out 3-polytope they describe below, but for the present let us continue to think of them as the 12-point (hexad of 6 orthogonal reflex edges forming a Jessen's icosahedron), and consider their situation in the 120-cell. Those 3 pairs of parallel edges lie a bit uncertainly, infinitesimally mobile and behaving like a tensegrity icosahedron, so we can wiggle two of them a bit and make the whole Jessen's icosahedron expand or contract a bit. Expansion and contraction are Stott's operators of dimensional analogy, so that is a clue to their situation. Another is that they lie in 3 orthogonal planes embedded in 4-space, so somewhere there must be a 4th plane orthogonal to them, in fact more than just one more orthogonal plane, since 6 orthogonal planes (not just 4) intersect at the center of the 3-sphere. The hexad's situation is that it lies orthogonal to another hexad, and its 3 orthogonal planes (xy, yz, zx) lie Clifford parallel to its orthogonal hexad's planes (wz, wx, wy).{{Efn|name=Six orthogonal planes of the Cartesian basis}} There is a 24-point (hexad) here, and we know what kind of right-side-out 4-polytope that is. The 24-point (24-cell) has 4 Hopf fibrations of 4 great circle fibers, each fiber a great hexagon, so it is a complex of 16 great hexads, generally not orthogonal to each other, but containing at least one subset of 6 orthogonal great hexagons, because each of the 6 [[w:Borromean_rings|Borromean link]] great rectangles in our pair of Jessen's hexads must be inscribed in one of those 6 great hexagons. 

If we look again at a single Jessen's hexad, without considering its orthogonal twin, we see that it has 3 orthogonal axes, each the rotation axis of a plane of rotation that one of its Borromean rectangles lies in. Because this 12-point (tensegrity icosahedron) lies in 4-space it also has a 4th axis, and by symmetry that axis too must be orthogonal to 4 vertices in the shape of a Borromean rectangle: 4 additional vertices. We see that the 12-point (Jessen's vertex figure) is part of a 16-point (8-cell 4-polytope) containing 4 orthogonal Borromean rings (not just 3), which should not be surprising since we already knew it was part of a 24-point (24-cell 4-polytope), which contains 3 16-point (8-cells). In the 120-cell the 8-point (cube) shown inscribed in the Jessen's illustration is one of 8 concentric cubes rotated with respect to each other, the 8 cubes of a 16-point (8-cell tesseract). Each 12-point (6-reflex-edge Jessen's) is 8 Jessens' rotated with respect to each other, [[W:Snub 24-cell|96 disjoint]] {{radic|6}} reflex edges. We find these tensegrity icosahedron struts in the 24-point (24-cell) as well, which has 96 {{radic|6}} chords linking every other vertex under its 96 {{radic|2}} edges. From this we learned that the hexad's situation is that it occurs in bundles of 8, close together in the sense of being concentric rotations of each other. 

....

Augmentation-diminishment and expansion-contraction are entirely different operation pairs. Augmentation and diminishment just add or remove a vertex without moving any of the other vertices. Expansion and contraction add or remove a vertex and move all the other vertices, to keep them uniformly distributed on the surface of the 3-sphere. Augmentation and diminishment commute; you can lop off several vertices all at once, or one at a time in any order, and the results are the same. The expansion-contraction operators do not commute, generally. If you remove several vertices by contraction, the choice of vertices and the order in which you do it usually matters. Except in special cases, the result produces a left-handed or right-handed chiral polytope, depending on how you pair the vertices you remove.

....
...through the duality of the quasi-regular 4-polytopes (11-cell and 57-cell)....

...and the jitterbug contraction-expansion relation through the 4-polytope sequence...

...to dimensional analogy and a distinct sample rotation that has a dimensional relativity to it...

This nondescript abstract 60-point 4-polytope...
(This picture is of the corresponding even-more-abstract 60-point 3-polytope)...
Actually contains all 600 vertices of the 120 cell which conflates 10 into one...
They are the vertices of two orthogonal regular five cells...
The vertices of the hexad cell are the 12 vertices of an icosahedron, a section of the 120 cell...

The sport of making an 11-cell is a matter of parabolic orbits. It's golf.
&lt;blockquote&gt;&quot;A regulation golf course usually consists of 18 holes of varied length. There are generally four short holes, 130 to 200 yards (par 3); ten average holes 350 to 400 yards (par 4); and four long holes 450 to 550 yards (par 5).&quot;
&lt;/blockquote&gt;

...most fibrations exist only as potentials, all fibrations being equivalent... and all the cells being identical, picking out a cell ring is arbitrary, they all exist all the time... except when the object is actually rotating... then one distinct fibration is real, and all the others are mere abstractions... the 11-cells are the same, but opposite... they exist in any rotation, the same fibration no matter what the rotation is... they are never abstract except in the one special case when there is no rotation at all.

...to the relativity of rotations generally... how the sun and stars either revolve over us and/or we rotate under them but not both in any distinct place and time which is always some distinct measurable double rotation which is the distinguished (proper) rotation ''for that object'' no matter who is looking at it -- no other object viewing it has a different, and as proper, a view of it -- though it may ''appear'' to be another rotation from another reference frame, which may ascribe a different rotation to it from to its own point of view, that is not the distingushed (proper) rotation of the object, it's own view is -- from its own viewpoint an object can always tell if it is rotating or not, and how, and can tell for certain that it is a satellite of some larger object -- so can the larger object -- their viewpoints correllate, and they reach the same conclusion re: who is going around who 

...how that's what a double rotation is, it's flying through a chiral hose in 4-space, and the hose only exists in that shape when the whole space is rotating at once, though you can make it exist for you by looking at it from your proper reference frame instead if from inside it's proper reference frame...
...spherical space S4 is just universal space R4 seen from a particular central point (an arbitrarily chosen point except to the observers there)...
...the space has a different geometric subdivision (we find different equivalent objects in it) when it is rotating this way versus that way... so in a non-isoclinic double rotation a 120-cell contains a mix of dodecahedral and tetrahedral cell rings! ...and that is what the quasi-regular dual 4-polytope (really singular) with its mix of cell types represents....

....
The most striking congruence of this dimensional analogy is what ''doesn't'' move, even if the 120-cell were to actually flex. One important invariant of the [[Kinematics_of_the_cuboctahedron#Duality_of_the_rigid-edge_and_elastic-edge_transformations|tensegrity icosahedron transformation]] is that the 3 orthogonal planes are not displaced at all, throughout the cycle. The 3 pairs of 2 parallel reflex-edge struts move toward and away from each other, but all 6 stay in their respective orthogonal planes throughout, which do not go anywhere, analogous to the way all the points in completely orthogonal invariant rotation planes stay in their rotating invariant planes during an isoclinic rotation in 4-space. If the 120 60-{{radic|5}}-edged 60-point (truncated icosahedra) were to actually shrink into the 120 60-{{radic|5}}-chorded 60-point (rhombicosadodecahedra), as if a monster 4-dimensional shadfish the 600-point (Shad) had bit down with his 120 10-jawed mouths, even then 55 of the 60 {{radic|5}} (5-cell edges) in each 60-point would not move, because they are already congruent, and remain so throughout all stages of the jitterbug transformation cycle.

The jitterbug dimensional analogy is complete, and includes 4-polytope analogues of all the kinematic 3-polytopes (flexings of the cuboctahedron). The largest wide-open-mouthed phase, the unstable inflection point in the cycle, is the 120 constructs of 12 5-point (regular 5-cells) the 60-point (truncated icosahedral pyramids), which are also a construct of 25 24-point (24-cells the cuboctahedrons' analogues). 120 compounds of 5 12-point (regular icosahedral pyramid vertex figures of the 600-cell) come next. Then the stable equilibrium point: 120 compounds of 5 12-point (Jessen's icosahedral pyramid vertex figures). After them follows contraction to compounds of (expanded octahederon 16-cell) building blocks. 

...irregular 5-point (pentad 4-orthoscheme)&lt;math&gt;^4&lt;/math&gt; building blocks....

As usual, Coxeter was the first to describe this transformation cycle in the 3-polytopes, and then at the same time the first to describe it in the 4-polytopes,{{Sfn|Coxeter|1973|pp=50-52|loc=§3.7: Coordinates for the vertices of the regular and quasi-regular solids|ps=; describes the [[Kinematics of the cuboctahedron|cuboctahedron transformation in Euclidean 3-space]]; separately (pp=150-152, §8 Truncations) Coxeter also describes its 4-dimensional analogue, in which the cuboctahedron transforms in the curved 3-space of the [[W:3-sphere|3-sphere]] embedded in [[W:Four-dimensional space|Euclidean 4-space]]; it includes a further stage of the cuboctahedron transformation (which is not reached in the tensegrity icosahedron transformations): from the octahedron the vertices continue moving along the same helical paths, separating again into the 12 vertices of the [[W:Regular icosahedron#Symmetries|snub octahedron]], a smaller regular icosahedron nested inside the octahedron.}} though others have also studied the cycle in 4-space more recently.{{Sfn|Itoh &amp; Nara|2021|loc=Abstract|ps=; &quot;This article addresses the [[24-cell|24-cell]] and gives a continuous flattening motion for its 2-skeleton [the cuboctahedron], which is related to the Jitterbug by Buckminster Fuller.&quot;}} He noticed the presence of the quasi-regular 96-point (snub 24-cell) in the 4-space version of cycle, and also something more mysterious: the 4-space cycle continues to contract beyond the octahedral stage into a second, smaller regular icosahedron (snub octahedron). Coxeter did not say if the cycle ever reaches the simplex (regular 5-cell), or if the smallest 4-polytope (5-cell) and the largest (120-cell) being adjacent to each other ''on both sides'' at that point in the short-circuited loop would imply that the cycle could repeat with objects of a different size the next time around. But Fuller did include the tetrahedron in the contraction cycle after the octahedron; he saw deeper than the [[Kinematics of the cuboctahedron#Cyclical cuboctahedron transformations|orientable double cover of the octahedron]] (which does not go there), and folded his vector equilibrium down finally into the quadruple cover of the tetrahedron, with four cuboctahedron edges coincident at each edge. Fuller's cyclic design must be a cycle (in 4-space at least) which loops through the simplex, and finds objects of an adjacent size on each iteration. It is interesting to consider that in such a cycle the 24-point (24-cell) shad must make another non-deterministic decision whether to get larger or smaller (or perhaps to stay the same size) at the unstable inflection point. 

We should not expect to find a family of such cycles, one in each dimension, since the 24-cell is its unstable inflection point, and as Coxeter observed at the very end of ''Regular Polytopes'', the 24-cell is the unique thing about 4-space, having no analogue above or below. But perhaps Fuller's cyclic design is also trans-dimensional, a single cycle that loops through all dimensions, with the 4-dimensional 24-point (24-cell) as its unstable center, and the ''n''-simplexes at its stable minimums, and the shad has a third decision to make, whether to go up or down a dimension (or stay in the same space). Fuller himself saw only the shadow of the shad in 3-space, the 12-point (cuboctahedron shadowfish), not its operation in 4-space, or the 24-point (24-cell shadfish) which is the real vector equilibrium, of which the 12-point (cuboctahedron) is only a lossy abstraction, its Hopf map. So Fuller's cyclic design is trans-dimensional, at least between dimensions 3 and 4. Some dimensional analogies are realized in only a few dimensions, like the case of the [[w:Demihypercube|demihypercube]]&lt;nowiki/&gt;s, but perhaps any correct dimensional analogy is fully trans-dimensional in the sense that at least an abstract polytope can be found for it in every dimension.

== Legendre's 12-point hexad binds the compounds together ==
[[File:Distances between double cube corners.svg|thumb|Distances between vertices of a 12-point double unit cube are square roots of the first six natural numbers due to the Pythagorean theorem. {{radic|7}} is not possible due to [[W:Legendre's three-square theorem|Legendre's three-square theorem]].]]
Fuller's perfectly realized 12-point tensegrity icosahedron and Coxeter's abstracted 6-point hemi-icosahedron are the same polytope, Jessen's curious icosahedron with its three orthogonal pairs of reflex edges, [[W:Jessen's icosahedron|Douady's six-beaked shaddock]] with its mouths normally open exactly 90 degrees. Its cell-complement (vertex figure) is the 12-point (truncated cuboctahedron). Its contraction by removal of one point is the 11-point (11-cell). Constructed exclusively from [[W:Legendre's three-square theorem|Legendre's distinguished roots]] of the first 6 natural numbers, this 2-hexad expresses the [[W:Pyritohedral symmetry|pyritohedral]] symmetry. 

The 12-point (Legendre hexad) is embedded in Euclidean 4-space to remarkable effect. In 3-space [[Kinematics of the cuboctahedron|it flexes kinematically]] to capture any of the regular polyhedra, since they are all related by its pyritohedral symmetry operations of expansion and contraction.{{Efn|name=Concentric polyhedra}} In 4-space it does something simpler, but more astonishing, with its pyritohedral symmetry: it lies in each of the regular 4-polytopes ''at the same time'', without flexing at all, and binds them together. 

=== ''n''-simplexes ===
....

The regular 5-cell has only digon central planes (intersecting two vertices, a 5-cell edge). The 120-cell with 120 inscribed 5-cells contains great rectangles whose longer edges are these 1200 5-cell edges. They occur in the 120-cell's 200 irregular great dodecagon central planes, which also contain the 120-cell's own 1200 edges (not shown), its 400 great hexagons (not shown), and its 800 great triangles (one shown). The 5-cell's edges but not its faces lie in these 200 central planes, which occur as the 3 orthogonal rectangular central planes of 600 Jessen's icosahedra vertex figures. Each disjoint {{radic|5}} 5-cell face triangle (each 11-cell face) spans three Jessen's icosahedra, one edge in each. Equilateral {{radic|6}} Jessen's &quot;face&quot; triangles are actually great triangles, inscribed two in a great hexagon, which lies in a central plane of another Jessen's icosahedron....

....do opposing faces of a Jessen's lie in completely orthogonal planes...or do they lie in opposing pairs in the same plane of some other Jessen's...or both?

...(The hexagons lie two in a dodecagon central plane of 3 other Jessen's icosahedra.)

The 120-cell contains 600 inscribed instances of the 12-point (Jessen's), and 1200 disjoint {{radic|5}} equilateral triangle faces of its 120 disjoint regular 5-cells. A pair of opposing 5-cells occupies each cubic axis of the 12-point (Jessen's), the four axes running through the equilateral face centers, and the pair of 5-cells contributes a pair of opposing {{radic|5}} hemi-icosahedron faces. The pair of opposing {{radic|6}} triangular Jessens' &quot;faces&quot; that also occupy the axis are actually opposing great triangles in the central plane orthogonal to the axis.

...another non-convex regular octahedron like the hemi-cuboctahedron is the [[w:Schönhardt_polyhedron|Schönhardt polyhedron]]....

=== ''n''-orthoplexes ===
....
[[File:Great 5-cell √5 digons rectangle.png|thumb|{{radic|5}} x {{radic|3}} rectangles found in 200 central planes (3 rectangles in each plane) of the 600-point (120-cell) of radius {{radic|2}}, and in its 600 inscribed Jessen's icosahedra (3 orthogonal planes in each Jessen's). Each plane intersects {12} vertices in an irregular great dodecagon which contains 4 {{radic|6}} great triangles inscribed in 2 {{radic|2}} ''regular'' great hexagons. These are the same 200 dodecagon central planes illustrated above, which also contain 6 120-cell edges and two opposing ''irregular'' great hexagons (truncated {{radic|6}} triangles). The 1200 {{radic|5}} 5-cell face triangles occupy 1200 other, non-central planes.]]

...the chopstick geometry of two parallel sticks that grasp...the Borromean rings...
&lt;blockquote&gt;Geometrically, the Borromean rings may be realized by linked [[W:ellipse|ellipse]]s, or (using the vertices of a regular [[W:icosahedron|icosahedron]]) by linked [[W:golden rectangle|golden rectangle]]s. It is impossible to realize them using circles in three-dimensional space, but it has been conjectured that they may be realized by copies of any non-circular simple closed curve in space. In [[W:knot theory|knot theory]], the Borromean rings can be proved to be linked by counting their [[W:Fox n-coloring|Fox {{mvar|n}}-colorings]]. As links, they are [[W:Brunnian link|Brunnian]], [[W:alternating link|alternating]], [[W:algebraic link|algebraic]], and [[W:hyperbolic link|hyperbolic]]. In [[W:arithmetic topology|arithmetic topology]], certain triples of [[W:prime number|prime number]]s have analogous linking properties to the Borromean rings.&lt;/blockquote&gt;

...600 vertex icosahedra...

The vertex figure of the 120-point (600-cell) is the icosahedron, so the 600-point (120-cell) contains 600 [[600-cell#Icosahedra|instances of the regular icosahedron]] in its curved 3-space envelope, one surrounding each vertex. In that curved space each icosahedron is exactly filled by 20 regular tetrahedra, one behind each face, that meet at the vertex. This is not true of regular icosahedra in flat Euclidean 3-space! The 6-point (hemi-icosahedron) in the 11-cell represents parts of at least 5 such curved, regular-tetrahedra-filled icosahedra, since each of its 6 vertices ''identifies'' five icosahedral vertices. That means it picks them out, wherever they are in the 120-cell (no two are the same vertex at the center of the same regular icosahedron), and represents all five as if they were coincident.

[[File:Jessen's icosahedron with dimensions.png|thumb|400px|[[W:Jessen's icosahedron|Jessen's icosahedron]], known to Buckminster Fuller as the [[W:tensegrity icosahedron|tensegrity icosahedron]], on a 2-sphere of radius {{radic|5}}. All dihedral angles are 90°. The polyhedron is a construct of the lengths {{radic|1}} {{radic|2}} {{radic|3}} {{radic|4}} {{radic|5}} {{radic|6}} of [[W:Legendre's three-square theorem|Legendre's three-square theorem]] and the angles {{sfrac|𝝅|2}} {{sfrac|𝝅|3}} {{sfrac|𝝅|4}}. It is related to the regular icosahedron, octahedron and cuboctahedron by a [[Kinematics of the cuboctahedron|cyclic pyritohedral symmetry transformation]] that is not a rotation.]]
Each hemi-icosahedron in the 120-cell is real, and has a distinct object that is its concrete realization somewhere in the 120-cell, which we have seen is a 60-point (Moxness's rhombicosadodecahedron). In fact each abstract 6-point (hemi-icosahedron) represents various distinct real icosahedra in the 120-cell, but not any one of the 600 distinct regular icosahedra surrounding the 600 vertices. Its 6 conflated vertices are the 12 vertices of distinct Jessen's icosahedra. The [[W:Jessen's icosahedron|Jessen's icosahedron]] is a non-convex polyhedron with 6 reflex edges. [[W:Adrien Douady|Adrien Douady]] was the first to study this polyhedron as one of a family of strange fish; in its normal form, it has concave faces with a dihedral angle of 90°.

...

Each 11-cell is a real object that consists of 11 sets of 5 identified vertices (Todd's perfect pentads), each of which pentads we can identify as an actual 5-point object, ''not'' a 5-cell this time, but a 120-cell face pentagon. Each 11-cell also conflates the 55 vertices of 11 disjoint 5-cells that we can identify into its own 11 vertices, 5 vertices into 1, forming its 11 non-disjoint 6-point (hemi-icosahedron) cells. Each 6-point (hemi-icosahedron) in the 11-cell conflates the 60 vertices of the 5 disjoint 12-point (Jessen's icosahedra) contained in one distinct 60-point (Moxness's Hull #8) illustrated above, 10 vertices into 1 (identified with 10 disjoint 5-cells), by conflating 12 pentagons (6 completely orthogonal pairs of 120-cell face pentagons) into 6 vertices.

The Jessen's icosahedra are isomorphic to the 5-cell's 10 {{radic|5}} edges at their own {{radic|5}} reflex edges. There are 1200 of these {{radic|5}} edges in the 120-cell, and they form disjoint triangles. Each triangle is a face of a completely disjoint 5-cell (there is only room for one 5-cell around each reflex edge). Each is also a face of an 11-cell. Each is also a triangle with one {{radic|5}} edge in 3 distinct Jessen's icosahedra, in three distinct hemi-icosahedra, in three distinct 11-cells. Each Jessen's has 6 reflex edges. It turns out there are three ways to assemble 200 Jessens' out of 1200 5-cell edges, and the 120-cell contains 200 disjoint, but 600 distinct, Jessen's icosahedra.

Previously we saw how each of the 11 5-cells in an 11-cell shares just one of its 10 faces with the 11-cell, and its other 9 faces and 6 edges must be interior parts of the 11-cell (not part of its element set, which includes only its surface elements). By symmetry we should expect each 5-cell to share all 10 of its edges and faces with some 11-cell in exactly the same way, and now we see that it does that by identifying one edge in each of 10 different 11-cells, and each face with 3 of them.

... ...

...let us consider the 4-orthoplex as another 4-polytope which, like the 5-cell, occurs a distinct number of times in the 11-cell. The hemi-icosahedron is a [[W:Hopf fibration|Hopf map]] of the 11-cell, which means that each vertex of the hemi-icosahedral map lifts to a great circle of the 11-cell, in this case a great circle intersecting 4 vertices: a great circle square of edge length {{radic|4}}, radius {{radic|2}}. The [[16--cell|16-cell]] is the 8-point regular 4-orthoplex polytope which consists of 6 orthogonal great circle squares. That is, it consists of the 4 Cartesian orthogonal axes with vertices at +{{radic|2}} and −{{radic|2}} on each axis, joined by edges of length {{radic|4}}, and spanned by 6 (4 choose 2) mutually orthogonal central planes. 

The 16-cell and the hemi-icosahedron have a number in common, and that is they both have 6 of something (and this all began with hexads). In the 11-cell, each hemi-icosahedron is identified with a distinct 16-cell as its natural Cartesian basis frame, with the hemi-icosahedron's 6 vertices occurring as only 6 of the 16-cell's 8 vertices (and not as an octahedron), with each vertex lying in just one of [[16--cell#Coordinates|the 16-cell's 6 orthogonal central planes]]. The 16-cell does contain 4 orthogonal 6-point octahedra as central hyperplanes, but the hexad is not one of the octahedra; it is a symmetrical distribution of 6 points on the 3-sphere, one in each orthogonal central plane (which is what the 16-cell has 6 of). The octahedra's significance with respect to 11-cells is that Coxeter &amp; Petrie showed in 1938 that an octahedron can be circumscribed about an icosahedron 5 different ways, which will help us to see in another way how the five-fold conflation of 11-cell vertices arises.{{Efn|name=Concentric polyhedra}} A significant thing about the 6 orthogonal central planes is that not all of them intersect in an axis the way central planes do in three dimensions. Most pairs of them do, but there are 3 pairs of ''completely'' orthogonol planes that do not share an axis. In addition to being perpendicular, these pairs are also parallel (in the [[W:Clifford parallel|Clifford]] sense) and opposing, in a manner dimensionally analogous to the opposite edges of a tetrahedron.

In the 16-cell, each of the 3 pairs of completely orthogonal great circle squares is a [[16--cell#Rotations|discrete Hopf fibration covering all 8 vertices]], which is to say that the 16-cell has as a characteristic symmetry 3 distinct left (and right) isoclinic rotations in a pair of invariant great square planes. In the 11-cell, 11 non-intersecting great circle monogons are a discrete Hopf fibration covering all 11 vertices, and there are 11 such fibrations, each cell contributing one vertex to each fibration.

....

....The two triangles around the reflex edge are {{radic|5}},{{radic|3}}, {{radic|3}} isosceles triangles....

.... Each pair is a 120-cell axis lying in one of the 6 orthogonal central planes of the hemi-icosahedron's identified 16-cell, but each 11-cell contains only 55 of the 60 axes of the 120-cell....

....pyritohedral symmetry group....

....

=== ''n''-taliesins ===
The 4-simplex pentad building block combines with the 4-orthoplex hexad building block to make the 4-taliesin heptad building block. The heptad is the essential building block of the ''taliesin'' family of polytopes. The 11-cell and 57-cell hemi-polytopes and the 137-cell and 120-cell regular 4-polytopes are nuclear members of this trans-dimensional family. The 24-cell and 600-cell are extended members of it, because they also contain instances of the heptad building block.

'''Taliesin''' ({{IPAc-en|ˌ|t|æ|l|ˈ|j|ɛ|s|ᵻ|n}} ''tal YES in'')

* [[W:Celtic nations|Celtic]] for ''[[W:Taliesin (studio)#Etymology|shining brow]]''.
* An early [[W:Taliesin|Celtic poet]]  whose work has possibly survived in a [[W:Middle Welsh|Middle Welsh]] manuscript, the ''[[W:Book of Taliesin|Book of Taliesin]]''.
* A [[W:Taliesin (studio)|house and studio]] designed by [[W:Frank Lloyd Wright|Frank Lloyd Wright]].
* Wright's fellowship of architecture, or [[W:Taliesin West|one of its headquarters]].
* The architectural principle that if you build a house on the top of a hill, you destroy its spherical symmetry, but there is a circle just below the top of the hill that is the proper site for a rectilinear structure, its shining brow.
* [[W:Taliesin Myrddin Namkai-Meche|Taliesin Myrddin Namkai-Meche]], a young Reed College scholar who was stabbed to death by a crazy man in 2017 on a Portland, Oregon light rail train, when he rose to the nonviolent defense of two black teenagers who were being violently assaulted in racist and anti-Muslim language by a right-wing extremist.
* The [[W:Symmetry group|symmetry]] relationship expressed by the mapping between a geometric object in [[W:n-sphere|spherical space]] and the dimensionally analogous object in orthogonal [[W:Euclidean space|Euclidean space]] obtained by an expansion or contraction operation, such as by removing one point from the sphere in [[W:stereographic projection|stereographic projection]].
* '''Taliesan polytope''', an (''n+1)''-polytope whose ''n''-polytope dimensional analogue inscribed concentrically in the ''n''-sphere forms a convex honeycomb of itself which is the (''n+1)''-polytope.{{Efn|name=Embedding point}}
* The taliesan polytope's honeycomb, a Hopf fibration of its great circles.
* The ''n''-dimensional family of dimensionally analogous polytopes related to a taliesan polytope by expansion and contraction operations.
* The characteristic [[w:Orthoscheme|orthoscheme]] of a teliesan polytope, an irregular simplex that is the complete gene for the entire expression of trans-dimensional symmetry relationships among the polytopes of the ''n''-dimensional family, as well as the complete gene for the teliesan polytope's rotational symmetry.{{Efn|name=Characteristic orthoscheme}}

=== ''n''-cubics ===
Two 8-point 16-cells compounded form the 16-point (8-cell 4-cubic), but this construction is unique to 4-space....

=== ''n''-equilaterals ===
[[File:24-cell vertex geometry.png|thumb|Vertex geometry of the radially equilateral{{Efn|The long radius (center to vertex) of the 24-cell is equal to its edge length; thus its long diameter (vertex to opposite vertex) is 2 edge lengths. Only a few uniform polytopes have this property, including the four-dimensional 24-cell and [[W:Tesseract#Radial equilateral symmetry|tesseract]], the three-dimensional [[W:Cuboctahedron#Radial equilateral symmetry|cuboctahedron]], and the two-dimensional [[W:Hexagon#Regular hexagon|hexagon]]. (The cuboctahedron is the equatorial cross section of the 24-cell, and the hexagon is the equatorial cross section of the cuboctahedron.) '''Radially equilateral''' polytopes are those which can be constructed, with their long radii, from equilateral triangles which meet at the center of the polytope, each contributing two radii and an edge.|name=radially equilateral|group=}} 24-cell, showing the 3 great circle polygons and the 4 vertex-to-vertex chord lengths.|alt=]]
The 24 vertices of the 24-cell are distributed at four different [[#n-leviathon|chord]] lengths from each other. In the {{radic|2}}-radius polytopes we have using as examples in this paper they are {{radic|2}}, {{radic|4}}, {{radic|6}} and {{radic|8}}, but in a ''unit-radius'' polytope they are: {{sqrt|1}}, {{sqrt|2}},  {{sqrt|3}} and {{sqrt|4}}.

Each vertex is joined to 8 others by an edge of length 1, spanning 60° = &lt;small&gt;{{sfrac|{{pi}}|3}}&lt;/small&gt; of arc. Next nearest are 6 vertices located 90° = &lt;small&gt;{{sfrac|{{pi}}|2}}&lt;/small&gt; away, along an interior chord of length {{sqrt|2}}. Another 8 vertices lie 120° = &lt;small&gt;{{sfrac|2{{pi}}|3}}&lt;/small&gt; away, along an interior chord of length {{sqrt|3}}. The opposite vertex is 180° = &lt;small&gt;{{pi}}&lt;/small&gt; away along a diameter of length 2. Finally, as the 24-cell is radially equilateral, its center is 1 edge length away from all vertices.

A radially equilateral uniform polytope exists only in Euclidean 0-space, 2-space, and 4-space. In 2-space there is the hexagon. In 3-space there is the cuboctahedron (Fuller's vector equilibrium), but of course it is not quite regular. Uniquely, in 4-space there are two, the 16-point ([[W:8-cell#Radial equilateral symmetry|8-cell 4-cubic]]) and the 24-cell, both regular. The 24-cell is an analogue of the cuboctahedron, but regular.

Three 16-cells compounded form a 24-point 4-polytope, the [[24-cell|24-cell]]. In addition to its three disjoint 16-cells, the 24-cell also contains three distinct but ''not'' disjoint 16-point (8-cell 4-cubics), which share 16-cells. The 24-cell is [[24-cell#Geometry|a full-course meal]] by itself, and we will not begin to consume it here, but only mention its relationship to the 11-cells. Like the 5-cell and the 11-cell the 24-cell has triangular faces, but they are smaller triangles, of edge length {{radic|2}}, equal to the radius. The 24-cell has no {{radic|5}} chords, and has never heard of the regular 5-cell. Its relationship to the 11-cell is entirely indirect, by way of the 16-cell....

In the 120-cell the 11 vertices of the 11-cell represent .. actual vertices conflated from 25 disjoint 24-cells that are all rotated 15.5~° (12° isoclinically) with respect to each other, so that their corresponding vertices are &lt;small&gt;&lt;math&gt;\tfrac{1}{\phi^2\sqrt{2}} \approx 0.270&lt;/math&gt;&lt;/small&gt; apart, joined by a 120-cell edge. The 11-cell equilateral triangle faces which appear to meet at an edge are actually faces of completely disjoint regular 5-cells, and the 5 conflated 11-cell vertices are separated by these short 120-cell edges. Each 11-cell vertex is actually a 120-cell face pentagon which has been conflated to a point in the abstraction. But the face pentagon is a concrete object, which can be found in the 120-cell joining the ends of 5 {{radic|5}} reflex edges of 5 distinct Jessen's icosahedra.

....the four great hexagon planes of the 4-Jessen's icosahedron....

....the 25 24-cells and the 25/5 600-cell: Squares and 4𝝅 polygrams

=== ''n''-pentagonals ===

[[File:600-cell.gif|thumb|The 120-point (600-cell), &lt;math&gt;H^4&lt;/math&gt;.]]
...

[[File:600-cell vertex geometry.png|thumb|Vertex geometry of the 600-cell, showing the 5 regular great circle polygons and the 8 vertex-to-vertex chord lengths with angles of arc. The golden ratio{{Efn|name=golden chords|group=}} governs the fractional roots of every other chord,{{Efn|name=fractional root chords}} and the radial golden triangles which meet at the center.|alt=|400x400px]]
The 120 vertices of the 600-cell are distributed{{Sfn|Coxeter|1973|p=298|loc=Table V: The Distribution of Vertices of Four-dimensional Polytopes in Parallel Solid Sections (§13.1); (iii) Sections of {3, 3, 5} (edge 2𝜏&lt;sup&gt;−1&lt;/sup&gt;) beginning with a vertex; see column ''a''}} at eight different [[#n-leviathon|chord]] lengths from each other. These edges and chords of the 600-cell are simply the edges and chords of its five great circle polygons. In ascending order of length in a ''unit-radius'' polytope (not in a {{radic|2}} radius polytopes like our usual examples), they are {{radic|0.𝚫}}, {{radic|1}}, {{radic|1.𝚫}}, {{radic|2}}, {{radic|2.𝚽}}, {{radic|3}}, {{radic|3.𝚽}}, and {{radic|4}}.{{Efn|1=Square roots of non-integers are given as decimal fractions:
{{indent|7}}&lt;math&gt;\text{𝚽} = \phi^{-1} \approx 0.618&lt;/math&gt; 
{{indent|7}}&lt;math&gt;\text{𝚫} = 1 - \text{𝚽} = \text{𝚽}^2 = \phi^{-2} \approx 0.382&lt;/math&gt;
{{indent|7}}&lt;math&gt;\text{𝛆} = \text{𝚫}^2/2 = \phi^{-4}/2 \approx 0.073&lt;/math&gt;
&lt;br&gt;
For example:
{{indent|7}}&lt;math&gt;\phi = \sqrt{2.\text{𝚽}} = \sqrt{2.618\sim} \approx 1.618&lt;/math&gt;
|name=Decimal fractions of square roots|group=}}

Notice that the four [[User:Dc.samizdat/A symmetrical arrangement of 120 11-cells#''n-''equilaterals|hypercubic chords]] of the 24-cell ({{radic|1}}, {{radic|2}}, {{radic|3}}, {{radic|4}}) alternate with the four new chords of the 600-cell's additional great circles, the decagons and pentagons. The new ''golden chord'' lengths are necessarily square roots of fractions, but very special fractions related to the golden ratio{{Efn|1=The fractional-root ''golden chords'' are irrational fractions that are functions of {{radic|5}}. They exemplify that the [[W:golden ratio|golden ratio]] &lt;big&gt;𝜙&lt;/big&gt; {{=}} {{sfrac|1 + {{radic|5}}|2}} ≈ 1.618 is a circle ratio related to &lt;big&gt;𝜋&lt;/big&gt;:&lt;br&gt;
: {{sfrac|𝜋|5}} = arccos (𝜙/2)
is one decagon edge, the 𝚽 = {{radic|0.𝚫}} = {{radic|0.382~}} ≈ 0.618 chord.
Reciprocally, in this function discovered by Robert Everest expressing &lt;big&gt;𝜙&lt;/big&gt; as a function of &lt;big&gt;𝜋&lt;/big&gt; and the numbers 1, 2, 3 and 5 of the Fibonacci series:&lt;br&gt;
: &lt;big&gt;𝜙&lt;/big&gt; = 1 – 2 cos ({{sfrac|3𝜋|5}})
{{sfrac|3𝜋|5}} is the arc length of the &lt;big&gt;𝜙&lt;/big&gt; = {{radic|2.𝚽}} = {{radic|2.618~}} ≈ 1.618 chord.|name=golden chords|group=}} including the two [[W:golden section|golden section]]s of {{radic|5}}, as shown in the diagram.{{Efn|The 600-cell edges are decagon edges of length {{radic|0.𝚫}}, which is 𝚽, the ''smaller'' golden section of {{radic|5}}; the edges are in the inverse [[W:golden ratio|golden ratio]] {{sfrac|1|φ}} to the {{radic|1}} hexagon chords (the 24-cell edges).
The other fractional-root chords exhibit golden relationships as well. The chord of length {{radic|1.𝚫}} is a pentagon edge.
The next fractional-root chord is a decagon diagonal of length {{radic|2.𝚽}} which is &lt;big&gt;𝜙&lt;/big&gt;, the ''larger'' golden section of {{radic|5}}; it is in the golden ratio{{Efn|name=golden chords|group=}} to the {{radic|1}} chord (and the radius).{{Efn|Notice in the diagram how the &lt;big&gt;𝜙&lt;/big&gt; chord (the ''larger'' golden section) sums with the adjacent 𝚽 edge (the ''smaller'' golden section) to {{radic|5}}, as if together they were a {{radic|5}} chord bent to fit inside the {{radic|4}} diameter.}}
The last fractional-root chord is the pentagon diagonal of length {{radic|3.𝚽}}.
The [[W:Pentagon#Side length is given|diagonal of a regular pentagon]] is always in the golden ratio to its edge, and indeed &lt;big&gt;𝜙&lt;/big&gt;{{radic|1.𝚫}} is {{radic|3.𝚽}}.|name=fractional root chords|group=}}

....


...the tetrahelix in the 600-cell is another of Coxeter's discoveries, &quot;a necklace of tetrahedral beads&quot; found by the wisest child who ever built with these blocks....

...fibration of 30/11 regular triacontagram great circles.... Sadoc was not so rash as to proclaim that the 11-fold must also have a distinct polyhedron somewhere as its Hopf map, but he named it distinct, calling it &quot;an example of another kind of symmetry&quot;. It is remarkable that he could see the 11-fold symmetry in the 120-point (600-cell), where it isn't manifest yet except in the twist of the tetrahelix, since there can be no 5-cells, {{radic|5}} chords or 11-cells until there is a compound of five 600-cells; those objects emerge only in the 600-point (120-cell). But as van Ittersum stressed, the shadows of the other four 600-cells are always there, even when only a singleton 600-cell is manifest; both of the &lt;math&gt;H_4&lt;/math&gt; polytopes possess all the same symmetry.

...Borromean rings in the compound of 5 (10) 600-cells...

=== ''n''-leviathon ===
...some diagrams of special subsets of these chords should be the illustration at the top of these preceding sections:

...great dodecagon/great hexagon-triangle/irregular hexagon central plane diagram from [[120-cell_§_Chords|120-cell § Chords]]......belongs at the beginning of the n-taliesins section above...

...great decagon/pentagon central plane diagram of golden chords from [[600-cell#Chords|600-cell § Chords]]......belongs at the beginning of the n-pentagonals section above....

...radially equilateral hypercubic chords from [[24-cell#Chords|24-cell § Chords]]......belongs at the begining of the n-equilaterals section above...

600 chords converge at each of the 600 vertices of the 120-cell. The vertex figure of the 120-cell is a compound of all the vertex figures of all its inscribed 4-polytopes, including the 11-cell. Because the vertex figure of a polytope is the complement of its cell, the 120-cell's vertex figure has something to say about the cells of every inscribed 4-polytope, in particular, all the relationships between the regular 4-polytopes and the regular 3-polytopes. These relationships are precisely the symmetries that we look for when we look for the correct set of dimensional analogies between objects in 3-space and objects in 4-space. The complex 120-cell vertex figure expresses them all; in a sense it says everything there is to say about them. Because Euclidean 4-space is representative of Euclidean ''n''-space generally up to the heptagon, in that every regular polytope in existence in any dimension has an analogue in a regular 4-polytope, this expressive power of the 120-cell vertex figure with respect to dimensional analogies extends to the ''n''-polytopes generally. This distinct object, the 120-cell vertex figure, embodies in itself all the dimensional analogy symmetries of the ''n''-polytopes, though it does not express all those symmetries, not all by itself. It is but one expression of them, but it is one expression of all of them.

== The 11-cells and the identification of symmetries by women ==
The 12-point (Legendre vertex figure) is embedded in Euclidean 4-space, as representative of Euclidean ''n''-space generally, to remarkable effect. The symmetrical arrangement there of its &lt;math&gt;10^2&lt;/math&gt; disjoint instances magically creates a strange object, the 11-cell, which never occurs alone, but only as a symmetrical arrangement of 11 of its instances. We should resist calling the 11-cell either a singleton polytope or a honeycomb, unless we call it both. We should not even firmly ascribe to it a distinct dimensionality (4), since its 137-cell honeycomb contains instances of every distinct regular convex ''n''-polytope and ''all their distinct inter-dimensional relationships'', except one. The great whale the 120-cell is larger than this school of strange fish, as it is larger than everything else, and swallows it whole. 

There is a great deal more to discover about the 11-cells, but the central point is this: the 11-cells are the expression of all the distinct symmetries of dimensional expansion and contraction characteristic of the various families of ''n''-polytopes, in one distinct 11-point ''n''-polytopes (plural). What these combined symmetries generate, as expressed in their entirety in the 11-cells, is not a family of regular ''n''-polytopes for some ''n,'' as the rotational ''n''-symmetries do. In the 11-cells they operate inter-dimensionally, to generate the symmetrical families of dimensional analogies we find between ''n''-polytopes of different ''n''. 

Their inter-dimensional symmetry operations were identified and named by [[W:Alicia Boole Stott|Alicia Boole Stott]], as her operations of expansion and contraction, before they were identified as operations of the general dimensional analogy between 2-sphere and 3-sphere by [[W:Heinz Hopf|Heinz Hopf]].{{Sfn|Boole Stott|1910|loc=''Geometrical deduction of semiregular from regular polytopes and space fillings''|pp=12-45|ps=; presents two cyclical sequences of regular and semi-regular 4-polytopes linked by expansion-contraction operations to their embedded 3-polytopes, comprising a large trans-dimensional polytope family that includes 6 regular 4-polytopes and their 3-polytope dimensional analogues, and 45 Archimedean 4-polytopes and their 13 Archimedean 3-polytope analogues.}} .... It is only because she did not have the 11-cell to start with that she did not discover the regular 137-cell {3, 3, ..} by her method.

Between Alicia Boole Stott and her identification of dimensional analogy symmetry with polytopes (a word she coined), and [[W:Emmy Noether|Emmy Noether]] and her identification of symmetry groups with conservation laws (in her 1st theorem), we owe the deepest mathematics of geometry and physics originally to two pioneering women, colleagues of their contemporaries the greatest men of the academy, but not recognized then or now as even more than their equals.

== Conclusion ==
Thus we see what the 11-cell really is: not just an [[W:abstract polytope|abstract 4-polytope]], not just a singleton convex 4-polytope, and not just a honeycomb.{{Sfn|Coxeter|1970|loc=''Twisted Honeycombs''}} The 11-point (11-cell) has a concrete ...regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}, {{smaller|{{sfrac|5|2}}}}&lt;nowiki/&gt;} as its identified element sets, which are subsets of the 120-cell's element sets just as all the convex 4-polytopes' element sets are. Eleven 11-cells have a concrete regular realization {&lt;nowiki/&gt;{{smaller|{{sfrac|5|2}}}}, 3, 3} as the 137-point (137-cell).

The 120 11-cells' realization as 600 12-point (Legendre vertex figures) captures precisely the geometric relationship between the regular 5-cell and 16-cell (the 4-simplex and the 4-orthoplex), which are both inscribed in the 11-cell, 137-cell and 120-cell, but are so distantly related to each other that they are not found together anywhere else. More generally, the 11-cells capture the geometric relationships between the regular convex ''n''-polytopes of different ''n''.

The symmetry groups of all the regular 4-polytopes are expressed in the 11-cells, paired in a special way with their analogous 3-symmetry groups. It is not simple to state exactly what relates 3-symmetry groups to 4-symmetry groups (there is Dechant's induction theorem),{{Sfn|Dechant|2021|loc=''Clifford Spinors and Root System Induction: H4 and the Grand Antiprism''}} but the 11-cells seem to be the expression of their dimensional analogies.{{Efn|name=Apology}}

== Build with the blocks ==

&lt;blockquote&gt;&quot;The best of truths is of no use unless it has become one's most personal inner experience.&quot;{{Sfn|Duveneck|1978|loc=Carl Jung, quoted in ''Life on Two Levels''|p=ii|ps=.{{Sfn|Jung|1961|ps=: &quot;The best of truths is of no use unless it has become one's most personal inner experience. It is the duty of everyone who takes a solitary path to share with society what he finds on his journey of discovery.&quot;}}}}&lt;/blockquote&gt;

&lt;blockquote&gt;&quot;Even the very wise cannot see all ends.&quot;{{Sfn|Tolkien|1954|loc=Gandalf}}&lt;/blockquote&gt;

{{Regular convex 4-polytopes|columns=9|wiki=W:|radius={{radic|2}}}}

== Acknowledgements ==
...

== Notes ==
{{Regular convex 4-polytopes Notelist}}

== Citations ==
{{Reflist}}

== References ==
{{Refbegin}}
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{{Refend}}</text>
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    </revision>
  </page>
  <page>
    <title>Draft:Nigerian Pidgin/Main Page</title>
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    <revision>
      <id>2692190</id>
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      <timestamp>2024-12-16T14:53:37Z</timestamp>
      <contributor>
        <username>Elías Fortaleza de la Fuerza Sánchez</username>
        <id>2962785</id>
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      <comment>/* Introduction */</comment>
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      <text bytes="1731" sha1="1ogyuhrq3dhphvnq53nd2yg4qbyvg6u" xml:space="preserve">==Introduction==
The box gives a summary of Nigerian Pidgin.{{languages}}{{Linguistics}}
{{Infobox language|name=[[wikipedia:Nigerian Pidgin|Nigerian Pidgin]]|altname=Naijá (languej)|ethnicity=[[wikipedia:Igbo|Igbo]], [[wikipedia:Yoruba|Yoruba]], [[wikipedia:Hausa|Hausa]], etc.|region=[[wikipedia:Nigeria|Nigeria]]|speakers=120 million+ ([[wikipedia:First language|L1]]+[[wikipedia:Second language|L2]])|date=2020|ref=&lt;ref&gt;[https://ethnologue.com/language/pcm Ethnologue:Nigerian Pidgin]&lt;/ref&gt;&lt;ref&gt;[https://apics-online.info/surveys/17 Apics-Online Services:Nigerian Pidgin]&lt;/ref&gt;|family_color=#FF0000|fam1=[[wikipedia:English Creole|English Creole]]|fam2=Antlantic|fam3=[[wikipedia:West African Pidgin English|West African Pidgin English]]|fam4=[[wikipedia:Nigerian Pidgin|Nigerian Pidgin]]|boxsize=22em|script=[[wikipedia:LatinScript|Latin Script]]|iso2=|iso3=pcm}}
This course focuses on the pidgin of [[wikipedia:Nigerian pidgin|Nigeria]] '''not elsewhere''', even if '''similar yet distinct pidgins may exist'''.

==Contents==
#{{75Percent}} [[Draft:Nigerian Pidgin/part1|Introduction, Alphabet (Unaccented, Accented &amp; Total) &amp; Sounds]]
#[[Draft:Nigerian Pidgin/part2|Nouns, Pronouns &amp; Determiners]]
#[[Draft:Nigerian Pidgin/part3|Verbs, Adverbs &amp; Adjectives]]
#[[Draft:Nigerian Pidgin/part4|Prepositions, Conjunctions &amp; Interjections]]
#[[Draft:Nigerian Pidgin/part5|Phrases, Proverbs &amp; Slangs]]
#[[Draft:Nigerian Pidgin/part6|Numbers, Date and Time &amp; Seasons]]
#[[Draft:Nigerian Pidgin/part7|Questions, Exclamations &amp; Vocabulary]]
#[[Draft:Nigerian Pidgin/part8|Addendum, Glossary &amp; Notes]]
#[[Draft:Nigerian Pidgin/part9/|Revision, Test, Questions]]
#[[Draft:Nigerian Pidgin/part10|Read More]]

==References==
{{reflist}}</text>
      <sha1>1ogyuhrq3dhphvnq53nd2yg4qbyvg6u</sha1>
    </revision>
  </page>
  <page>
    <title>African Arthropods/Arthropods on ''Ficus burkei''</title>
    <ns>0</ns>
    <id>306078</id>
    <revision>
      <id>2692281</id>
      <parentid>2684646</parentid>
      <timestamp>2024-12-17T11:16:22Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
      </contributor>
      <comment>/* Fig Wasps */</comment>
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      <model>wikitext</model>
      <format>text/x-wiki</format>
      <text bytes="3204" sha1="4kzqbtpoud443psd336qemlj52e8abg" xml:space="preserve">==Hymenoptera==
===Agaonidae (Fig wasps)===
&lt;gallery mode=packed heights=200&gt;
Elisabethiella stueckenbergi 41850656.jpg|''Elisabethiella stueckenbergi'', the pollinator of ''Ficus burkei''
&lt;/gallery&gt;
===Pteromalidae (Fig wasps)===
&lt;gallery mode=packed heights=200&gt;
Otitesella tsamvi 2023 01 23 16 31 08 5877.jpg|Female ''Otitesella tsamvi'' ovipositing into a syconium of ''Ficus burkei''
Otitesella tsamvi 122353646.jpg|Male ''Otitesella tsamvi'' on a syconium of ''Ficus burkei''
Philotrypesis_2019_06_29_4560.jpg|''Philotrypesis parca''
Seres barbarus iNat 123397793.jpg|Male ''Seres barbarus''
Seres barbarus iNat 226725647.jpg|Female ''Seres barbarus'' attempting to enter a ''Ficus burkei'' syconium
Sycoscapter cornutus 2022 06 04 11 47 20 9999.jpg|''Sycoscapter cornutus'' ovipositing into a syconium of ''Ficus burkei''
Watshamiella alata 2022 06 04 12 25 06.jpg|''Watshamiella alata'' ovipositing into a syconium of ''Ficus burkei''
&lt;/gallery&gt;
===(Fig wasps)===
&lt;gallery mode=packed heights=200&gt;
Sycophila_2019_08_24b.jpg|''Sycophila'' sp.
Sycophila_2019_08_24c.jpg|''Sycophila'' sp.
Lachaisea_brevimucro_2022_06_26_11_06_44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;
===Encyrtidae==
&lt;gallery mode=packed heights=200&gt;
Homalotylus iN 228280717.jpg
Encyrtidae lateral view with annotations.jpg
&lt;/gallery&gt;

===Chalcididae===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2024 06 30 15 13 13 0532 iN 227105442.jpg
&lt;/gallery&gt;

==Beetles==
&lt;gallery mode=packed heights=200&gt;
Harmonia axyridis 2022 06 04 14 58 48 iN 121601499.jpg|''Harmonia axyridis''
Microfreudea cyclica iNat 226950451.jpg|''Microfreudea cyclica''
Sciobius pullus 2022 06 04 11 40 06 iNat 120358257.jpg|''Sciobius pullus''
&lt;/gallery&gt;

==True Bugs, Hoppers, Aphids, and Allies (Order Hemiptera)==
&lt;gallery mode=packed heights=200&gt;
Pseudoeriopsylla 2024 06 30 14 53 24 0450 iN 227101062.jpg|A homotomid psylloid; ''Pseudoeriopsylla'' sp.
Uhlunga typica 2023 02 02 07 51 41 iN 148445638.jpg|Mating pair of ''[[w:Uhlunga typica|Uhlunga typica]]'' with eggs.
Greenidea iNat 227097030.jpg|Aphids; ''Greenidea'' sp.
Pauropsylla 2024 08 23 248754368 a.jpg|Adult leaf-rolling psylloids, ''Pauropsylla'' sp. (Triozidae) on a ''Ficus burkei'' leaf.
Pauropsylla 2024 08 23 248755613 c.jpg|An emerging leaf-rolling psylloid, ''Pauropsylla'' sp. (Triozidae) with a recently emerged adult
&lt;/gallery&gt;

==Moths and butterflies==
&lt;gallery mode=packed heights=200&gt;
Myrina silenus ssp. ficedula iN 46961472.jpg|''[[w:Myrina silenus|Myrina silenus]]'' (Common fig tree blue)
Myrina dermaptera iN 228280728 a.jpg|''[[w:Myrina dermaptera|Myrina dermaptera]]'' (Caterpillar of the lesser fig tree blue)
Naroma varipes 2024 06 29 15 09 34 iNat 226958889.jpg|''[[w:Naroma varipes|Naroma varipes]]'' mating pair
&lt;/gallery&gt;

==Thrips==
&lt;gallery mode=packed heights=200&gt;
Thrips Pietermaritzburg 2021 01 17.jpg|Tube-tailed thrips (Family [[w:Phlaeothripidae|Phlaeothripidae]])
&lt;/gallery&gt;

==Spiders==
&lt;gallery mode=packed heights=200&gt;
Gephyrota glauca 2024 06 30 15 49 46 iN 227105452.jpg|''[[w:Gephyrota|Gephyrota glauca ]]''
Vicirionessa mustela 2024 07 09 12 46 50 0886 iN 228280721.jpg|Female ''[[w:Vicirionessa|Vicirionessa mustela]]''
&lt;/gallery&gt;


{{BookCat}}</text>
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    <revision>
      <id>2692282</id>
      <parentid>2692281</parentid>
      <timestamp>2024-12-17T11:16:45Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
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      <comment>/* =Encyrtidae */</comment>
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      <text bytes="3205" sha1="b3qxi6sa7j8w2voiuhd0fqbvpwm9vaj" xml:space="preserve">==Hymenoptera==
===Agaonidae (Fig wasps)===
&lt;gallery mode=packed heights=200&gt;
Elisabethiella stueckenbergi 41850656.jpg|''Elisabethiella stueckenbergi'', the pollinator of ''Ficus burkei''
&lt;/gallery&gt;
===Pteromalidae (Fig wasps)===
&lt;gallery mode=packed heights=200&gt;
Otitesella tsamvi 2023 01 23 16 31 08 5877.jpg|Female ''Otitesella tsamvi'' ovipositing into a syconium of ''Ficus burkei''
Otitesella tsamvi 122353646.jpg|Male ''Otitesella tsamvi'' on a syconium of ''Ficus burkei''
Philotrypesis_2019_06_29_4560.jpg|''Philotrypesis parca''
Seres barbarus iNat 123397793.jpg|Male ''Seres barbarus''
Seres barbarus iNat 226725647.jpg|Female ''Seres barbarus'' attempting to enter a ''Ficus burkei'' syconium
Sycoscapter cornutus 2022 06 04 11 47 20 9999.jpg|''Sycoscapter cornutus'' ovipositing into a syconium of ''Ficus burkei''
Watshamiella alata 2022 06 04 12 25 06.jpg|''Watshamiella alata'' ovipositing into a syconium of ''Ficus burkei''
&lt;/gallery&gt;
===(Fig wasps)===
&lt;gallery mode=packed heights=200&gt;
Sycophila_2019_08_24b.jpg|''Sycophila'' sp.
Sycophila_2019_08_24c.jpg|''Sycophila'' sp.
Lachaisea_brevimucro_2022_06_26_11_06_44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;
===Encyrtidae===
&lt;gallery mode=packed heights=200&gt;
Homalotylus iN 228280717.jpg
Encyrtidae lateral view with annotations.jpg
&lt;/gallery&gt;

===Chalcididae===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2024 06 30 15 13 13 0532 iN 227105442.jpg
&lt;/gallery&gt;

==Beetles==
&lt;gallery mode=packed heights=200&gt;
Harmonia axyridis 2022 06 04 14 58 48 iN 121601499.jpg|''Harmonia axyridis''
Microfreudea cyclica iNat 226950451.jpg|''Microfreudea cyclica''
Sciobius pullus 2022 06 04 11 40 06 iNat 120358257.jpg|''Sciobius pullus''
&lt;/gallery&gt;

==True Bugs, Hoppers, Aphids, and Allies (Order Hemiptera)==
&lt;gallery mode=packed heights=200&gt;
Pseudoeriopsylla 2024 06 30 14 53 24 0450 iN 227101062.jpg|A homotomid psylloid; ''Pseudoeriopsylla'' sp.
Uhlunga typica 2023 02 02 07 51 41 iN 148445638.jpg|Mating pair of ''[[w:Uhlunga typica|Uhlunga typica]]'' with eggs.
Greenidea iNat 227097030.jpg|Aphids; ''Greenidea'' sp.
Pauropsylla 2024 08 23 248754368 a.jpg|Adult leaf-rolling psylloids, ''Pauropsylla'' sp. (Triozidae) on a ''Ficus burkei'' leaf.
Pauropsylla 2024 08 23 248755613 c.jpg|An emerging leaf-rolling psylloid, ''Pauropsylla'' sp. (Triozidae) with a recently emerged adult
&lt;/gallery&gt;

==Moths and butterflies==
&lt;gallery mode=packed heights=200&gt;
Myrina silenus ssp. ficedula iN 46961472.jpg|''[[w:Myrina silenus|Myrina silenus]]'' (Common fig tree blue)
Myrina dermaptera iN 228280728 a.jpg|''[[w:Myrina dermaptera|Myrina dermaptera]]'' (Caterpillar of the lesser fig tree blue)
Naroma varipes 2024 06 29 15 09 34 iNat 226958889.jpg|''[[w:Naroma varipes|Naroma varipes]]'' mating pair
&lt;/gallery&gt;

==Thrips==
&lt;gallery mode=packed heights=200&gt;
Thrips Pietermaritzburg 2021 01 17.jpg|Tube-tailed thrips (Family [[w:Phlaeothripidae|Phlaeothripidae]])
&lt;/gallery&gt;

==Spiders==
&lt;gallery mode=packed heights=200&gt;
Gephyrota glauca 2024 06 30 15 49 46 iN 227105452.jpg|''[[w:Gephyrota|Gephyrota glauca ]]''
Vicirionessa mustela 2024 07 09 12 46 50 0886 iN 228280721.jpg|Female ''[[w:Vicirionessa|Vicirionessa mustela]]''
&lt;/gallery&gt;


{{BookCat}}</text>
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    <revision>
      <id>2692284</id>
      <parentid>2692282</parentid>
      <timestamp>2024-12-17T11:57:09Z</timestamp>
      <contributor>
        <username>Alandmanson</username>
        <id>1669821</id>
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      <comment>/* Hymenoptera */</comment>
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      <text bytes="3199" sha1="h4mv0ertwcnhntrh6k4ycykogbko05t" xml:space="preserve">==Wasps==
===Agaonidae (Fig wasps)===
&lt;gallery mode=packed heights=200&gt;
Elisabethiella stueckenbergi 41850656.jpg|''Elisabethiella stueckenbergi'', the pollinator of ''Ficus burkei''
&lt;/gallery&gt;
===Pteromalidae (Fig wasps)===
&lt;gallery mode=packed heights=200&gt;
Otitesella tsamvi 2023 01 23 16 31 08 5877.jpg|Female ''Otitesella tsamvi'' ovipositing into a syconium of ''Ficus burkei''
Otitesella tsamvi 122353646.jpg|Male ''Otitesella tsamvi'' on a syconium of ''Ficus burkei''
Philotrypesis_2019_06_29_4560.jpg|''Philotrypesis parca''
Seres barbarus iNat 123397793.jpg|Male ''Seres barbarus''
Seres barbarus iNat 226725647.jpg|Female ''Seres barbarus'' attempting to enter a ''Ficus burkei'' syconium
Sycoscapter cornutus 2022 06 04 11 47 20 9999.jpg|''Sycoscapter cornutus'' ovipositing into a syconium of ''Ficus burkei''
Watshamiella alata 2022 06 04 12 25 06.jpg|''Watshamiella alata'' ovipositing into a syconium of ''Ficus burkei''
&lt;/gallery&gt;
===(Fig wasps)===
&lt;gallery mode=packed heights=200&gt;
Sycophila_2019_08_24b.jpg|''Sycophila'' sp.
Sycophila_2019_08_24c.jpg|''Sycophila'' sp.
Lachaisea_brevimucro_2022_06_26_11_06_44.jpg|''Lachaisea brevimucro''
&lt;/gallery&gt;
===Encyrtidae===
&lt;gallery mode=packed heights=200&gt;
Homalotylus iN 228280717.jpg
Encyrtidae lateral view with annotations.jpg
&lt;/gallery&gt;

===Chalcididae===
&lt;gallery mode=packed heights=200&gt;
Brachymeria 2024 06 30 15 13 13 0532 iN 227105442.jpg
&lt;/gallery&gt;

==Beetles==
&lt;gallery mode=packed heights=200&gt;
Harmonia axyridis 2022 06 04 14 58 48 iN 121601499.jpg|''Harmonia axyridis''
Microfreudea cyclica iNat 226950451.jpg|''Microfreudea cyclica''
Sciobius pullus 2022 06 04 11 40 06 iNat 120358257.jpg|''Sciobius pullus''
&lt;/gallery&gt;

==True Bugs, Hoppers, Aphids, and Allies (Order Hemiptera)==
&lt;gallery mode=packed heights=200&gt;
Pseudoeriopsylla 2024 06 30 14 53 24 0450 iN 227101062.jpg|A homotomid psylloid; ''Pseudoeriopsylla'' sp.
Uhlunga typica 2023 02 02 07 51 41 iN 148445638.jpg|Mating pair of ''[[w:Uhlunga typica|Uhlunga typica]]'' with eggs.
Greenidea iNat 227097030.jpg|Aphids; ''Greenidea'' sp.
Pauropsylla 2024 08 23 248754368 a.jpg|Adult leaf-rolling psylloids, ''Pauropsylla'' sp. (Triozidae) on a ''Ficus burkei'' leaf.
Pauropsylla 2024 08 23 248755613 c.jpg|An emerging leaf-rolling psylloid, ''Pauropsylla'' sp. (Triozidae) with a recently emerged adult
&lt;/gallery&gt;

==Moths and butterflies==
&lt;gallery mode=packed heights=200&gt;
Myrina silenus ssp. ficedula iN 46961472.jpg|''[[w:Myrina silenus|Myrina silenus]]'' (Common fig tree blue)
Myrina dermaptera iN 228280728 a.jpg|''[[w:Myrina dermaptera|Myrina dermaptera]]'' (Caterpillar of the lesser fig tree blue)
Naroma varipes 2024 06 29 15 09 34 iNat 226958889.jpg|''[[w:Naroma varipes|Naroma varipes]]'' mating pair
&lt;/gallery&gt;

==Thrips==
&lt;gallery mode=packed heights=200&gt;
Thrips Pietermaritzburg 2021 01 17.jpg|Tube-tailed thrips (Family [[w:Phlaeothripidae|Phlaeothripidae]])
&lt;/gallery&gt;

==Spiders==
&lt;gallery mode=packed heights=200&gt;
Gephyrota glauca 2024 06 30 15 49 46 iN 227105452.jpg|''[[w:Gephyrota|Gephyrota glauca ]]''
Vicirionessa mustela 2024 07 09 12 46 50 0886 iN 228280721.jpg|Female ''[[w:Vicirionessa|Vicirionessa mustela]]''
&lt;/gallery&gt;


{{BookCat}}</text>
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  <page>
    <title>Social Victorians/Diamond Jubilee Garden Party</title>
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        <username>Scogdill</username>
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      <text bytes="124623" sha1="1i0ifwhmwkj98zjuo11lstrc9jjw829" xml:space="preserve">=Event=
On Monday 28 June 1897, Queen Victoria hosted a garden party at Buckingham Palace, inviting between 5,000 and 6,000 people. This party was the final official event of the London Diamond Jubilee celebrations. The Queen released to the press the names of people invited, which means the newspapers could print some or all of this list. The very long article in the London ''Morning Post'', for example, prints what may be the comprehensive list of those invited, although two columns are illegible in some places.

The original newspaper account seems to have been published by the ''Court Circular'', and then the popular newspapers reprinted pieces of that story, many adding contextualizing paragraphs of their own. Some of these later reports are quite long, perhaps 5 or more full columns. Sometimes the newspapers included short descriptions of the women's dresses, suggesting that for the list of people invited, the source was the ''Court Circular'', but the parts of the stories devoted to context, history or fashion might have been written by a reporter present at the event.

==Logistics==
* 28 June 1897, Monday, in the gardens at Buckingham Palace, hosted by Queen Victoria.
* Between 5,000 and 6,000 guests were invited.
* Many visitors from the empire who were in town for the Jubilee celebrations were invited to this garden party.
* The weather was fine, having improved since the day before.
* The garden party was held in the grounds around Buckingham Palace, and the Palace itself was open and available for guests to visit:&lt;blockquote&gt;Great preparations had been made in the splendid grounds adjoining the Royal Palace for the party, the whole scene presenting a fascinating appearance. The beautifully-kept grounds were partially covered with tents and marquees for the convenience of the many guests, and the lovely lake was really in the hands of the Queen’s bargemen, who had charge of the many boats which had been placed on the extensive ornamental waters for the use of guests. There was also plenty of music, several regimental bands being in attendance, while for those who wished to become acquainted with the valuable pictures and works art which are to be found at the Royal residence, all the State and reception rooms of the Palace were thrown open.&lt;ref name=&quot;:2&quot;&gt;“The Queen’s Garden Party. Brilliant Scene at Buckingham Palace.” ''Globe'' 29 June 1897, Tuesday: 6 [of 8], Col. 3a–c [of 5]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0001652/18970629/050/0006. Print p. 6.&lt;/ref&gt; (6, Col. 3a)&lt;/blockquote&gt;
*The streets around the entrances to Buckingham Palace were lined with spectators beginning hours before the Queen was to arrive:&lt;blockquote&gt;Although the Garden Party was not timed to commence until after five o’clock, the Mall from Marlborough House to Buckingham Palace was well lined by two o’clock, and an hour afterwards large crowds, for the most part composed of ladies, had taken up their positions. This was also the case along Constitution-hill, where the assembly which had gathered to witness the Queen’s arrival at the Palace from Windsor nad [sic] to a large extent remained. The heat was somewhat oppressive, but the trees along the Mall and the Green Park afforded welcome shelter. Many ladies had evidently come prepared for a long wait, as they had provided themselves with the now familiar camp stool, which is always prominent on these occasions. On the other hand, the police were waging war against the men who frequent such places with stools and forms, and as soon as any of them put in an appearance they were quickly pounced upon by the officers, who at once proceeded to destroy the intended stands before the eyes of the helpless owners. Among the sightseers were several of the Indian visitors in gorgeous coloured coats, tight-fitting trousers, and turbans, as well as some of the Australian and New Zealand troops.&lt;ref name=&quot;:2&quot; /&gt; (6, Col. 3a)&lt;/blockquote&gt;
==Related Events==
This garden party was the culminating event of the official celebrations for Queen Victoria's Diamond Jubilee, and more specific events led up to it:

# Trip from Windsor to Paddington Station Queen Victoria and a large retinue traveled by train from Windsor to Paddington Station the day before, preceded on an earlier train by &quot;the royal equipages sent from Buckingham Palace for the use of the Queen and her suite,&quot; which were&lt;blockquote&gt;First came the splendied semi-state landau in which the Queen made her now famous journey on June 22d. It was preceded by scarlet-coated outriders, and horsed by four magnificent bays driven by postilions in navy blue and white uniforms. Two similar carriages followed, and these were in turn succeeded by a number of pair-horse clarences for the conveyance of the household and suite, and several breaks and ‘buses for luggage. A captain's escort, furnished by the 2d Life Guards, and commanded by Captain Ellison, clattered along in rear of the carriages, and took up a position opposite the spot where, by prior arrangement, Her Majesty’s saloon was to be brought to a standstill. These magnificent troops, riding their great black horses, and with the sunlight dancing upon their nodding plumes, and reflected by their burnished helmets, cuirasses, and trappings, made a very fine show indeed. The escort did not carry the colour, as it did on the 21st, nor was it accompanied by the regimental trumpets.&lt;ref name=&quot;:0&quot;&gt;&quot;Jubilee Festivities. The Queen Again in London. Interesting Functions. A Visit to Kensington. The Garden Party.&quot; ''North British Daily Mail'' 29 June 1897, Tuesday: 5 [of 8], Col. 3a–7b [of 9]. ''British Newspaper Archive'' https://www.britishnewspaperarchive.co.uk/viewer/bl/0002683/18970629/083/0005. Print p. 5.&lt;/ref&gt;{{rp|5, Col. 3b}}&lt;/blockquote&gt;
# Reception at Paddington
# Visit to Kensington
# Kensington to Buckingham Palace
# The Garden Party
# Return to Windsor by Way of Paddington

=== Foreign Admirals ===
On 29 June 1897, the day after the garden party, the ''North British Daily Mail'' reports that, after the Queen's garden party, the foreign admirals would return to Spithead for a tour around the dockyard and luncheon:&lt;blockquote&gt;THE FLEET AT SPITHEAD&lt;p&gt;
The fleet at Spithead was again illuminated last night, the railway companies having duplicated the ordinary train service to bring visitors down. The Koenig Wilhelm was to have sailed on Sunday evening, but her departure has been deferred, and last night her officers gave a private dinner party aboard for the anniversary of the Queen’s Coronation. All the commissioned ships in the harbour were dressed at noon. A royal salute was fired. The [Col. 6c–7a] foreign admirals will return from their visit to London on the occasion of the Queen’s garden party to be conducted round the dockyard to-day, and they will be entertained to luncheon.&lt;ref name=&quot;:0&quot; /&gt;{{rp|5, Col. 6c–7a}}&lt;/blockquote&gt;

=== Colonial Premiers ===
The day of the garden party the colonial premiers attended a meeting with Secretary of State for the Colonies, [[Social Victorians/People/Chamberlain|Joseph Chamberlain]]:&lt;blockquote&gt;THE COLONIAL PREMIERS

The whole of the Colonial Premiers went to the Colonial Office yesterday for further conference with Mr Chamberlain, who received them in his private room, attended by Mr F. H. Wilson, legal assistant, Mr Reid and the Hon. T. Cochrane, M.P., assistant private secretaries. The conference lasted hours, and was of a strictly private and confidential character, the matters discussed involving several points of high State policy.

Premiers will be entertained at Warwick Castle by the Earl and Countess of Warwick on July 15th. On the same occasion the Attorney General of Queensland will present a loving cup from Warwick, in Queensland, to the old county town of Warwick, from which it takes its name. He will be accompanied by the Colonial troopers.&lt;ref name=&quot;:0&quot; /&gt;{{rp|5, Col. 7a}}&lt;/blockquote&gt;

For these visitors to London during the Diamond Jubilee, the next major social event was on 15 July, at Warwick Castle, hosted by [[Social Victorians/People/Warwick|Daisy, Countess of Warwick and Francis, 5th Earl of Warwick]], although perhaps some attended the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 2 July 1897 fancy-dress ball]].

== Who Was Present ==
In the absence of a copy of the report about the garden party in the ''Court Circular'', the newspaper account with the fullest list of names is from the ''Morning Post'', although people further down the list can be impossible to identify, and two full columns are damaged (Col. 7 on p. 4 and Col. 1 on p. 5).&lt;ref name=&quot;:1&quot;&gt;“The Queen’s Garden Party.” ''Morning Post'' 29 June 1897, Tuesday: 4 [of 12], Cols. 1a–7c [of 7] and 5, Col. 1a–c. ''British Newspaper Archive''  https://www.britishnewspaperarchive.co.uk/viewer/BL/0000174/18970629/032/0004 and https://www.britishnewspaperarchive.co.uk/viewer/bl/0000174/18970629/032/0005.&lt;/ref&gt; Whenever possible, then, what is here has been amended with other newspaper reports that have names to help decipher the illegible ones in the ''Morning Post'' account. The names in the Morning Post are grouped, mostly by rank and name.

=== People of Color at This Event ===

One purpose of a closer look at this event is to get a more precise list of names of people of color from the various countries in the empire, who were not recognized and thus not named in newspaper descriptions of other events. For example, the [[Social Victorians/1897 Fancy Dress Ball|Duchess of Devonshire's 2 July 1897 fancy-dress ball]] was said to include a number of South Asian dignitaries, but because the Duchess did not release to the newspapers the names of those who were invited, those dignitaries went mostly unnamed in the newspaper reports, if their presence was noted at all. Besides the South Asian guests invited to this garden party, some South Asian visitors to London were spectators as well:&lt;blockquote&gt;Among the sightseers were several of the Indian visitors in gorgeous coloured coats, tight-fitting trousers, and turbans, as well as some of the Australian and New Zealand troops.&lt;ref name=&quot;:2&quot; /&gt; (6, Col. 3a)&lt;/blockquote&gt;In a section on what people — mostly women — wore, the reporter for the ''Daily News'' said,&lt;blockquote&gt;Suffice to say, the modistes had done their best, and that their achievements excited general admiration. Here and there, however, was an Eastern beauty whose golden lace drapery, loosely enveloping a figure that owed nothing to the corset, challenged comparison, we will not say with what success, with the European model. In the almost entire absence of uniforms or Court dress, the costumes of the East Indian notables lent colour to the assemblage, while their pearls and diamonds, the wealth of Ormuz and of Ind, were not allowed to pass unobserved.&lt;ref name=&quot;:3&quot; /&gt; (5, Col. 6b)&lt;/blockquote&gt;

=== People Invited ===
# Queen Victoria, with escort and attendants
## Captain's Escort of the 2nd Life Guards
## The Duchess of Buccleuch, Mistress of the Robes
## The Dowager Lady Churchill, Lady in Waiting
## The Hon. Harriet Phipps, Woman of the Bedchamber
## Maids of Honour in Waiting
### The Hon. Mary Hughes
### The Hon. Aline Majendie
## the Earl of Kintore, Lord in Waiting
## Captain Drummond, Groom in Waiting
## Equerries in Waiting
### Major-General Sir John M'Neill, V.C.
### Lieutenant Colonel Davidson, M.V O. [sic]
#Grand Duke and Grand Duchess Serge of Russia
#Princess Henry of Battenberg, with attendants
##Miss Minnie Cochrane
##Colonel John Clerk, C.S.I., C.V.O.
#Her Imperial Majesty the Empress Frederic, attended by 
##the Dowager Lady Ampthill
##Lord Harris
##Colonel S. Waller
##Princess Hatzfeldt Trachenberg
##Count Seckendorff
##Baron and Baroness Reischach
#Their Royal Highnesses the Prince and Princess of Wales, with attendants
##Lady Suffield, Lady in Waiting
##Miss Knollys, Woman of the Bedchamber
##Lord Colville of Culross, K.T., G.C.V.O., Chamberlain to the Princess of Wales
##The Earl of Gosford, K.P., Lord in Waiting
##General Sir D. Probyn, G.C.V.O., K.C.B., K.C.S.I., V.C, Comptroller
##Sir Francis Knollys, K.C.M.G., C.B., Groom in Waiting
##Major-General Stanley Clarke, C.M.G., Equerry in Waiting
#Princess Victoria of Wales
#Their Royal Highnesses Prince and Princess Charles of Denmark
#Their Royal Highnesses the Grand Duke and Grand Duchess of Mecklenburg-Strelitz, attended by
##Lady Caroline Cust
##Mr. Hugo Erskine Wemyss
##Count Reventlow Criminil
##Baron von der Wense
#Their Royal Highnesses Prince and Princess Christian, attended by
##Baroness von und zu Egloffstein
##Colonel the Hon. Charles Eliot
#Her Highness Princess Victoria
#His Highness Prince Christian Victor
#His Highness Prince Albert of Schleswig-Holstein
#Her Royal Highness Princess Louise Marchioness of Lorne and the Marquis of Lorne, attended by
##Lady Sophia Macnamara
##[[Social Victorians/People/Arthur Collins|Colonel Arthur Collins]], M.V.O.
#Their Royal Highnesses Prince and Princess Henry of Prussia, attended by
##Admiral of the Fleet Sir Edmund Commerell
##Baron and Baroness Seckendorff
##Count Hahn
##Captain Muller
#Their Royal Highnesses the Duke and Duchess of Saxe-Coburg and Gotha, attended by
##The Hon. Mrs. Monson
##His Excellency Herr von Schön
##Captain the Hon. D. J. Mouson [sic, s/b Monson?], M.V.O.
##Mr. A. D. J. Monson
##Captain von Ruxleben
#Princess Beatrice of Saxe-Coburg and Gotha
#The Hereditary Prince of Saxe-Coburg and Gotha
#Their Royal Highnesses the Duke and Duchess of Connaught and Strathearn, attended by
##Colonel and the Hon. Mrs. A. Egerton
#Her Royal Highness the Duchess of Albany, attended by
##Sir Robert and Lady Collins
##Miss Potts
#Her Royal Highness Princess Frederica of Hanover and Baron von Pawel Raminingen, attended by
##Mr. and Mrs. Charles Wood
#His Royal Highness the Duke of Cambridge, attended by
##Colonel A. C. FitzGeorge, C.B.
#Her Royal Highness the Duchess of Teck and his Highness the Duke of Teck, attended by
##Lady Katherine Coke
##The Hon. A. Nelson Hood
#Her Royal Highness Princess Louise Duchess of Fife and the Duke of Fife
#His Highness the Prince and her Royal Highness Princess Frederic Charles of Hesse, attended by
##The Hon. A. Hay
##Fraulein von Tasmund
##Baron von Kotwitz
#Their Highnesses Prince and Princess Aribert of Anhalt, attended by
##Miss Deverell
##Major Evan Martin
#Her Royal Highness the Hereditary Princess of Saxe-Meiningen and her Serene Highness Princess Feodore of Saxe-Meiningen, attended by
##The Hon. Aubrey FitzClarence
##Miss von Dreskan
##Baron von Roeder
#His Serene Highness the Prince of Schaumburg-Lippe
#Their Highnesses Prince and Princess Edward of Saxe-Weimar
#Her Serene Highness Princess Victor of Hohenlohe
#Countess Gleichen (x2)
#Their Serene Highnesses Prince and Princess Adolphus of Teck
#The Prince Francis and Prince Alexander of Teck
#His Highness Prince Augustus Leopold of Saxe-Coburg
#Their Serene Highnesses Prince and Princess Blucher von Wahlstatt
#Their Serene Highnesses Prince and Princess Joachim Murat
#Their Serene Highnesses [[Social Victorians/People/Pless|Prince and Princess Hans Henry Pless]]
#Prince and Princess Loewenstein
#Their Serene Highnesses the Duke and Duchess of Arenberg
#Prince Victor Duleep Singh
#Prince Frederick Duleep Singh
#Princess Duleep Singh (x2)
#ARGENTINE REPUBLIC — M. Florencio Dominguez and M. Carlos Dominguez
#BADEN — Herr yon Brauer, Mr. Brook Taylor, and Baron Bohlen Halbach
#BAVARIA — His Royal Highness the Prince Rupert, General Sir L. Gardiner, K.C.V.O., C.B., Major Fairholme, Lieutenant-Colonel Emile von le Bret Nucourt, and Captain Othon von Stettin
#BELGIUM — His Serene Highness the Prince Charles de Ligne, Princess de Ligne, Madlle. de Ligne, Mr. C. lnnes Ker, Count de Jonghe d'Ardoye, and the Marquis d’Asshe
#BOLIVIA — M. Caso, Mr. Conway Seymour, M. Pedro Suarez, Madame Suarez, and M. Adolfo Bolivian
#BRAZIL — M. [[Social Victorians/People/Souza Correa|de Souza Correa]] [Corréa?]
#BULGARIA — Their Royal Highnesses the Prince and Princess of Bulgaria, Colonel J. R. Slade, C.B., Madame Petrow Tchomakoff, Count Robert de Bourboulon, Lieutenant-Colonel Marcoff, Major Petrew, Captain Stoïanow, and Mr. Martin Furth
#CENTRAL AMERICA (Greater Republic) — M. Medina and Miss Medina
#CHILI — M. Ramon Subercasseaux and Mr. Raglan Somerset
#CHINA — His Excellency Chang Yen Hoon, Colonel Mark Bell, V.C,. Mr. Liang, Mr. Jui, and Mr. Koo
#COREA — His Excellency Min Young Hwan, Major A. Cavendish, Mr. Min Young Chan, Mr. Min Shangho, and Mr. von Rautenfeld
#COSTA RICA — Senor Don Demetrio Iglesias, Mr. C. Alban Young, Dona Eudoxia Castro, Señorita Maria Iglesias, Don Ricardo Fernandez Guardia, and Dona Christina Castro Keith
#DENMARK — His Royal Highness the Prince Waldemar, Major-General Arthur Ellis, C.S.I., M. Charles Rothe, and Captain Evers
#EGYPT — Prince Mohammed Ali Pasha, Colonel Larking, Tigrane Pasha, Colonel Aziz Bey, Mr. George Smart, Said Zoulfikar Bey
#ECUADOR — M. Navares, Colonel Concha
#FRANCE — General Davoust, Duc d'Auerstadt, Duchesse d'Auerstadt, and Madlle. Davoust, Colonel Brabazon, Colonel Dawson, General Hagron, M. Crozier, Colonel Humbert, and Captain Riviers de Mauny
#GERMANY — His Royal Highness the Prince Albert of Prussia, Prince Regent of Brunswick, Major-General Sir C. du Piat, K.C.B., Colonel Grierson, Lieutenant-General von Plessen, Colonel von Arnim, Captain Fischel, Count von der Schulenberg (Hofmarschall), Major Freiherr von Stein, Dr. Schreibe, Captain von Unzer
#GREECE — M. Rangabi, Mr. R. D. Norton
#GUATEMALA — Dr. Cruz, Madlles. Cruz (2), Señor Estrada
#HAWAIIAN ISLANDS — Mr. S. M. Damon, Captain the Hon. H. Napier, Major Curtis P. Jaukea
#HESSE — Their Royal Highnesses the Grand Duke and Grand Duchess of Hesse, Colonel the Hon. H. Byng, C.B., Baroness de Grancy, Baron Riedesel zu Eisenbach, Baron de Genadius Grancy
#ITALY — Their Royal Highnesses the Crown Prince and Princess, the Earl of Clarendon, Colonel Needham, Countess Giulia Trigona, Lieutenant-General Terzaghi, Major Cavaliere Viganoni, Captain Cavaliere Merli Miglietti, Count Romnaldo Trigona, Cavaliere F. Comotto
#JAPAN — His Imperial Highness the Prince Arisugawa, Mr. R. F. Synge, Captain Beaumont, R.N., Marquis Ito, Mr. S. Saito, Marquis Kido, Captain Funaki, Lieutenant-Colonel Murata, Lieutenant Kato, Mr. Nabeshima
#LIBERIA — Mr. H. Hayman
#LUXEMBURG — His Royal Highness the Hereditary Grand Duke of Luxemburg, Colonel H. D. Browne, Baron Ritter yon Grünstein
#MECKLENBURG-SCHWERIN — His Excellency Herr D. yon Vietinghoff, Mr. Eyre A. Crowe
#MEXICO — Don Antonio Mier y Celis, Mr. Arnold Royle, C.B., Don Francisco R. Gallardo, Don Eustagino dc Escaudon, and Captain Don Ponfirio Diaz
#MONTENEGRO — His Highness the Prince Danilo, Major the Hon. C Harbord, Colonel Djurcovitch, and Captain Pejanovitch
#NETHERLANDS — Count van Lynden, Countess van Lynden, Mr. Horace West, and Count W. de Bylandt
#PARAGUAY — M. E. Machain and Madame Machain
#PERSIA — His Imperial Highness the Prince Amir Khan, General Sir Thomas Gordon, K.C.I.E., C.B., C.S.I.[,] Mr. Harry Churchill, General Karim Khan, Mirza Ahmad Khan, Mirza Ohaness Khan, Mirza Mohamad Ali Khan
#PERU — Senor Canevaro, Duchesse de Zoagli Canevaro, Dr. Don A. N. Puente, Don Alfredo Elster, and Don Carlos von der Heyde
#PORTUGAL — His Royal Highness the Duke of Oporto, Major the Hon. H. C Legge, M.V.O., Colonel Duval Telles, Captain Moreira de Sà, Major d'Albuquerque, and Lieutenant Jose de Melie[?]
#ROME — Right Rev. Monsignore Sambucetti, [[Social Victorians/People/Stonor|Hon. Harry Stonor]], Right Rev. Monsignore Belmont, the Right Rev. Monsignore de Vaz, Marchesi and Marchesa Muccioli, of the Noble Guard
#ROUMANIA — General Pancovici, Colonel G. P. Georgescu
#RUSSIA — Their Imperial Highnesses the Grand Duke Serge and Grand Duchess Feodrowna, the Grand Duke Cyril, Lord Churchill, Lieutenant-Colonel Waters, Countess Olsouffiew, Princess Youssoupoff, Princess Lobanoff de Rostow, General Stépanoff, Colonel Gadon, and Prince Youssoupoff, Colonel Clements, Mr. Alexander Gordon Ross, and Sub-Lieutenant N. Coubé (A.D.C. to Grand Duke Cyril)
#SAXE-COBURG — His Royal Highness the Prince Philip of Saxe-Coburg, Captain Walter Campbell, and Herr von Schön
#SAXE-WEIMAR — His Highness the Prince Hermann of Saxe-Weimar, Mr. Frederick Campbell, and Count Zeppelin
#SAXONY — His Royal Highness the Prince Frederick Augustus, Duke of Saxony, Colonel Howard, Freiherr yon Reitzenstern, First Lieutenant von Metzsch, and Baron von Oppell
#SERVIA — M. Mijatovich and Madame Mijatovich
#SIAM — His Royal Highness the Crown Prince and the Prince Mahit of Siam, Colonel E. H. Sartorius, V.C., Lieutenant-Colonel Rajavallabha, Lieutenant-Colonel C. Vernon Hume, Colonel Indaraty, Surgeon-Major Yarr
#SPAIN — Duke of Sotomayor, Captain the Hon. A. Greville, Señor José Caro, Señor Alfonso Merry del Val, and Señor Benitez al Villar
#SWEDEN AND NORWAY — His Royal Highness the Prince Eugène of Sweden and Norway, Captain G. L. Holford, Count G. Gyldenstolpe, Captain Roeder, Captain Baron Cederstrom
#TURKEY — Munir Pasha, Major Surtees, Brigadier-General Nassir Pasha, Captain Enver Bey, Colonel Gordon Ponsonby
#UNITED STATES — His Excellency the Hon. Whitelaw Reid, Mrs. Whitelaw Reid, Colonel Hallam Parr, Major-General Nelson A. Miles, Mrs. Nelson Miles, Rear-Admiral Joseph N. Miller, Captain M. P. Maus, Mr. Ogden Mills, Mrs. Ogden Mills, Mr. G. Creighton Webb, Mr. Erskine Hewett, Commander W. H. Emory, Lieutenant Philip Andrews, Lieutenant T. S. Rogers
#URUGUAY — Dr. Alberto Nin, Madlle. Nin, Don Alfonso Saenz de Zumaran, Don Luis Posadas, Colonel C. Robido
#WURTEMBURG— His Royal Highness the Duke Albert of Wurtemburg, Colonel C. Swaine, Lieutenant-General von Bilfinger, First Lieutenant Count von Degenfeld- Schonburg; five officers of the Queen's German Regiment: Major C. R. Burn (in attendance), Lieutenant-Colonel von Falkenhayn, Major von Arnim, First Lieutenant Baron von Moeller-Lilienstern, First Lieutenant von Gerlach, Second Lieutenant von Studnitz
#&quot;Native Princes, and gentlemen and ladies accompanying them&quot;&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 2b)
##His Highness the Raja of Kaparthala
##His Highness the Thakur Sahib of Morvi, K.C.I.E.
##His Highness the Thakur Sahib of Gondal, C.I., and her Highness the Maharani of Gondal, C.I.
##Colonel Maharaj Dhiraz
##Sir Pratab Singh, K.C.S.I.
##Thakur Hari Singh[,?]
##Kunwar Dhokal Singh
##Rajah Ajit Singh of Khetri, attended by
##Rajkumar Unmaid Singh of Shahpura, attended by
###Colonel Trevor (in attendance upon the Rajah Ajit Singh of Khetri and the Rajkumar Unmaid Singh of Shahpura)
##Bijey Singh
##Sir Jamaetjee Jejeebhoy, Bart., C.S.I., Miss Jejeebhoy, Mr. Jejeebhoy
##Mr. and Mrs. Powrala
##Major J. G. Turner and Mrs. Turner
##Mr. A. R. Wood and Mrs. Wood
#The &quot;officers of the Imperial Service Troops, with British officers and ladies&quot;&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 2b)
##Captain Mir Hashim Ali Khan Hyderabad-Resaldar
##Major Sunayat Singh, Kashmir
##Commandant Abdul Ganny, Gwalior
##Commandant Gooind, Rao Matkar, Indore
##Commandant Mirza Kurim Beg, Bhopal
##Rai Bahadur Dhunpat Rai, Jeypore
##Commandant Nand Singh, Patiala
##Commandant Rai Bahadur Thakur Dip Sing, Bikanir
##Commandant Chatru Singh, Bhartpur
##Resaldar Abdul Majid Khan, Babawalpur
##Commandant Daud Khan, Ulwar
##Commandant Nazir Khan, Rampur
##Risalda-Major Didar Singh, Sindi
##Risaldar-Major Kishan Singh, Nabha
##Risaldar Hara Singh, Karpurthala
##Risaldar Dhan Singhi, Bhavnagar
##Colonel H. Melliss, C.S.I., and Mrs. Melliss
##Major F. H. R. Drummond and Mrs. Drummond
##Captain F. Angelo
##Lieutenant H. Coape-Smith
##Captain G. F. Chenevix-Trench
#The &quot;officers of Native Cavalry Corps with British officers and ladies&quot;&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 2b)
##Risaldar-Major Baha-ud-din-Khan
##Sardar Bahadur, A.D.C. to Viceroy
##Risaldar-Major Sayyid Abdul Aziz
##Risaldar-Major Khan Bahadur
##Risaldar-Major Izzat Khan
##Risaldar-Major Hukam Singh
##Risaldar-Major Sher Singh
##Risaldar-Major Husain Khan
##Risaldar-Major Mangal Singh
##Risaldar-Major Kesar Singh
##Risaldar- Major Faiz Khan
##Risaldar-Major Muhammad Umar Khan
##Risaldar-Major Ali Mahomed Khan
##Risaldar-Major Mihrab Ali Khan
##Risaldar Kaddam Khan
##Risaldar Jahanzir Khan
##Risaldar Nadir Khan
##Risaldar Mir Haidar Shah Khan
##Risaldar Makbul Khan
##Risaldar Net Ram
##Ressaidar Gurdatt Singh
##Subadar Muhammed Beg Junadar
##Abdul Karin Khan
##Lieutenant-Colonel J. C. H. Gordon and Mrs. Gordon
##Major A. Phayre and Mrs. Phayre
##Captain C. F. Campbell
##Captain P. Melville, in attendance on his Highness Thakur Sahib of Morvi
##Captain M'Cartney Filgate, in attendance on their Highnesses the Thakur Sahib and Maharani of Gondal
##Mr. Nowroz
##M. Parveez
##Sir M. Mansherjee Bhownaggree, M.P.
##Mr. Percy Armytage and Mrs. Armytage
##Mr. Frank Cook, C.I.E., and Mrs. Frank Cook
#The &quot;commanding officers of Colonial contingents, with the ladies accompanying them&quot;&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 2b)
##Colonel the Hon. M. and Mrs. Aylmer, Canada
##Colonel and Mrs. Lassetter, New South Wales
##Major Reay, Victoria
##Colonel Pitt, New Zealand
##Major and Miss King, Queensland
##Lieutenant and Mrs. Phillips, Cape of Good Hope
##Lieutenant-Colonel Rowell, South Australia
##Major Strickland, Western Australia
##Captain Shepstone, Natal
##Major and Miss Reeves, Ceylon
##Mr. Badeley, Hong Kong
##Colonel Walker, C.M.G., and Mrs. Walker, Straits Settlements
##Captain Lucie Smith, Jamaica
##Lieutenant-Colonel E. B. M'lnnis, C.M.G., and Mrs. M'lnnis, British Guiana
##Major Rooks, Trinidad
##Captain Bernard, Malta
##Captain Kershaw, Cyprus
##Captain and Mrs. Middlemist, Gold Coast
##Inspector Hook, Lagos
##Captain Blakeney, Sierra Leone
##Lieutenant Festing, Royal Niger Company
##Captain Flint, British North Borneo Company
##The Hon. M. Gifford, Rhodesian Horse
##The following British officers attached: Lieutenant-Colonel Boulton, Lieutenant-Colonel Prior, Lieutenant-Colonel Tucker, Lieutenant-Colonel Domville, Lieutenant-Colonel Gibson, and Lieutenant-Colonel Tyrwhitt
#The &quot;gentlemen representing the various races in the Island of Ceylon&quot;&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 2c)
##Maha Mudaliyar don Solomon Dias Bandaranaihe
##The Hon. Alexander Dealius Sonewiratne
##M. E. Rowland Goonoratne
##M. Charles de Soysa Dessanayaka
##Panabokko Jikiri Banda
##Nugawela Kuia Banda
##Kobbokeduwe Loku Banda
##M. E. S. W. Senathi rajah [sic] and Mrs. Senathi
##M. J. H. de Saram and Miss de Saram
##M. P. Ramanathan
##M. Saunders and Miss Saunders
#The &quot;members of the Corps Diplomatique and other foreigners of distinction&quot;&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 2c)
##The Russian Ambassador, Madame de Staal, Madlle. de Staal, Madame de Stoeckl, Princess de San Donato, Madame Yermoloff, Madlle. Yermoloff, the Councillor, three Secretaries, and four Attachés of Embassy
##The German Ambassador, Countess Paul Hatzfeldt-Wildenburg, her Serene Highness Princess Hans Hohenlohe-Oehringen, Baroness yon Eckardtstein, the Councillor, two Secretaries, three Attachés of Embassy, and the Director of the Chancery
##The Austro-Hungarian Ambassador, Countess Deym, Countess Isabella Deym, Countess Clary Aldringen, Baroness Ferstel, the Councillor, two Secretaries, and four Attachés of Embassy
##The French Ambassador, Baroness de Courcel[,] Madlle. de Courcel, Madame Geoffray, the Minister Plenipotentiary, five Secretaries, and three Attachés of Embassy
##The Italian Ambassador, Princess Ruspoli, three Secretaries, and three Attachés of Embassy
##The Spanish Ambassador, Countess de Casa Valencia; Mesdlles. de Alcala Galiano (2), Marquise de Guiria, Donna de Zea Bermudez, Countess de Morella, Donna de Ia Camara y Livermore, three Secretaries, and four Attachés of Embassy
##The Turkish Ambassador, Madame Antbopoulos, the Councillor, and two Secretaries of Embassy
##The United States Ambassador, Mrs. Hay, Miss Hay, Mrs. Henry White, Mrs. Carter, Mrs. Colwell, two Secretaries, one Attaché of Embassy, and the Private Secretary to the Ambassador
##The Argentine Minister, Madame Dominguez, Mesdlles. Dominguez (3), and the Secretary of Legation
##The Persian Minister, and one Secretary of Legation
##The Danish Minister, Madame de Bille, Madame Gosch, and the Secretary of Legation
##The Siamese Minister, Mrs. Verney, Miss Verney, Mrs. Loftus, the Councillor, the Secretary, the Attaché, and the Interpreter to the Legation
##The Liberian Minister
##The Roumanian Minister and the Councillor of the Legation
##The Netherlands Minister, Baroness de Goltstein d'Oldenaller, Baroness Schimmelpenninck van der Oye, and the Councillor of Legation
##The Belgian Minister, the Councillor, and two Secretaries of Legation
##The Mexican Minister, Madame Yturbe, Madame Romero, Madame Farias, Madame Garcia, two Secretaries and three Attachés of Legation
##The Japanese Minister, Madame Kato, two Secretaries, and three Atachés [sic] of Legation
##The Minister for Sweden and Norway, Countess Lewenhaupt, and the Attaché of Legation
##The Chinese Minister, Lady Macartney, the English Secretary, three Secretaries, and four Attachés of Legation
##The Portuguese Minister, Madlle. de Quilinan, three Secretaries, and one Attaché of Legation
##The Swiss Minister, Madame Bourcart, Madame de Salis, the Secretary, and the Attaché of Legation
##The Haytian Chargé d’Affaires
##The Chargé d’Affaires of Greece, Madame Metaxas, and the Attaché
##The Chargé d’Affaires of Chile and Madame Bascunan
##Two Secretaries and one Attaché of the Brazilian Legation
##Count E. van Rosen
##Mr. Hippolyte de Aranjo
##Vice-Admiral Montt
##Mr. Pinto, Mrs. Pinto
##Mr. and Mrs. Scaramanga
##Vicomte de Galard
##Dr. Arnold, and Madlle. von Rappoport
##Mrs. John Meiggs, Miss Meiggs
##Miss Margaret Butler
##Mrs. Henry Morgan
##Hon. Chauncey Depew
##Mr. and Mrs. James Taylor
##Mr. and Mrs. Charles Marshall
##Mr. and Mrs. Edmund Bayliss
##Mrs. Colgate
##Miss Furniss
##Miss Wells
##Miss Harris
##Hon. Levi P. Morton, Mrs. Morton, and the Misses Morton
##The Bishop of Illinois and Mrs. Leonard, Miss Leonard
##The Bishop of Albany and Mrs. Doane
##The Bishop of New York and Mrs. Potter
##the Bishop of Minnesota and Mrs. Whipple
##Mr. and Mrs. Walter Burns
##Mrs. Douglas Grant
##Miss Scott
##Mrs. Grace, Miss Margarita Grace
##Mrs. Wentworth
##Miss van Wart
##M. Valentin de Courcel
##Madame la Marquise de Talleyrand Perigord
##Comte Boson de Perigord
##Vicomte d'Espenilles
##Madame and Madlle. Thierry Delanoue
##Madlle. de la Cherè
##M. Cellerier
##M. and Madame Delawarre
##Madame Evelina Fenzi
##Count A. Zannini
##M. and Madame Jules Cottran
##Chevalier E. Mazzuechi
##Signor A. Tedeschi
##Signor A. Mariotti
##Captain Lucian von Ziegler
##Chevalier Lieutenant von Barry
##Baron Georg Rothschild
##Privy Councillor Count Berchtold
##Baron G. E. Levi, Baroness Levi
##Commander E. Philipson, Mrs. E. Philipson
##The Duke and Duchess of San Germano Calabritto
##The Marquis of San Vito
##Donna Lidia Serramezzana
##Donna Margherita Chigi
##Marchioness Vitelleschi
##Chevalier Elia
##Count de Franqueville
##Count Urbain Chevrau
##M. Marcel Fonquier
##M. Baudon de Mony, Madame Baudon de Mony
##Duchess de Rohan
##Marquis de Lastorgrie, Marchioness de Lastorgrie
##Count de Boisgelin, Countess L. de Boisgelin
##M. Stern, Madame Stern, Madlle. Stern
##Count Charles du Luart
##General de Saucy
##M. E. Seydoux
##Count Jean de Madre
##M. de Monbrison
##Baron de la Chevrelière
##Count de la Villestreux, Countess de la Villestreux
##Count Urbain de Maille, Countess Urbain de Maille
##General Faveret de Kerbrich
##Monsieur de la Haye Jousselin
##Baronne Faveret de Kerbrich
##Colonel Matton
##M. Ferinier Didet
##Madame Ferinier Didet
##Donna Isabella Colonna, Donna Victoria Colonna
##Pom-k-Soh
##Madame Reyntiens
##Marquis de Fuente Hermosa
##Herr Rudolf Swobody
##M. Lauritz Tuxen
##Duchesse de Baiten
##M. de Marcoarti
##Comte de Heeren, Madlle. de Heeren
##Monsieur M. de Mauny Talvande
##Senor Don Nicolas Campero
##Lieutenant Charny
##Lieutenant Sanders
##Madame and Madlle. de Mouni
##Comtesse de Montsoulmin
#&quot;Foreign Admirals and Commanding Officers and Staffs&quot;&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 3a / Col. 3b)
##Austrian Admiral Baron von Spaun, Commander von Ziegler, Lieutenant Retter yon Barry, Lieutenant Mitchell, R.N. (attached)
##Danish Admiral H. H. Koch, Captain Waudel, Lieutenant Middelboc, Lieutenant Majendie, R.N. (attached)
##French Admiral C. F. E. De Courthille, Captain Germinet, Commander Poidlone, Lieutenant Perdriel, Sub-Lieutenant de Caqueray, Lieutenant Phillimore, R.N. (attached)
##Italian Admiral C. E. Morin, Commander Count Prasca, Lieutenant Lunghetti, Lieutenant Count Morano, Lieutenant Henderson. R.N. (attached)
##German Admiral his Royal Highness Prince Henry of Prussia, Captain Muller, Lieutenant von Spee, Sub-Lieutenant Wittman, Lieutenant Garforth, R.N. (attached)
##Japanese Admiral H.I.H. Prince Arizugawa, Captain Miura, Commander Tsuda, Lieutenant Stewart, R.N. (attached)
##Netherlands Admiral F. K. Englebrecht, Captain de Groot, Lieutenant Baron von Hardenbrock, Lieutenant Woolcombe, R.N. (attached)
##Norwegian Rear-Admiral von Krogh, Captain Muller, Lieutenant Petersen, Lieutenant Kerr Pearse, R.N. (attached)
##Portuguese Captain Barreto de Vascomellos, Captain de Cartillo, Lieutenant Trye, R.N. (attached)
##Russian Admiral Nicholas Skrydloff, Captain Domojiroff, Lieutenant Stetsenkoff, Lieutenant Twisleton Wykeham Fiennes, R.N. (attached)
##Spanish Admiral Don Segismundo Bermijo y Merelo, Captain Don Antonio Eulate y Fery, Lieutenant Don Juan Romero, Lieutenant Don Antonio Romero, Lieutenant Fair, R.N. (attached)
##Swedish Admiral A. F. H. Klintberg, Captain Ingelman, Commander Flack, Lieutenant Alton, R.N. (attached)
##United States Admiral J. N. Miller, Lieutenaut Richmond (attached)
##Captain de Mar E. Guerra
##Captain R. S. D. Cumins
#The Lord Lieutenant of Ireland and Countess Cadogan
#The Right Hon. the Speaker and Mrs. Gully, Miss Gully, and Miss Shelly Gully
#Cardinal Vaughan
#Right Hon. the Lord Mayor and Lady Mayoress, and Misses Faudel Phillips (2)
#The Gold Stick in Waiting, Silver Stick in Waiting, Silver Stick Adjutant in Waiting
#Officer Commanding 1st Life Guards and five officers
#Officer Commanding 2nd Life Guards and four officers
#Officer Commanding Royal Horse Guards and four officers
#Officer Commanding 2nd Dragoons and three officers
#Field Officer in Brigade Waiting, Adjutant in Brigade Waiting
#Commanding Officer Grenadier Guards
#Commanding Officer Coldstream Guards
#Commanding Officer Scots Guards, a Regimental Adjutant
#Commanding Officer 1st, 2nd, and 3rd Battalions Grenadier Guards and three officers of each Battalion
#Commanding Officer 1st and 2nd Battalions Coldstream Guards and three officers of each Battalion
#Commanding Officer 1st and 2nd Battalions of Scots Guards and three officers of each Battalion
#Commanding Officer Woolwich District and six officers
#Commanding Officer R.H.A. Home District and two officers
#Commanding Officer R.E. and four officers
#Commanding Officer 2nd Battalion Lincolnshire Regiment and three officers
#Commanding Officer Royal Marines (Chatham) and four officers
#Commanding Officer Royal Marines (Portsmouth) and two officers
#Four officers of the Honourable Corps of the Gentlemen at Arms
#Archbishops — Canterbury, York, Armagh, Ontario, Rupertsland
#Dukes and Duchesses
##The Duke and Duchess of [[Social Victorians/People/Argyll|Argyll]]
##The Duke and Duchess of [[Social Victorians/People/Abercorn|Abercorn]]
##The Duchess of De Baileu
##The Duke and Duchess of [[Social Victorians/People/Buccleuch|Buccleuch]]
##The Duchess of [[Social Victorians/People/Cleveland|Cleveland]]
##The Duke and Duchess of [[Social Victorians/People/Devonshire|Devonshire]]
##The Duchess of [[Social Victorians/People/Douglas-Hamilton Duke of Hamilton|Hamilton]]
##The Duke and Duchess of [[Social Victorians/People/Leeds|Leeds]]
##The Duke and Duchess of [[Social Victorians/People/Marlborough|Marlborough]]
##The Duke and Duchess of [[Social Victorians/People/Manchester|Manchester]]
##The Duke and Duchess of [[Social Victorians/People/Montrose|Montrose]]
##The Duke and Duchess of [[Social Victorians/People/Newcastle|Newcastle]]
##The Duke of [[Social Victorians/People/Norfolk|Norfolk]]
##The Duke of [[Social Victorians/People/Northumberland|Northumberland]]
##The Duke and Duchess of [[Social Victorians/People/Portland|Portland]]
##The Duke of [[Social Victorians/People/Richmond and Gordon|Richmond and Gordon]]
##The Duke and Duchess of [[Social Victorians/People/Roxburghe|Roxburghe]]
##The Duke and Duchess of [[Social Victorians/People/Somerset|Somerset]]
##The Duke and Duchess of [[Social Victorians/People/Sutherland|Sutherland]]
##The Duke and Duchess of St. Albans
##The Duke and Duchess of Wellington
##The Duchess of [[Social Victorians/People/Westminster|Westminster]]
#Marquises and Marchionesses
##The Marquis of Abergavenny
##The Marchioness of Ailesbury
##The Marquis and Marchioness of Ailsa
##The Marquis of Anglesey
##The Marquis and Marchioness of [[Social Victorians/People/Breadalbane|Breadalbane]]
##The Marchioness of [[Social Victorians/People/Marlborough#Marchioness of Blandford|Blandford]]
##The Marquis and Marchioness of Bristol
##The Marquis of [[Social Victorians/People/Camden|Camden]]
##The Marquis and Marchioness of Conyngham
##Dowager [Marchioness of] Conyngham
##The Marchioness of Cassar de Sai[n]
##The Marquis and Marchioness of Cholmondeley
##The Marquis of D'Auerstadt
##The Marquis and Marchioness [[Social Victorians/People/Stonor|D'Hautpoul]]
##The Marquis and Marchioness of Downshire
##Dowager [Marchioness of] Downshire
##The Marquis and Marchioness of [[Social Victorians/People/Hamilton Temple Blackwood|Dufferin and Ava]]
##The Marquis and Marchioness of [[Social Victorians/People/Exeter|Exeter]]
##The Marquis and Marchioness of Granby
##The Marchioness of [[Social Victorians/People/Florence Rawdon-Hastings Chetwynd|Hastings]]
##The Marquis and Marchioness of [[Social Victorians/People/Bective|Headfort]]
##The Marquis and Marchioness of Hertford
##The Marquis and Marchioness of Huntly
##The Marquis and Marchioness of [[Social Victorians/People/Abercorn#James Hamilton, Marquess of Hamilton|Hamilton]]
##The Marquis and Marchioness of [[Social Victorians/People/Lansdowne|Lansdowne]]
##The Marquis and Marchioness of Lothian
##Dowager (Marchioness of) [[Social Victorians/People/Londonderry|Londonderry]]
##The Marquis and Marchioness of [[Social Victorians/People/Londonderry|Londonderry]]
##The Marquis and Marchioness of [[Social Victorians/People/Ormonde|Ormonde]]
##The Marchioness of [[Social Victorians/People/Queensberry|Queensberry]]
##The Marquis and Marchioness of [[Social Victorians/People/Ripon|Ripon]]
##The Marquis and Marchioness of [[Social Victorians/People/Salisbury|Salisbury]]
##The Marquis and Marchioness of [[Social Victorians/People/Tweeddale|Tweeddale]]
##Dowager (Marchioness of) [[Social Victorians/People/Tweeddale|Tweeddale]]
##John Stewart-Murray, [[Social Victorians/People/Atholl|Marquess of Tullibardine]]
##Lawrence, [[Social Victorians/People/Zetland|Marquess of Zetland]] and Lilian, [[Social Victorians/People/Zetland|Marchioness of Zetland]]
#Earls and Countesses 
##Countess of Aberdeen and Dowager Countess of Aberdeen
##Earl and Countess of Albemarle and Dowager Countess of Albemarle
##Earl and Countess of Ancaster
##Earl and Countess of Amherst
##Earl of Ava
##Earl and Countess of Antrim
##Earl and Countess of Aylesford
##Earl and Countess of Annesley
##Earl and Countess of Airlie
##Earl and Countess of Arran
##Earl of Aberdeen
##Earl and Countess of Bandon
##Countess of Bantry
##Earl and Countess of Beauchamp
##Earl and Countess of Bathurst and Dowager Countess of Bathurst
##Countess of Bective
##Earl and Countess of Belmore
##Earl of Bradford
##Countess of Bremer
##Earl and Countess of Brownlow
##Earl and Countess of Buckinghamshire
##Earl of Burford
##Earl and Countess of Cairns
##Earl and Countess of Caledon
##Earl of Camperdown
##Earl of Cardigan
##Earl and Countess of Carnarvon and Dowager Countess of Carnarvon
##Earl of Carnwath
##Earl and Countess of Carrington
##Earl and Countess of Carysfort
##Earl and Countess of Castlestuart
##Earl and Countess of Cathcart
##Earl and Countess of Cavan
##Earl and Countess of Chesterfield
##Earl and Countess of Chichester
##Dowager Countess of Clancarty
##Countess of Clanwilliam
##Earl and Countess of Compton
##Countess of Cottenham
##Earl of Courtown
##Earl and Countess of Cowper
##Earl and Countess of Cranbrook
##Earl and Countess of Craven and Dowager Countess of Craven
##Earl and Countess of Crawford
##Earl of Crewe
##Earl and Countess of Cork and Orrery
##Earl and Countess of Coventry
##Countess of Cromartie and Dowager Countess of Cromartie
##Earl and Countess of Dalkeith
##Earl and Countess of Dartmouth
##Earl and Countess of De Grey
##Dowager Countess of De La Warr
##Earl and Countess of Denbigh
##Earl and Countess of Derby
##Earl and Countess of Donoughmore
##Earl and Countess of Drogheda
##Earl of Ducie
##Earl and Countess of Dudley and Dowager Countess of Dudley
##Earl and Countess of Dundonald
##Earl and Countess of Dunmore
##Earl and Countess of Dunraven
##Earl of Durham
##Earl and Countess of Eglinton and Winton
##Earl of Eldon
##Earl and Countess of Ellesinere
##Earl and Countess of Enniskillen
##Earl and Countess of Erne
##Earl and Countess of Errol
##Earl and Countess of Essex and Dowager Countess of Erroll
##Earl of Euston
##Earl and Countess of Feversham
##Earl and Countess of Fingall
##Earl of Fortescue
##Earl and Countess of Gainsborough
##Earl and Countess of Galloway
##Earl and Countess of Glasgow
##Countess of Gosford
##Earl and Countess of Granard
##Countess of Granville
##Earl and Countess of Grey
##Countess of Grosvenor
##Countess of Guilford
##Earl and Countess of Harewood and Dowager Countess of Harewood
##Earl and Countess of Harrington
##Earl and Countess of Hopetoun
##Earl and Countess of Huntingdon
##Earl and Countess of Harrowby
##Countess of Hohenau
##Countess of Howe
##Earl and Countess of Iddesleigh
##Earl and Countess of Jersey
##Earl and Countess of Kenmare
##Earl of Kerry
##Earl and Countess of Kilmorey
##Earl of Kimberley
##Earl and Countess of Kingston
##Earl of Kinnoull
##Josephine, Countess Kinsky
##Earl and Countess of Kintore
##Countess of Leitrim
##Earl and Countess of Lanesborough
##Countess of Lathom
##Earl and Countess of Lauderdale
##Countess of Leicester
##Earl and Countess of Leven and Melville
##Earl and Countess of Lichfield
##Earl and Countess of Limerick
##Earl and Countess of Lindsay
##Earl and Countess of Lisburne
##Earl and Countess of Listowel
##Earl and Countess of Londesborough
##Earl and Countess of Longford
##Earl and Countess of Lonsdale and Dowager Countess of Lonsdale
##Earl and Countess of Loudoun
##Earl and Countess of Lovelace
##Earl and Countess of Lucan
##Countess of Lytton
##Countess of Macclesfield
##Earl and Countess of Malmesbury and Dowager Countess of Malmesbury
##Earl and Countess of Mar
##Earl and Countess of Mar and Kellie and Dowager Countess of Mar and Kellie
##Earl and Countess of Mayo and Dowager Countess of Mayo
##Countess of Meath
##Countess of Metaxas
##Earl and Countess of Mexborough
##Earl and Countess of Minto
##Earl of De Montalt
##Earl and Countess of Morley
##Earl and Countess of Morton and Dowager Countess of Morton
##Earl of Nelson
##Earl and Countess of Norbury
##Earl of Northbrook
##Earl and Countess of Northesk and Dowager Countess of Northesk
##Earl and Countess of Onslow
##Earl of Orford
##Countess of Oxford
##Earl and Countess of Pembroke
##Countess of Percy
##Earl and Countess of Portarlington
##Earl and Countess of Portsmouth
##Earl and Countess of Powis
##Earl and Countess of Radnor
##Earl and Countess of Ravensworth
##Earl and Countess of Roden
##Earl and Countess of Romney
##Lawrence, [[Social Victorians/People/Zetland|Earl of Ronaldshay]]
##Earl of Rosebery
##Earl and Countess of Rosse
##Earl and Countess of Rosslyn and Dowager Countess of Rosslyn
##Earl of Sandwich
##Earl of Scarbrough
##Earl and Countess of Selborne
##Countess of Selkirk
##Countess of Shaftesbury
##Dowager Countess of Shrewsbury and Talbot
##Earl and Countess of Spencer
##Earl and Countess of Stamford
##Earl and Countess of Stanhope
##Earl and Countess of St. Germans
##Earl of Stradbroke
##Earl of Strafford
##Earl and Countess of Suffolk and Berkshire
##Earl and Countess of Temple (of Stowe)
##Earl and Countess of Verulam
##Earl and Countess of Waldegrave
##Earl and Countess of Warwick
##Earl and Countess of Westmeath
##Earl and Countess of Wharncliffe
##Elizabeth, Dowager Countess of Wilton and Isabella, Dowager Countess of Wilton
##Earl and Countess of Winchilsea and Nottingham
##Earl and Countess of Winterton
##Earl and Countess of Yarborough and Dowager Countess of Yarborough
#Viscounts&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 3c / Col. 4a) and Viscountesses
##Viscount and Viscountess of Boyne
##Viscountess of Cantelupe
##Viscount and Viscountess of Castlerosse
##Viscount and Viscountess of Chelsea
##Viscount and Viscountess of Chetwynd
##Viscountess of Chewton
##Viscount and Viscountess of Clifden
##Viscount and Viscountess of Cobham
##Viscount and Viscountess of Coke
##Viscount of Corry
##Viscount and Viscountess of Cranborne
##Viscount of Crichton
##Viscount and Viscountess of Cross
##Viscount of Curzon
##Viscount and Viscountess of Dalrymple
##Viscount and Viscountess of Deerhurst
##Viscount and Viscountess of De Vesci
##Viscount and Viscountess of Dillon
##Viscount of Doneraile
##Viscount and Viscountess of Duncannon
##Viscount of Dungarvan
##Viscount and Viscountess of Ebrington
##Viscount and Viscountess of Emlyn
##Viscount of Encombe
##Viscount and Viscountess of Exmouth
##Viscount and Viscountess of Falkland
##Viscount and Viscountess of Falmouth
##Viscount of Fitz Harris
##Viscount and Viscountess of Folkestone
##Viscount and Viscountess of Frankfort de Montmorency
##Viscount and Viscountess of Gage
##Viscount and Viscountess of Galway
##Viscount and Viscountess of Garnock
##Viscount and Viscountess of Gough
##Viscount of Gort
##Viscount and Viscountess of Halifax
##Viscount and Viscountess of Hardinge
##Viscount of Harrington
##Viscount and Viscountess of Hood
##Viscount and Viscountess of Kilcoursie
##Viscount and Viscountess of Knutsford
##Viscount and Viscountess of Lifford
##Viscount of Llandaff
##Viscount and Viscountess of Maitland
##Viscount and Viscountess of Marsham
##Viscount and Viscountess of Massereene and Ferrard
##Viscount and Viscountess of Melville
##Viscount and Viscountess of Midleton
##Viscount and Viscountess of Milton
##Viscount and Viscountess of Monck
##Viscount and Viscountess of Morpeth
##Dowager Viscountess of Mountmorres
##Viscount and Viscountess of Newark
##Viscount and Viscountess of Newport
##Viscount and Viscountess of Oxenbridge
##Viscount of Parker
##Viscount of Peel
##Viscount and Viscountess of Portman
##Viscount and Viscountess of Powerscourt
##Viscount and Viscountess of Raincliffe
##Viscountess of Sherbrooke
##Viscount of Sidmouth
##Viscount of St. Cyres
##Viscount of Southwell
##Viscount of Suirdale
##Viscount and Viscountess of Templetown
##Viscountess of Torrington
##Viscount and Viscountess of Trafalgar
##Viscount and Viscountess of Valentia
##Viscount of Valletort
##Viscount of Villiers
##Viscountess of Wolseley
#Bishops — Auckland, Barry, Bath and Wells, British Colombia, Chichester, Durham, Ely, Exeter, Gloucester and Bristol, Gibraltar, Hereford, London, Lichfield, Lincoln, Manchester, Newcastle, Norwich, Oxford, Peterborough, Rochester, Ripon, Stepney, Southwark, St. Albans, Salisbury, Sodor and Man, Southwell, Sydney, Sierra Leone, Worcester, Winchester, Wellington
#Baronesses — Burdett-Coutts, Macdonald
#Lords and Ladies&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 4b / Col. 5a) — 
##Lord and Lady Abercromby
##Lord and Lady Aberdare
##Lord Aberdour
##Lady Abinger
##Lady Alexandra Acheson
##Lady Adam
##Lady Adderley
##Lord and Lady Addington
##Lady Adye
##Lady Agnew
##Lady Alderson
##Lord and Lady Alington
##Lady Alison
##Lady Mildred Allsopp
##Lord and Lady Amherst of Hackney
##Lady Heathcoat Amory
##Lord and Lady Ampthill
##Lady Agnes Anderson
##Lady Bertha Anson
##Lady Arbuthnot
##Lady Alice Archer Houblon
##Lord Ardee
##Lord and Lady Ardilaun
##Lady Armstrong
##Lady Arnold
##Lady Arnott
##Lord and Lady Ashbourne
##Lord and Lady Ashburton and Dowager Ashburton
##Lord and Lady Ashcombe
##Lady Alice Ashley
##Lady Edith Ashley
##Lady Ashmead-Bartlett
##Lord and Lady Ashton
##Lord and Lady Ashtown
##Lady Florence Astley
##Lady Gertrude Astley-Corbett
##Lady Austin
##Lord Bagot
##Lady Bailey
##Lady Blanche Baillie
##Lady Baird
##Lady Baker
##Lord Balcarres
##Lord and Lady Balfour of Burleigh, Lady Nina Balfour and Lady Betty Balfour
##Lord Balvaird
##Lord Bangor
##Dowager Lady Barclay
##Lord and Lady Barnard
##Lady Florence Barnardiston
##Lady Constance Barne
##Lady Barran
##Lady Barrington
##Lord and Lady Basing
##Lord and Lady Bateman
##Lady Evelyn Bathurst
##Lord and Lady Battersea
##Lady Steuart Bayley
##Lady Violet Beauchamp
##Lord Osborne Beauclerk and Lady Beauclerk (2)
##Lady A. Beaumont
##Lady Bedford
##Lord and Lady Belhaven and Stenton and Dowager Belhaven and Stenton
##Lord and Lady Bellew and Dowager Bellew
##Lord and Lady Belper
##Lady Charles Beresford
##Lady William Beresford (Lilian Duchess of Marlborough)
##Lady Bergne
##Lord and Lady Bertie and Lady Elizabeth Bertie
##Lady Biddulph, Lady Elizabeth Biddulph and Lady Wilfreda Biddulph
##Lady Bigge
##Lord and Lady Bingham
##Lord and Lady Binning
##Lord Blackwood, Lord Basil Blackwood. Lady Hermione Blackwood and Lord Terence Blackwood
##Lady Bloomfield
##Lady Blythswood
##Lord and Lady Bolton
##Lady Maud Bootle-Wilbraham, Lady Bertha Bootle-Wilbraham and Lady Edith Bootle-Wilbraham
##Lord Borthwick
##Lady Margaret Boscawen
##Lord and Lady Boston
##Lady Boughey
##Lady Albreda Bourke and Lady Florence Bourke
##Lady Bowen
##Lady Bower
##Lady Muriel Boyle and Lady Boyle (2)
##Lady Mary Brabazon
##Lady Brackenbury
##Lady Braddon
##Lady Bramwell
##Lady Bramston
##Lord Brassey, Lady Idina Brassey and Lady Violet Brassey
##Lord and Lady Braye
##Lady Mary Bridgeman
##Lady Eleanor Brodie
##Lady Hilda Brodrick
##Lady De Capel Brooke and Dowager Brooke
##Lady Cunliffe Brooks
##Lord and Lady Brougham and Vaux
##Lord and Lady Ulick Browne, Lady Browne and Lady Crichton Browne
##Lady Brownlow
##Lord and Lady F. Brudenell-Bruce
##Lady Brunner
##Dowager Buchanan-Riddeil
##Lady Audrey Buller
##Lady Burdett
##Lord and Lady Burghclere
##Lord Burghley
##Lady Agnes Burne
##Lady Burrell
##Lord and Lady Burton
##Lady Butler and Lady Butler (2)
##Lord and Lady Arthur Butter
##Lady Buxton and Lady Victoria Buxton
##Lady Susan Byng
##Lord and Calthorpe
##Lady C. Cameron and Lady Margaret Cameron
##Lord and Lady Archibald Campbell and Lady A. Campbell
##Lord and Lady George Campbell
##Lady Campbell-Bannerman
##Lord and Lady Camoys
##Lord and Lady Carbery and Dowager Carbery
##Lady Carbutt
##Lady Cardon
##Lord and Lady Cardross
##Lord and Lady Carew
##Lady Carmichael
##Lord and Lady Carnegie
##Lord and Lady Castlemaine
##Lord and Lady Castletown
##Lady Eva Cathcart and Lady R. Cathcart
##Lady Frederick Cavendish, Lady Myra Cavendish, Lady Evelyn Cavendish and Lady Harriet Cavendish
##Lord Charles Cavendish-Bentinck, Lord and Lady Henry Cavendish-Bentinck, Lord William Cavendish-Bentinck, Lady Ottoline Cavendish-Bentinck
##Lord and Eustace Cecil, Lord Hugh Cecil, Lord and John Cecil, Lord and Edward Cecil, Lord and Lady Robert Cecil, Lord W. Cecil, Lady Gwendolen Cecil, Lady Florence Cecil, Lady William Cecil, Lady Louisa Cecil
##Lady Francis Cecil-Dallas
##Lady Chamberlain
##Lady Chelmsford
##Lord and Lady Chesham
##Lady Chetwode
##Lord Cheylesmore
##Lord and Lady Fitzwarine Chichester
##Lady Chitty
##Lady Cholmeley
##Lady Henry Cholmondeley
##Lady Clements (2)
##Lady Churchill, Lady Randolph Churchill, Dowager Churchill, Lady Spencer Churchill (2)
##Lord Edward Spencer-Churchill, Lady Alfred Spencer-Churchill
##Lord and Lady Churston
##Lord and Lady Clifford of Chudleigh
##Lady Marshal Clarke, Lady E. Clarke
##Lady Isabel Clayton
##Lord and Lady Clinton
##Lord and Lady Clonbrock
##Lord Cloncurry
##Lady Muriel Close
##Lady Evelyn Cobbold
##Lady Cochrane, Lady Gertrude Cochrane, Lady Adela Cochrane
##Lady Coddington
##Lady Mabel Coke
##Lord and Lady Colchester
##Lady Cole (2)
##Lady Colebrooke
##Lord and Lady Coleridge
##Lady Collins
##Lady Colomb
##Lady Colvile, Lady Colville
##Lord and Lady Colville of Culross
##Lady Jane Seymour Combe, Lady Constance Combe
##Lady Commerell
##Lord and Lady Alwyne Compton
##Lady Dowager Congleton
##Lord and Lady Connemara
##Lady Conyers
##Lady Blanche Conyngham
##Lady Cooper
##Lady Evelyn Cotterell
##Lord and Lady Cottesloe
##Lady Couch
##Lord and Lady Courtenay
##Lady Coventry (2)
##Lady Cowell
##Lady Helen Craven
##Lord and Lady Crawshaw
##Lady Evelyn Crichton, Lady Emma Crichton
##Lord Crofton
##Lady Cromer
##Lady Mary Crosse
##Lady Crossley
##Lady Mary Cuffe
##Lady Culme-Seymour
##Lady Cunliffe
##Lady Georgiana Curzon
##Lady Elizabeth Cust
##Lady Ida Dalzell
##Lady Mary Dashwood
##Lord and Lady Davey
##Lady Victoria Dawnay, Lady Evelyn Dawnay, Lady Adelaide Dawnay
##Lady Decies
##Lord and Lady De Freyne
##Lord and Lady De L’Isle and Dudley
##Lord De Manley
##Lady Mildred Denison, Lady Elinor Denison
##Lord Deramore
##Lord and Lady De Ramsey
##Lady Dering
##Lady De Ross
##Lord and Lady De Saumarez
##Lady Des Voeux
##Lady De Trafford, Lady Agnes De Trafford
##Lady De Winton
##Lord and Lady Digby
##Lady Dorchester
##Lady Dorington
##Lady Margaret Douglas, Lady Edith Douglas
##Lady H. Douglas-Hamilton
##Lady Dowell
##Lady Drummond, Lady Edith Drummond
##Lady Du Cane
##Lady Duckworth
##Lady Eva Dugdale
##Lord Dunally
##Lady Florence Duncombe, Lady Ulrica Duncombe, Lady Caroline Duncombe
##Lady Alice Dundas
##Lord and Lady Dunleath
##Lord Dunglass
##Lady Dunn
##Lord Dunsandle and Clanconal
##Lady Durand
##Lord Dynevor
##Lord Ebury
##Lady Edmonstone
##Lady Edwards, Lady J. B. Edwards, Lady Blanche Edwards
##Lady Ernestine Edgcumbe
##Lady Egerton (2)
##Lord Egerton of Tatton
##Lady Grey-Egerton
##Lord and Lady Elcho
##Lord and Lady Elibank
##Lady Ellenborough
##Lady Ellis
##Lord and Lady Elphinstone
##Lady Winifred Cary-Elwes
##Lady Engleheart
##Lord Erskine, Lady Erskine (2), Lady Horatia Erskine, Lady Erskine
##Lord and Lady Esher
##Lady Evans
##Lady Evelyn Ewart, Lady Mary Ewart
##Lady Evelyn Eyre
##Lady Fairbairn
##Lady Fairfax
##Lady Anne Fane, Lady Augusta Fane
##Lady Farquhar
##Lord and Lady Farrer
##Lady Fayrer
##Lady Louisa Feilding
##Lady Helen Munro Ferguson
##Lady Fergusson
##Lady Ffolkes
##Lady Finlay
##Lady Fisher
##Lady Dorothea Fitz-Clarence, Lady Maria Fitz-Clarence, Lady Dorothy Fitzclarence
##Lord and Lady Henry Fitz-Gerald, Lady B. Fitz Gerald, Lady M. FitzGerald, Lord Seymour Fitz-Gerald
##Lady Beatrix Fitzmaurice
##Lord and Lady F. FitzRoy, Lady C. Fitz-Roy
##Lady Mary Fitzwilliam
##Lady FitzWygram
##Lady Fletcher
##Lady Flower, Lady Flower
##Lord Foley, Lady Mary Foley
##Lady Gertrude Foljambe
##Lady Angela Forbes, Lady Forbes (2), Dowager Helen Forbes
##Lord and Lady Forester
##Lady Forrest
##Lady Susan Fortescue
##Lady Forwood
##Lady Foster
##Lady Fowler
##Lady Edith Franklin
##Lady Fremantle, Lady Fremantle
##Lady Frere
##Lady Fulton
##Lady Gardiner, Lady Lynedoch Gardiner
##Lord Garioch
##Lady Galton
##Lady Katharine Gathorne-Hardy
##Lady Garvagh
##Lord and Lady Gerard
##Lady Gilbey
##Lady Gillford
##Lady Susan Gilmour
##Lady Gipps
##Lord and Lady Glamis
##Lord and Lady Glenesk
##Lady Glyn, Lady Mary Carr Glyn
##Lady D'Arcy Godolphin-Osborne
##Lady Gordon
##Lady Margaret Ormsby Gore, Lady Constance Gore
##Lady Gore Langton (2)
##Lord Walter Gordon-Lennox, Lord Algernon Gordon-Lennox
##Lady Evelyn Goschen
##Lord R. S. Gower
##Lady Graham, Lady Margaret Graham, Lady Helen Graham
##Lady Charlotte Graham-Toler
##Lady Grant, Lady Florence Grant
##Lady Grant-Duff
##Lady Green
##Lord Greenock
##Lady Grenfell
##Lady Frances Gresley
##Lady Victoria Grey, Lady Grey
##Lady Jane Grey-Trefusis
##Lady Griffin
##Lady Helen Grimston
##Lord and Lady Arthur Grosvenor, Lady Grosvenor (2)
##Lady Gull
##Lady Haldon
##Lady Haliburton
##Lady Basil Hall
##Lady Halle
##Lord and Lady Halsbury
##Lord and Lady E. Hamilton, Lord F. Hamilton, Lady F. Douglas Hamilton, Lady Alexandra Hamilton, Lady Baillie Hamilton (2), Lady C. Hamilton, Lady Victoria Hamilton, Lady George Hamilton
##Lady Hanson
##Lady Harcourt
##Lady Cicely Hardy, Lady Hardy
##Lady Beatrice Hare
##Lord Harlech
##Lady Constance Harris, Lady Harris
##Lady Harrison, Lady Harriet Harrison
##Lady Hart
##Lady Emily Hart-Dyke
##Lady Dixon-Hartland
##Lady Hartopp
##Lord and Lady Hastings
##Lord and Lady Hatherton
##Lady Alice Havelock-Allan
##Lady Hawke
##Lord and Lady Hawkesbury
##Lady John Hay, Lady Hay
##Lady Blanche Haygarth
##Lady Hayter
##Lady Hely-Hutchinson (2)
##Lady Hemming
##Lord and Lady Heneage
##Lord and Lady Henley
##Lord Henniker
##Lady Beatrix Herbert, Lady Herbert (2)
##Lord and Lady Herries
##Lord and Lady Herschell
##Lord Francis Hervey, Lady Augustus Hervey
##Lady Hervey-Bathurst
##Lady Fermor Hesketh
##Lady Hibbert
##Lady Lucy Hicks-Beach
##Lord and Lady Arthur Hill, Lady Clement Hill, Lady Stock Hill
##Lord and Lady Hillingdon
##Lord and Lady Hindlip
##Lord and Lady Hobhouse
##Lady Norah Hodgson
##Lady Holdich
##Lady Mary Holland
##Lady Beatrix Douglas Home
##Lady Maria Hood
##Lady Hood of Avalon
##Lady Hooker
##Lady Mary Hope
##Lady Hoskins
##Lord and Lady Hotham
##Lord and Lady Hothfield
##Lady Houldsworth
##Lady Eleanor Howard, Lady Agnes Howard, Lady Howard (2), Lady Mabel Howard, Lady Rachel Howard
##Lord and Lady Howard of Glossop
##Lady Howarth
##Lady Mary Hozier
##Lady Florentia Hughes
##Lady Seager Hunt
##Lady Hunter
##Lord Hyde
##Lady Hylton
##Lord and Lady Inchiquin
##Lord Inverurie
##Lord and Lady Iveagh
##Lady Jackson
##Lord James of Hereford
##Lady Margaret Jenkins, Lady Jenkins
##Lady Jenner
##Lady Jephson
##Dowager Jessel, Lady Jessell
##Lady Jeune
##Lady Hill Johnes
##Lady Joicey
##Lady Alice Jolliffe
##Lady Burn Jones
##Lady Caroline Lister Kaye, Lady Beatrice Lister Kaye, Lady Lister Kaye
##Lady Isabella Keane
##Lady Keith-Falconer (2)
##Lord and Lady Kelvin
##Lady Kemball
##Lady Beatrice Kemp
##Lady Kennard
##Lady Kennaway
##Lady Aline Kennedy
##Lady Kennett-Barrington
##Lord Kenyon
##Lady Mabel Kenyon-Slaney
##Lord Kensington
##Lady Mary Stuart Keppel
##Lady Innes-Ker (2)
##Lady Kerr (2)
##Lord Kilmarnock
##Lady King
##Lady Florence King King
##Lady Emily Kingscote
##Lady Edith King-Tenison
##Lord and Lady Kinnaird
##Lady Kitson
##Lady Laking
##Lady Frances Lambart, Lady Ellen Lambart
##Lady Victoria Lambton
##Lady Adela Larking
##Lady Isabel Larnach
##Lady Mary Lascelles
##Lord and Lady Lawrence
##Lady Lawson
##Lord and Lady Leconfield
##Lady Elliott Lees, Lady Lees
##Lady Leese
##Lady Legard
##Lord and Lady Leigh
##Lady Henry Gordon-Lennox, Lady Walter Gordon-Lennox, Lady Algernon Gordon-Lennox, Lady Caroline Gordon-Lennox
##Lady Katharine Le Poer Trench
##Lady Constance Leslie
##Lady Susan Leslie-Melville
##Lady Lewis
##Lady Lilian Liddell
##Lady Lindley
##Lady Harriet Lindsay, Lady Jane Lindsay, Lady Jane Lindsay
##Lord and Lady Lingen
##Lord and Lady Lister
##Lady Gwendolen Little
##Lady Margaret Littleton
##Lord and Lady Llangattock
##Lady Llewelyn
##Lord and Lady Loch
##Lady Lockwood
##Lady Louise Loder
##Lady Catherine Loftus
##Lady Doreen Long
##Lady Longley
##Lady Albertha Lopes
##Lady Loraine
##Lord and Lady Lovat
##Lady Drury Lowe, Lady Lucy Drury Lowe
##Lady Lowry-Corry (2)
##Lady Mary Loyd
##Lady Lubbock
##Lord and Lady Lurgan and Dowager Lurgan
##Lady Lyall
##Lady Lyell
##Lady Mary Lygon
##Lady Lyons
##Lady Lysons
##Lady Lyttelton
##Lady Emily Lytton
##Lady MacCormac
##Lord and Lady Macdonald
##Lady Macgregor, Lady MacGregor, Lady Helen MacGregor
##Lady Mackenzie, Lady Mackenzie
##Lady Mackworth
##Lady Maclean
##Lord and Lady Macnaghten
##Lady Macpherson-Grant
##Lady Caroline Madden, Lady Madden
##Lady Louisa Magenis
##Lady Magheramorne, Dowager Magheramorne
##Lady Nora Maitland
##Lady Margaret Crichton-Maitland
##Lady Margaret Majendie
##Lord Cecil Manners, Lord Edward Manners, Lord Manners, Lady Victoria Manners, Lady Manners
##Lady Blundell Maple
##Lady Mappin
##Lady Marjoribanks
##Lady Markham
##Lady Marriott
##Lady Martin, Lady Martin
##Lady Evelyn Mason
##Lady Maude (2)
##Lady H. Maxwell, Lady Maxwell, Lady Maxwell, Lady Maxwell
##Lady Heron-Maxwell
##Lady M'Clintock
##Lady Evelyn M'Donnell
##Lady Meade (2)
##Lord and Lady Medway
##Lady Methuen
##Lady Meysey-Thompson
##Lord and Lady Middleton, Lady Middleton
##Lady Mary Milbanke
##Lady Miller
##Lady Milner
##Lady Clementina Mitford
##Lady Lady M'lver
##Lady Hilda M'Neile
##Lady Monckton
##Lord Moncreiff, Lady Scott Moncrieff
##Lady Moncreiffe
##Lord and Lady Monkswell
##Lady Monson
##Lord Charles Montagu, Lady Cecil Scott Montagu, Lady S. Montagu, Lady Agneta Montagu
##Lord Montagu of Beaulieu
##Lord and Lady Monteagle
##Lady Edith Montgomerie, Lady Sophia Montgomerie
##Lady Charlotte Montgomery
##Lady More-Molyneux
##Lord and Lady Moreton
##Lady Morgan
##Lord and Lady Morris
##Lady Blanche Morris
##Lady Mary Morrison
##Lady Moseley
##Lord and Lady Mostyn
##Lord and Lady Mowbray and Stourton, Dowager Mowbray and Stourton, Lady Mowbray
##Lord and Lady Muncaster
##Lady Anne Murray
##Lady Murray (2)
##Lady Georgiana Mure, Lady Georgiana Mure [sic]
##Lord and Lady Napier and Ettrick
##Lord and Lady Napier of Magdala and Dowager Napier of Magdala
##Lady Naylor-Leyland
##Lady Nelson
##Lord and Lady Henry Nevill
##Lord and Lady Newton
##Lord and Lady Newtown-Butler
##Lady Nicolson
##Lady Augusta Noel, Lady Agnes Noel
##Lady Norman
##Lord and Lady Norreys
##Lord and Lady North, Lady Muriel North
##Lady Northcote, Lady Northcote (2)
##Lord Norton
##Lady Elizabeth Nugent
##Lady O'Brien, Lady O'Brien [sic]
##Lady O'Hagan
##Lady Olpherts
##Lord and Lady O'Neill
##Lady Gwendoline O'Shee
##Princep [sic] Alice Packe
##Lord and Lady Berkeley Paget
##Lady Alfred Paget
##Lady Paget of Cranmore
##Lady Katherine Pakenham
##Lady Palgrave
##Lady Sophia Palmer, Lady Palmer
##Lady Evelyn Parker
##Lady Parratt
##Lady Maude Parry
##Lady Muriel Parsons
##Lord and Lady Pearson, Lady Pearson
##Lady Peel, Lady Georgiana Peel
##Lady Constance Childe-Pemberton
##Lord and Lady Penrhyn
##Lady Mary Pepys
##Lady Perceval
##Lady Percy (2)
##Lady Petre
##Dowager Lady Peyton
##Lady Phillimore
##Lady William Phipps
##Lord and Lady Pirbright
##Lord and Lady Playfair
##Lady Chichele Plowden
##Lady Anna Chandos-Pole
##Lady Pollock
##Lord and Lady Poltimore
##Lady Pontifex
##Lady Alice Portal
##Lady Powell, Lady Powell [sic]
##Lady Baden-Powell
##Lady Dickson-Poynder
##Lady Poynter
##Lord and Lady George Pratt
##Lady Priestley
##Lady Probyn
##Lady Eva Wyndham-Quin, Lady Wyndham-Quin (2)
##Lord and Lady Raglan, Dowager Raglan
##Lady Ramsay
##Lord and Lady Rathdonnell
##Lady Rathmore
##Lord and Lady Rayleigh, Dowager Rayleigh
##Lord and Lady Reay
##Lady Reid
##Lord and Lady Rendel
##Lord Rendlesham
##Lady Jane Repton
##Lord Revelstoke
##Lord and Lady Ribblesdale
##Lady Laura Ridding
##Lord and Lady Robartes
##Lady O. Roberts
##Lady Roberts of Kandahar
##Lady Robinson
##Lord and Lady Rodney
##Lord Romilly
##Lord and Lady Rookwood
##Lord and Lady Rossmore
##Lord Rowton
##Lady Roxburgh
##Lord and Lady Rothschild
##Lady Victoria Russell, Lady Arthur Russell, Lady G. Russell, Lady W. H. Russell, Lady Alexander Russell
##Lord and Lady Russell of Killowen
##Lord and Lady Ruthven
##Lady Jane Ryan
##Lady Mary Sackville
##Lady Salmon
##Lord and Lady Saltoun
##Lady Samuelson, Lady S. Samuel
##Lady Mary Saurin
##Lord and Lady Savile, Lady Marie Savile
##Lady Savory
##Lord George Scott, Lord Henry Scott, Lord Herbert Scott, Lady Sophie Scott, Lady Charles Scott, Lady Louisa Scott, Lady Scott (2)
##Lord and Lady Seaton
##Lord and Lady Settrington
##Lady Seymour, Lady Albert Seymour, Lady William Seymour, Lady Seymour (2)
##Lord and Lady Shand
##Lady Shaw
##Lady Constance Shaw-Lefevre
##Lady Octavia Shaw-Stewart, Lady Alice Shaw-Stewart
##Lady Mary Shelley
##Lord and Lady Sherborne
##Lady Shippard
##Lady Shute
##Lady Kay-Shuttleworth
##Lady Simeon
##Lady Simmons
##Lady Simpson of Windsor
##Lord and Lady Sinclair
##Lord and Lady Skelmersdale
##Lady Esther Smith, Lady Barbara Smith, Lady Smith, Lady Blanche Smith, Lady Sybil Smith, Lady Euan Smith, Lady D. Smith
##Lady Smyth
##Lady Catherine Somerset, Lady Geraldine Somerset, Lady Henry Somerset
##Lord and Lady Southampton, Dowager Southampton
##Lady Edward Spencer-Churchill
##Lady Margaret Spicer
##Lady Sprigg
##Lady Stafford
##Lord Stalbridge
##Lady Stanhope (2)
##Lord Stanmore
##Lord Stanley, Lady Alice Stanley, Lady Isobel Stanley
##Lady Stansfield
##Lord Stavordale
##Lady Stephenson
##Lady Stevenson
##Lady Helen Stewart, Lady Mary Stewart, Lady Mark Stewart, Lady Stewart, Lady Houston Stewart, Lady Stewart [sic], Lady Isabel Stewart
##Lady Stewart of Grantully
##Lady Edith St. Aubyn
##Lord and Lady St. Levan
##Lady St. Leonards
##Lord and Lady St. Oswald
##Lady Stone
##Lady Charlotte Stopford
##Lord and Lady Stratheden and Campbell
##Lady Mary Stuart-Richardson
##Lord Suffield
##Lady Sutherland
##Lady Evelyn Sutton, Lady Susan Sutton
##Lord and Lady Swansea
##Lady Swinnerton Dyer
##Lady Kathleen Swinnerton-Pilkington
##Lord and Lady E. Talbot, Lady Emma Talbot
##Lady Jane Taylor
##Lady Taylour (2)
##Lady Tatton Sykes
##Lord Herbert Vane-Tempest, Lord Henry Vane-Tempest
##Lord and Lady Templemore
##Lady Tennant
##Lord and Lady Tennyson
##Lady Tenterden
##Lord Tewkesbury
##Lord and Lady Teynham
##Lord and Lady Thring
##Lady E. Thornton
##Lady Thursby
##Lady Ulrica Thynne
##Lord and Lady Tollemache
##Lady Agnes Townshend
##Lady Mary Trefusis
##Lady Tredegar
##Lady Trevelyan, Lady Trevelyan [sic]
##Lord and Lady Trevor
##Lady Troubridge
##Lady Turner
##Lady Henrietta Turnor
##Lady Tuson
##Lord and Lady Tweedmouth
##Lady Tyler
##Lady Emily Van De Weyer
##Lady Jane Van Koughnet
##Lord and Lady Ventry
##Lady Villiers (2), Lady Edith Villiers
##Lady Howard Vincent, Lady Helen Vincent, Lady Vincent
##Lady Vivian, Lady Jane Vivian
##Lady Mary Waldegrave
##Lady F. F. Walker, Lady James Walker
##Lady Walrond
##Lady Clementine Walsh
##Lord Wandsworth
##Lady Wantage
##Lord Warksworth
##Lady Leucha Warner
##Lady Warrender
##Lord and Lady Watson
##Lady Cecilia Webb
##Lady Rose Weigall
##Lord Welby
##Lady Willes
##Lady Willis
##Lady Arthur Wellesley
##Lord and Lady Wenlock
##Lord and Lady Westbury and Dowager Westbury
##Lady Isabella Whitbread
##Lady White
##Lady Whitehead
##Lady Whiteway
##Lady Elizabeth Williamson
##Lady Williams-Wynn
##Lady Willoughby (2)
##Lord Willoughby de Broke
##Lord Willoughby de Eresby
##Lady Willshire
##Lady Wilson, Lady Sarah Gordon Wilson
##Lord and Lady Wimborne
##Lady Windeyer
##Lord and Lady Windsor
##Lady Winnington
##Lady Constance Wodehouse
##Lord and Lady Wolverton
##Lady Julia Wombwell
##Lady Wood, Lady Mary Wood
##Lady Woods
##Lord Wrottesley
##Lady Hugh Wyndham
##Lady Barbara Yeatman
##Lady Lilian Yorke
##Lord Zouche
#Right Honourables
##H. H. Asquith
##E. Ashley
##A. H. Dyke Acland
##J. Atkinson
##J. B. Balfour
##Sir G. Bowen
##G. W. Balfour
##Sir Hicks-Beach
##A. J. Balfour
##James Bryce
##Sir H. Campbell-Bannerman
##A. H. Smith-Barry
##E. Carson
##H. Chaplin
##Sir J. Chitty
##Jesse Collings
##Sir R. Couch
##G. N. Curzon
##J. Chamberlain
##L. Courtney
##Sir M. Grant-Duff
##A. Akers-Douglas
##Sir W. Hart Dyke
##Sir H. Elliot
##F. Foljambe
##Sir H. Fowler
##Sir A. B. Forwood
##Sir J. Fergusson
##Herbert Gladstone
##Sir J. Gorst
##G. J. Goschen
##W. E. Gladstone
##Sir G. Grey
##C. H. Hemphill
##Charles Seale-Hayne
##R. W. Hanbury
##Lord George Hamilton
##Staveley Hill
##Sir J. T. Hibbert
##Sir W. Harcourt
##lon Hamilton
##Sir Arthur Hayter
##Sir F. Jeune
##W. L. Jackson
##Sir John Kennaway
##G. Shaw-Lefevre
##W. Lidderdale
##Sir Massey Lopes
##James Lowther
##Sir J. Lubbock
##Sir H. Lopes
##Walter Long
##Sir N. Lindley
##J. W. Mellor
##Sir G. O. Morgan
##John Morley
##Arnold Morley
##Sir J. Mowbray
##A. J. Mundella
##J. H. Macdonald
##F. Max Müller
##Sir W. Marriott
##Graham Murray (the Lord Advocate)
##Sir E. Monson
##Sir P. O'Brien
##Sir A. Otway
##Sir F. Peel
##Sir R. Paget of Cranmore
##W. J. Pirrie
##J. P. Robertson
##Sir. J. Rigby
##C. T. Ritchie
##Sir S. H. Strong
##Sir B. Saunderson
##Sir J. Stansfeld
##Sir A. Smith
##C. R. Spencer
##Sir C. Kay-Shuttleworth
##Sir R. Temple
##Sir R. Thompson
##Sir E. Thornton
##Lord Henry Thynne
##Sir G. O. Trevelyan
##C. P. Villiers
##Sir Algernon West
##Sir C. L. Wyke
##C. B. Stuart-Wortley
##S. J. Way
#Honourables&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 5a / Col. 5b) and Honourable Ladies&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 5b / Col. 5c)
##Mrs. Acland
##Mrs. Alexander
##H. Allsopp, Mrs. Allsopp, George Allsopp
##Mrs. Anstruther
##Mrs. Armytage
##[Hon. Lady] Vere Annesley
##Mrs. Bagot, Mrs. Bagot [sic 2x]
##Mrs. Baillie of Dochfour
##Mrs. Balfour
##[Hon.] Coplestone and [Hon.] Mrs. Bampfylde
##John Baring, Susan Baring, Lilian Baring
##Mrs. Barker
##Mrs. Barlow
##Eric Barrington, Mrs. Barrington
##Mrs. Hamar Bass
##Misses Bateman-Hanbury (2)
##Allen B. Bathurst
##Mrs. Benyon
##[Hon. Lady] Beresford
##[Hon.] R. Chetwynd
##Arthur Chichester
##Lady Biddulph
##C. E. Bingham, Mrs. Bingham, Albert Bingham, Mrs. Bingham [sic x2]
##Lady Birkbeck
##Ivo Bligh, Mrs. Bligh
##Diana Sclater-Booth
##O. Borthwick
##J. Boscawen
##Henry Bourke, Mrs. H. Bourke, Charles Bourke, Terence Bourke, Mrs. T. Bourke, Algernon Bourke, Mrs. A. Bourke, Mrs. E. R. Bourke
##Charles Brand, Arthur Brand, Mrs. Brand, Mrs. T. Brand
##T. Brassey, Mrs. A. Brassey
##Mrs. Stapleton Bretherton
##Reginald Brett, Mrs. Brett
##Mrs. F. Bridgeman, Misses Bridgeman (2)
##Mrs. Britten
##W. St. John Brodrick, Albinia Brodrick
##Emmeline Brownlow
##Mrs. T. C. Bruce, Misses Bruce (2)
##Misses M'Clintock Bunbury (2)
##Mary Byng
##T. J. Byrnes
##Arthur Cadogan, Mrs. A. Cadogan, Mrs. C. Cadogan, Ethel Cadogan
##Mrs. Gough-Calthorpe, Rachel (Gough) Calthorpe, Misses Gough Calthorpe (2)
##Mrs. Candy
##G. H. Campbell, K. Campbell, Hugh Campbell, Mrs. H. Campbell, Mrs. Ronald Campbell, Misses Campbell (2), Mrs. J. B. Campbell, Mildred Campbell
##Mrs. Carington
##Mrs. Carpenter
##Emily Cathcart
##W. Cavendish, Mrs. W. Cavendish, Mrs. Cavendish
##Eleonora Chetwynd, Mrs. R. Chetwynd
##Mrs. A. Chichester, Hilda Chichester
##Mrs. Clowes
##T. H. Cochrane
##Audrey Coleridge
##George Colville
##Mrs. Corbett
##Mrs. H. Corry
##Caroline Courtenay
##Henry Coventry
##Osbert Craven
##Misses Cross
##Mrs. P. Crutchley
##Henry Cubitt, Mrs. Cubitt
##Hamilton Cuffe, Mrs. Otway Cuffe
##Lady Cunningham
##Montagu Curzon, Darea Curzon, Mrs. Curzon
##Hew Dalrymple
##John Dawnay, Eustace Dawnay, W. Dawnay, Mrs. Dawnay (2)
##Misses de Montmorency (2)
##Mrs. H. Dennison
##R. C. Devereux, Mrs. R. C. Devereux
##Mrs. Digby
##Conrad Dillon, Mrs. C. Dillon, Edith Dillon
##Misses Douglas-Pennant (2)
##A. Hay Drummond, Mrs. Hay Drummond, Frances Drummond, Mrs. M. Drummond
##Hubert V. Duncombe, Cecil Duncombe, Mrs. C. Duncombe
##C. T. Dundas, Mrs. C. T. Dundas, W. Dundas, Mrs. W. Dundas, Mrs. John Dundas
##Lady Du Cane
##Herbert Eaton, Mrs. H. Eaton
##F. Egerton, Mrs. A. F. Egerton, Lady Grey Egerton, Tatton Egerton, Mrs. T. Egerton
##Arthur Elliot, Mrs. Arthur Elliot, Lady Elliot, Mrs. Eliot
##Lilian Elphinstone
##Mrs. Ellis
##Muriel Erskine
##H. Escombe, Mrs. Escombe
##Mrs. Evans
##Mrs. C. Keith-Falconer
##Sir S. Ponsonby Fane
##Mrs. W. Farquhar
##Ailwyn Fellowes, Mrs. A. Fellowes
##Mrs. Ferguson of Pitfour
##Everard Fielding
##N. Fitzgerald, Mrs. N. Fitzgerald, Mrs. Fitzgerald, , Mrs. F. G. FitzGerald, Lady FitzGerald
##R. Fitzwilliam, W. H. Fitzwilliam
##Mary Forester
##Sir John Forrest
##Mrs. W. H. Forster
##Mrs. Lionel Fortescue
##Sir C. Fremantle, Mary Fremantle
##Sir Malcolm Fraser, Misses Fraser (2)
##Mrs. Charles Keith-Fraser
##Violet Gibson
##Evelyn Giffard
##Mrs. Henry Gladstone
##Lady Godley
##George Ormsby Gore
##F. Leveson-Gower
##Mrs. Gough
##Mrs. Alaric Grant
##Ronald Greville, Mrs. R. Greville, Louis Greville, Mrs. L. Greville, Sidney Greville, Mrs. A. Greville, Mrs. A. H. F. Greville
##Robert Grosvenor, Algernon Grosvenor, Mrs. A. Grosvenor, Maud Grosvenor, Elizabeth Grosvenor
##Lady Hamilton Gordon, [Hon. Lady] Nevil Gordon
##Misses Guest (2)
##Geoffrey Browne Guthrie
##Mrs. Gye
##Mrs. A. Haig
##Mrs. Halford
##, Misses Hamilton (2)
##Mrs. North Dalrymple-Hamilton
##Mrs. Hobart Hampden
##Mrs. Assheton Harbord, Mrs. C. Harbord, Judith Harbord, Bridget Harbord, Mrs. Harbord
##C. Hardinge, Mrs. C. Hardinge, A. Hardinge
##A. E. Gathorne-Hardy, Nina Gathorne-Hardy
##Misses Hawke (2)
##C. G. Hay
##Misses Heneage (2)
##Helen Henniker, Mrs. Henniker
##Robert Herbert, Sir Robert Herbert, Mrs. R. Herbert, Mrs. Herbert
##A. Holland Hibbert, Mrs. A. Holland Hibbert
##Lady Higginson
##Mrs. Hill
##Lionel Holland, Sydney Holland
##Grosvenor Hood, Dorothy Hood
##Lady Acland-Hood
##Fanny Hood of Avalon
##Mrs. Curzon Howe
##[Hon.] Evelyn Hubbard, Mrs. E. Hubbard, Alice Hubbard
##Mary Hughes
##Mrs. Meynell Ingram
##G. Jolliffe, Sydney H. Jolliffe, Mrs. Jolliffe
##Lady Johnston
##G. Keppel, Mrs. Keppel, Derek Keppel, Mrs. William Keppel
##Mrs. Alfred Ker
##Constance Kerr
##Mrs. Kingscote
##C. C. Kingston
##Lady Knollys
##Bertha Lambart
##F. W. Lambton, Mrs. Lambton
##Mary Lascelles
##Charles Laurence, Herbert Laurence
##Wilfrid Laurier
##Mrs. Lawley
##Mrs. C. Lawrence, Misses Lawrence (2), Mrs. H. Lawrence
##Mrs. Legge
##T. W. Legh, Mrs. Legh, Sybil Legh
##F. D. Leigh, Mrs. F. D. Leigh, E. Chandos Leigh, Mrs. E. C. Leigh, Cordelia Leigh
##C. Hanbury Lennox, Mrs. Hanbury Lennox
##G. W. Leslie
##R. l’Estrange
##Atholl Liddell, Mrs. A. Liddell
##Mrs. H. Gore-Lindsay
##Reginald Lister
##Henry Littleton, Misses Littleton (2)
##Misses Loch (2)
##William Lowther, Mrs. W. Lowther, L. Lowther, Mrs. L. Lowther
##Mrs. E. H. Loyd
##Mrs. Lumley
##Alfred Lyttelton, Mrs. A. Lyttelton, Misses Lyttelton (2), Mrs. Lyttelton
##Flora Macdonald, Lady Macdonald
##Mrs. Mackinnon
##Mrs. Maclagan
##Mrs. Magniac
##Mrs. Maguire
##W. Massey-Mainwaring, Mrs. Massey-Mainwaring
##Mrs. Fuller-Maitland
##Aline Majendie
##Misses Henniker Major (2)
##Mrs. Mallet
##Archibald Marjoribanks
##Misses Constable Maxwell (2)
##Mrs. M'Calmont
##Schomberg M'Donnell
##Charles Mills, Violet Mills, Mrs. Mills
##Mrs. Percy Mitford
##Maud de Moleyns
##Mrs. C. Molyneux
##Annette Monck, Mrs. Monck
##Violet Monckton
##Mrs. Monson
##John Scott Montagu
##[Hon.] Evelyn Moore
##R. Moreton, Mrs. R. Moreton
##Mrs. Mostyn, Misses Mostyn (2)
##Mrs. G. H. Murray, Alice Murray
##Lady Musgrave
##[Hon. Lady] Napier, Emilia Napier, Mrs. Scott Napier
##Mrs. Neeld
##Sir Hugh Nelson
##[Hon.] R. Nevill
##Mrs. Newdigate
##Sir H. S. Northcote
##Misses O'Brien (2)
##Mary O'Hagan
##Mrs. Okeover
##Mrs. Oliphant
##R. Terence O'Neill, Henrietta O'Neill
##Misses Palk (2)
##Cecil Parker, R. Parker, F. Parker, Mrs. F. Parker, Mrs. Parker
##Mabel Parnell
##[Hon.] C. B. Parsons, Mrs. Parsons
##Mrs. W. Paton
##[Hon.] Sydney Peel, Misses Peel (2)
##Mrs. Anderson Pelham
##E. S. Douglas-Pennant, Mrs. E. S. Douglas-Pennant
##Mrs. Heber Percy
##Albert Petre, Mrs. A. Petre
##Harriet Phipps
##Mrs. Pirie
##Thomas Playford
##Horace C. Plunkett
##[Hon.] Ashley Ponsonby, Mrs. Ponsonby, Misses Ponsonby (2)
##H. Orde Powlett, Mrs. Orde-Powlett, Myra Orde-Powlett
##E. W. B. Portman, Mrs. Portman, Mary Portman
##Mrs. Pretyman
##C. Ramsay, Mrs. C. Ramsay
##G. H. Reid
##Misses Rendel (2)
##Misses Rice (2)
##Lady White Ridley
##Mrs. Ritchie
##F. Roberts, Mrs. Phillips Roberts
##Misses Roberts (of Kandahar) (2)
##J. M. Rolls, Eleanor Rolls
##W. Rothschild, Evelina Rothschild
##W. Rowley, Mrs W. Rowley, Lady Thelluson Rowley
##A. Russell, Misses Russell (2)
##Gustavus Hamilton-Russell, Misses Hamilton Russell (2)
##the Master of Ruthven, Mrs. Ruthven
##Mrs. J. D. Ryder
##Sir Saul Samuel
##A. Saumarez, Mrs. A. Saumarez
##Mrs. E. J. Saunderson
##J. Maxwell Scott, Mrs. Maxwell Scott
##R. J. Seddon
##Mary Sidney
##Lady Simeon
##Misses Skeffington (2)
##Sir Donald Smith, Mrs. A. H. Smith, [Hon.] W. F. D. Smith
##Granville Somerset, Mrs. G. Somerset, Arthur Somerset, Mrs. A. Somerset, R. Somerset, Violet Somerset
##Mrs. C. R. Spencer
##Sir J. Gordon Sprigg
##Lyulph Stanley, F. C. Stanley, George Stanley, Mrs. E. J. Stanley, Mrs. Stanley, Mrs. V. A. Stanley, Maude Stanley
##Lady Cowell-Stepney
##Randolph Stewart, Mrs. R. Stewart, FitzRoy Stewart, Mrs. Stewart
##Mabel St. Aubyn
##Misses St. Clair (2)
##Mrs. Stirling
##Horatia Stopford
##[Hon. Lady] Alison Stourton
##Mrs. Strutt, Misses Strutt (2)
##Hilda Sugden
##Alfred Talbot, Mrs. Talbot, Mrs. R. A. J. Talbot
##Sir D. Tennant
##S. R. Thayer
##Misses Thellusson (2)
##Edward Thesiger, Mrs. E. Thesiger, Frederick Thesiger, Mrs. F. Thesiger, Mary Thesiger
##Lady Thorold
##Katharine Thring
##Misses Tollemache (2)
##R. Marsham-Townshend, Mrs. Marsham-Townshend
##Alice Hanbury-Tracy
##Charles Grey Trefusis, Misses Trefusis (2)
##Mrs. Trelawny
##Mrs Tremayne
##Mrs. W. le Poer Trench
##Charles Trevor
##George Hill-Trevor, Marcus Hill-Trevor, Mrs. Hill-Trevor, Misses Hill-Trevor (2)
##Mrs. C. W. Trotter
##Lady Tryon
##Rosamond Tufton
##Sir G. Turner
##Rev. L. Tyrwhitt
##Misses Tyssen Amherst (2)
##Misses Vereker (2)
##R. Greville-Verney, Mrs. R. G. Verney, Misses Verney (2)
##F. Villiers, Mrs. F. Villiers
##Misses Vivian (2)
##Arthur Walsh
##Mrs. P. E. Warburton
##Robert Ward, Mrs. Dudley-Ward
##Mrs. West
##Mrs. Whateley
##Sir W. Whiteway
##F. Bootle-Wilbraham
##Ella Williamson
##Tatton Willoughby
##Lady Wilson
##[Hon.] Armine Wodehouse, Mrs. Wodehouse
##Frances Wolseley
##F. Wood, Misses Wood (2)
##Mrs. G. Wrottesley, Evelyn Wrottesley
##Percy Wyndham, Mrs. P. Wyndham, Misses Wyndham (2)
##Maud Wynn
##Lois Yarde-Buller
##Alex. G. Yorke, Mrs. J. Yorke, Mrs. E. C. Yorke
#Sirs&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 5c–6a)
##Augustus Adderley
##Edwin Arnold
##John Austin
##George Arthur
##John Heathcoat-Amory
##A. Armstrong
##Andrew Agnew
##Frederick Abel
##Henry Acland
##A. Arnold
##Alexander Arbuthnot
##John Barran
##G. Bower
##J. W. Bonser
##J. Crichton-Browne
##Joseph Bailey
##E. Ashmead-Bartlett
##Henry Barkly
##R. Beauchamp
##Raymond Burrell
##Charles Barrington
##David Baird
##Arthur Birch
##Edward Birkbeck
##W. Cunliffe Brooks
##A. de Capel Brooke
##Courtenay Boyle
##F. Burton
##F. Buxton
##Steuart Bayley
##John Bramston
##John Baker
##H. Bullard
##J. T. Brunner
##H. Bellingham
##Henry Bergne
##Thomas Boughey
##F. J. Bramwell
##E. Burne-Jones
##James Blyth
##Seymour Blane
##Henry Chamberlain
##Roderick Cameron
##Hugh Cholmeley
##John Conroy
##Edward Clarke
##C. Cameron
##E. Carbutt
##W. Coddington
##Marshal Clarke
##Reginald Cathcart
##Savile Crossley
##Edward Colebrooke
##Reginald Cust
##Charles Crosthwaite
##John Colomb
##Daniel Cooper
##F. Astley-Corbett
##Donald Currie
##Henry Cunningham
##Robert Cunliffe
##Henry Cotterell
##T. D. Gibson Carmichael
##F. Curden,
##George Dallas
##James Drummond
##Mortimer Durand
##G. Des Vieux
##Henry Dering
##J. N. Dick
##Dyce Duckworth
##T. Swinnerton Dyer
##E. Hastings Doyle
##John Dorington
##William Dunn
##Humphrey de Trafford
##Charles Dalrymple
##G. Dashwood
##Gardner
##Engleheart
##Francis Evans
##A. Edmonstone
##Whittaker Ellis
##W. H. Flower
##Horace Farquhar
##Joseph Fayrer
##H. Fletcher
##William Ffolkes
##William Fraser
##Bartle Frere
##Gerald Seymour Fitz-Gerald
##Robert Finlay
##B. Walter Foster
##Gerald FitzGerald
##R. FitzGerald
##Maurice FitzGerald
##Forrest Fulton
##William Flower
##Andrew Fairbairn
##John Gilbert
##E. T. Gourley
##Edward Grey
##W. Gull
##Walter Gilbey
##Lepel Griffin
##G. Macpherson-Grant
##Reginald Graham
##Philip Grey Egerton
##Douglas Galton
##R. Glyn
##Arthur Godley
##Charles Grant
##R. Gresley
##Alexander Acland-Hood
##T. G. Fermor Hesketh
##Arthur Haliburton
##Brydges Henniker
##F. Dixon-Hartland
##R. Hanson
##Alfred Hickman
##W. Houldsworth
##Henry Howorth
##F. Seager Hunt
##Charles Hall
##E. W. Hamilton
##Reginald Hardy
##Clement Hill
##Basil Hall
##Joseph Hooker
##Charles Hunter
##Charles Hartopp
##Victor Houlton
##Augustus Hemming
##Henry Irving
##Frederic Johnstone
##W. Jenner
##J. Jenkins
##James Joicey
##Charles Jessell
##Harry Johnston
##Edward Jenkinson
##James Hill Johnes
##John Jackson
##H. Seymour King
##James Kitson
##J. Lister-Kaye
##V. Kennett-Barrington
##George Kekewich
##John Leslie
##Thomas Dick Lander
##T. Villiers Lister
##James Linton
##Charles Lees
##Charles Legard
##Thomas Lea
##Wilfrid Lawson
##Elliott Lees
##A. C. Lyall
##J. T. D. Llewelyn
##Joseph Leese
##Leonard Lyell
##F. Laking
##Godfrey Lushington
##F. Lockwood
##Henry Longley
##George Lewis
##F. Milner
##Herbert Maxwell
##Francis Montefiore
##Graham Montgomery
##Robert Moncreiffe
##Musgrave
##Colin Scott Moncrieff
##Francis Mowatt
##Evan MacGregor
##J. G. Miller
##F. D. Maclean
##J. Blundell Maple
##Allan Mackenzie
##Lewis M'lver
##F. Mappin
##Theodore Martin
##Samuel Montagu
##William MacCormac
##Hubert Miller
##Lewis Morris
##Clements Markham
##A. C. Mackenzie
##John Monckton
##J. Stirling-Maxwell
##J. Heron Maxwell
##Kenneth Matheson
##J. S. Montefiore
##Acquin Martin
##W. Maxwell
##Oswald Moseley
##Arthur Nicolson
##Terence O'Brien
##Reginald Ogilvy
##Herbert Oakeley
##Hush Owen
##G. G. Petre
##Walter Parratt
##Frederick Pollock
##Herbert Perrott
##Douglas Powell
##Weetman Pearson
##Joseph Pease
##Francis S. Powell
##Reginald Palgrave
##W. Priestley
##E. G. Poynter
##G. S. Baden-Powell
##Charles Pontifex
##J. Dickson-Poynder
##James Paget
##C. M. Palmer
##C. Lennox Peel
##James B. Peile
##Westby Perceval
##Charles Pigott
##John Puleston
##W. Plowden
##Richard Quain
##George Russell
##C. Lister Ryan
##W. H. Russell
##J. Ramsay
##Owen Roberts
##R. T. Reid
##Charles Robinson
##J. Thellusson Rowley
##James Reid
##C. Euan-Smith
##J. Barrington Simeon
##J. B. Stone
##M. Shaw-Stewart
##Edward Sieveking
##T. H. Sanderson
##Augustus K. Stephenson
##Thomas Sutherland
##Mark Stewart
##Andrew Scoble
##Joseph Savory
##Douglas Straight
##Charles Shelley
##S. Shippard
##E. Sassoon
##A. Condie Stephen
##E. Sullivan
##Arthur Sullivan
##S. Scott
##H. Simpson
##E. Stafford
##Ernest Satow
##Tatton Sykes
##John Tyler
##Charles Tennant
##John Tenniel
##J. Thorold
##John Thursby
##Thomas Troubridge
##Charles Turner
##H. Meysey-Thompson
##W. Vincent
##Edgar Vincent
##Arthur Vicars
##W. Williams-Wynn
##James Walker
##R. Webster
##George Wombwell
##C. Rivers Wilson
##W. H. Wills
##Donald Mackenzie Wallace
##George Warrender
##F. Winnington
##James Whitehead
##Arthur Willshire
##Henry Wood
##Hugh Wyndham
##W. White
##Sidney Waterlow
##Hedworth Williamson
##Jacob Wilson
##W. Windeyer
##Albert Woods (Garter)
##Allen Young
#Chairman of County Council (Dr. Collins)
#Counts and Countesses
##Count Cassini
##Count and Countess De Ganay
##Count Gurowski
##Count Hohenau
##Count Theodor Bolesta Koziebrodski
##Count Leon Mniszeek
##Count and Countess Potocki
##Count and Countess Raben
#Barons and Baronesses
##Baroness Emile Beaumont d'Erlanger
##Baroness De Brienen
##Baron De Onethau and Baroness D’Onethan [sic]
##Baron and Baroness Alphonse de Rothschild
##Baron Ferdinand Rothschild
##Baron and Baroness Schröder
##Baron and Baroness von Deichmann
##Baron von Heeckeren van Wassenaer
##Baroness von Hügel, Baroness Gertrud von Hügel [sic]
##Baron and Baroness Campbell von Laurentz
##Baroness Wilhelm von Rothschild
#Rev. the Moderator of the General Assembly of the Church of Scotland
#Deans — Christ Church, St. Paul's, Westminster, Windsor
#The Provost of Eton
#Master of Trinity (Mr. Butler)
#The Sub-Dean of the Chapels Royal
#Canons — Blundell, Dalton, Duckworth, Fleming, Hervey, Teignmouth Shore, Wilberforce
#Dr. Adler (Chief Rabbi)
#Dr. M'Cormick
#Chaplain of the Fleet
#Chaplain General
#Reverend Doctors — Edmund Warre, C. J. Welldon
#Reverends — Prebendary Hawkshaw, Albert Baillie, W. H. Bliss, M. Ebrington Bisset, Lord W. Cecil, Lord Charles Fitzroy, J. H. Ellison, H. Haweis, W. R. Jolly, G. J. Martin, Newton Mant, Marquis of Normanby, A. Robins. W. Gunion Rutherford, Clement Smith, Montagu Villiers
#Doctors — Lennox Browne, J. V. Bridge, Barlow, Robert Farquharson, J. F. Fox, Surgeon-Major Kilkelly, John Lowe, C. H. H. Parry, G. V. Poore, Dorrien Smith, S. Wilks
#Messieurs&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 6b–7a),  Mesdames (4, Col. 7a–b) and Misses&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 7c – 5, Col. 1a)
##Mme Abdy
##Mr C. T. Dyke-Acland, Mme A. H. Dyke Acland, Mme Dyke Acland
##Mme Adair
##Misses Adam (2)
##Mr and Mme Adeane
##Misses Adye [?] (2)
##Mme Agar
##Mr Hamilton Aidé
##Mr John Aird, Misses Aird (2)
##Miss Akers-Douglas
##Mr Edward Alderson
##Mr George Alexander, Mme Alexander, Miss Alexander
##Miss Alison
##Mr and Mme Allhusen
##Mme Alma-Tadema
##Mr W. Ambrose
##Miss Heathcoat-Amory
##Mr R. Anderson, Miss Florence Anderson
##Mr E. H. Anson
##Mr H. T. Anstruther, Miss Rosomond Anstruther
##Mme Antrobus
##Mr Arbuthnot, Miss Arbuthnott [sic]
##Miss Archer-Houblon
##Mme Argles
##Mme Arkwright, Miss Arkwright
##Misses Armytage (2)
##Miss Arnott
##Mr and Mme Ascroft, Miss Ascroft
##Mr Arthur Ash
##Mr A. Asher
##Mme Ashton
##Mme Asquith
##Mr Astor, Mr W. Astor
##Mr B. F. Astley
##Mme Evelyn Atherley
##Mr and Mme Alfred Austin, Misses Austin (2)
##Mr and Mrs C. H. Babington
##Mr and Mrs Bagge
##Mrs Charles Bagot, Mrs J. F. Bagot, Miss Alice Bagot
##Mr James Bailey, Mrs J. Bailey, Mrs Bailey, Misses Bailey (2)
##Mrs Duncan Baillie, Misses Duncan Baillie (2)
##Mr Baillie of Dochfour
##Mr and Mrs W. A. Baillie-Hamilton
##Mr E. Bainbridge
##Mr and Mrs H. R. Baird, Mr and Mrs J. G. A. Baird, Misses Baird (2)
##Mr and Mrs Baldwin
##Mr and Mrs E. Balfour, Mr and Mrs Charles Balfour, Miss Balfour
##Mr and Mrs Banbury, Miss Banbury
##Mr and Mrs S. B. Bancroft [actor &quot;Bancroft and his wife accepted with becoming grace the congratulations with which they were well-nigh overwhelmed&quot;&lt;ref name=&quot;:3&quot; /&gt; (5, Col. 6b)]
##Bandanaratke [?]
##Mrs Bankes
##Mr Banks
##Mr and Mrs Walter Baring, Miss Baring
##Miss Barker
##Mr J. Emmott Barlow, Mrs Barlow, Mrs Barlow [sic 2x]
##Misses Barnardiston (2)
##Miss Barne
##Mr and Mrs F. G. Barnes, Mr and Mrs Barnes, Misses Barnes (2)
##Miss Barran (2)
##Mr and Mrs J. Wolfe Barry, Mr and Mrs F. Tress Barry, Mrs A. Barry
##Misses Bartlett (2)
##Mr and Mrs D. P. Barton, Mr and Mrs Barton
##Mr Hamar Bass
##Mrs Bates, Miss Bates
##Mr and Mrs H. Bathurst, Misses Bathurst (2)
##Mr and Mrs Baxendale, Miss Baxendale
##Miss Mariot [?] Bayley
##Mr and Mrs W. W. Beach, Miss Beach
##Misses Hicks-Beach (2)
##Mr R. M. Beachcroft
##Mr and Mrs Wentworth Beaumont, Mr Wentworth B. Beaumont, Mrs Beaumont, Miss Hilda Beaumont
##Mr and Mrs Rupert Beckett, Mr E. W. Beckett
##Mr and Mrs Beer
##Mr and Mrs F. F. Begg
##Mr  Charles Bell, Mr and Mrs Bell, Misses Bell (2)
##Miss Bellingham
##Mr and Mrs R. Benson, Mr and Mrs Benson
##Miss Berens
##Mr and Mrs Beresford, Miss Beresford
##Miss Berkeley, Misses Berkeley (2)
##Mr and Mrs Bertier, Miss Bertier
##Mr and Mrs Cosmo Bevan, Mr and Mrs F. Bevan, Miss Bevan
##Mr M. M. Bhownaggree
##Mr and Mrs F. Bibby
##Mr Leonard Biddulph, Mr Biddulph, Mr Victor Biddulph, Mr M. Biddulph, Mrs H. M. Biddulph, Misses Biddulph (2), Miss Biddulph, Miss Freda Biddulph
##Mr and Mrs Bigham
##Mr Bigwood
##Mrs C. Bill, Miss Bill
##Miss Birch
##Mrs Birch-Reynardson, Misses Birch-Reynardson (2)
##Mr A. Birrell, Mrs Birrell
##Mr and Mrs Bischoffsheim
##Mrs Ebrington Bissett
##Misses Blackwood (2)
##Mr and Mrs R. G. Blennerhassett
##Mrs W. H. Bliss
##Mrs Blundell, Miss Blundell
##Misses Blyth (2)
##Mr and Mrs Bolitho, Miss Bolitho
##Mr H. C. O. Bonsor, Mrs Bonsor, Miss Bonsor
##Mrs W. Borsel
##Mrs Griffith-Boscawen
##Mr and Mrs Boulnois
##Miss Bourke
##Mr W. R. Bousfield
##Mrs Bowden-Smith, Misses Bowden-Smith (2)
##Miss Bowen (2)
##Mr T. G. Bowles, Mrs Bowles
##Mr Edmund R. Boyle
##Miss Mabel Brackenbury
##Mrs Bradley, Miss Bradley
##Miss Beryl Bradford
##Miss Braddon
##Miss Bramwell
##Mr H. L. C. Brassey, Mrs H. A. Brassey, Misses Brassey (2), Misses Brassey (2) [sic 2x]
##Mr Stapleton Bretherton, Misses Stapleton Bretherton (2), Mr F. Stapleton Bretherton
##Mrs Bridge
##Mr G. Bridgman, Mr and Mrs C. G. O. Bridgeman
##Mr Brigg
##Mrs Brocklehurst
##Misses Brodie (2)
##Mr and Mrs Brookfield, Miss Brookfield
##Miss Bromley-Davenport
##Miss Brooke
##Miss Rhoda Broughton
##Mr and Mrs A. H. Brown, Miss Brown
##Mrs Browne, Misses Browne (2), Misses Browne (2) [sic 2x]
##Mrs Brownrigg, Miss Brownrigg
##Mr A. O. Bruce, Mrs A. C. Bruce [sic], Misses Bruce (2)
##Miss Brunner
##Mrs Bryce
##Mr Brymer
##Mr and Mrs Buchanan
##Mrs C. E. Buckle
##Mr Bucknill
##Miss Budgett
##Miss Mary Bulteel
##Miss Burdett
##Mr and Mrs Burges, Misses Burges (2)
##Mrs C. K. Burn
##Mr and Mrs F. C Burnand
##Miss Evelyne Burne
##Mr and Mrs W. Burns, Miss Burns
##Misses Burrell (2)
##Mr J. G. Butcher
##Mrs Butler, Mrs Butler, Miss Butler
##Mr Sydney Buxton, Mrs S. Buxton, Misses Buxton (2)
##Mr P. H. Calderon
##Mrs Calley
##Mrs Archibald Calvert, Miss Calvert
##Mr Cameron, Miss Cameron, Misses Cameron (2)
##Mr and Mrs J. D. Campbell, Mr J. A. Campbell, Miss J. A. Campbell, Mrs F. Campbell, Mrs W. Campbell, Mrs Hastings Campbell, Mrs W. Campbell [sic 2x], Mrs F. L. Campbell, Mrs D. B. O. Campbell, Miss Lilah Campbell, Miss Campbell, Miss Ronald Campbell, Misses Campbell (2)
##Miss Grace de Capell-Brooke
##Miss Carden
##Miss Carleton
##Mr and Mrs W. W. Carlile, Miss Carlisle
##Mrs Rivett Carnac
##Mrs Carnegy
##Mrs Boyd Carpenter, Misses Boyd Carpenter (2)
##Mrs Carson
##Mr and Mrs D'Oyly Carte
##Mrs Carter
##Mrs Castance
##Mr R. K. Causton, Mrs Causton, Miss Causton
##Mrs Cavaye
##Mr and Mrs C. Tyrall Cavendish, Mr Victor Cavendish, Mr Henry Cavendish, Mr Cavendish, Mrs Cavendish
##Mr and Mrs F. Cavendish-Bentinck, Mr Cavendish-Bentinck, Mrs W. G. Cavendish-Bentinck
##Mr F. Cawley
##Mr and Mrs Cayzer, Miss Cayzer
##Mr and Mrs W. M. Cazalet
##Mr F. Cazenove
##Mr Evelyn Cecil, Miss Cecil
##Mrs Chaine
##Mrs Chaloner
##Mr Austen Chamberlain, Mrs Chamberlain, Misses Chamberlain (2)
##Misses Chaning (2)
##Mr and Mrs Channing
##Mr and Mrs Cecil Chaplin, Misses Chaplin (2), Miss Edith Chaplin, Miss Chaplin
##Mrs Chapman
##Misses Chetwode (2)
##Mrs W. Chetwynd, Miss Chetwynd (2)
##Mr Childe-Pemberton
##Miss Chitty
##Miss Leila Crichton
##Miss Cholmeley (2)
##Miss Cholmondeley
##Miss Chrichton-Maitland
##Mrs H. Churchill
##Miss Spencer Churchill
##Mr J. D. Clark, Mr and Mrs Atkinson Clark, Mr Clark, Mrs B. F. Clark, Mrs G. D. Clark, Stanley Clark, Miss Clark
##Mr Purdon Clarke, Mr Ernest Clarke, Miss Clarke, Miss Stanley Clarke
##Mrs Clerk
##Mr and Mrs Henry Pelham Clinton
##Mrs Clive, Misses Clive (2)
##Mrs Close
##Mr Clough
##Mr Clowes, Misses Clowes (2)
##Mr Cobbold
##Mr T. B. Cochrane, Miss Cochrane
##Mr and Mrs W. A. Cockerell, Miss Cockerell, Miss Cockerell [sic 2x]
##Mr and Mrs D. Coghill
##Mr B. Cohen
##Mr Wentworth Cole
##Miss Colomb
##Mr and Mrs Colston
##Miss Colville
##Mr Richard Combe
##Miss Commerell, Miss Commerell [sic 2x]
##Mr and Mrs Compton
##Mr and Mrs Consett, Miss Vera Consett
##Mr and Mrs F. L. Cook, Mr Ward Cook, Miss Cook
##Mr and Mrs Kinloch Cooke, Mr Cooke, Mr and Mrs C. Kinloch Cooke
##Mr and Mrs Daniel Cooper, Mrs E. H. Cooper, Misses Cooper (2), Miss Cooper
##Mr and Mrs Cameron Corbett, Miss Corbett
##Mr and Mrs V. Seymour Corkran, Miss Corkran
##Mr and Mrs F. S. W. Cornwallis
##Mr and Mrs Cory
##Mrs Armar Corry, Mrs Clifford Corry, Miss Corry
##Mr J. R. G. Cotterell, Miss Cotterell (2)
##Mrs Stapleton Coton
##Mr and Mrs George Courroux
##Mrs Courtney
##Mr Burdett-Coutts
##Mrs Coventry
##Miss Cowell
##Miss Cowell-Stepney
##Mr and Mrs R. Cox, Mrs Cox, Miss Cox
##Mrs Crabbe, Misses Crabbe (2)
##Mrs Craik
##Mr and Mrs Crawshay
##Mrs Creignton, Miss Lucia Creighton
##Mr C. A. Cripps, Mr and Mrs Wilfrid Cripps
##Mr and Mrs Critchett
##Mr and Mrs Croombie
##Mrs A. B. Crosbie
##Mr and Mrs Shepherd Cross, Mr A. Cross, Miss Crosse
##Mr and Mrs Cruddas, Misses Cruddas (2)
##Mr and Mrs Percy Crutchley, Misses Crutchley (2)
##Miss Cuffe
##Miss Culme-Seymour
##Mrs Cuninghame
##Miss Cunliffe
##Mrs Dick-Cunynghame
##Mrs Curzon
##Misses Cust (2)
##Miss Custance
##Mrs Dalbiac
##Miss Gladys Dalgety [?]
##Mr C. B. Dalison
##Miss Dalrymple
##Mrs Dalton
##Mrs Denis Daly
##Mr and Mrs Darling
##Miss Dashwood
##Mr W. Bromley-Davenport
##Miss Davey
##Mr and Mrs Louis Davidson, Mrs Randall Davidson
##Mr W. Rees Davies, Mr Ben Davies, Mr and Mrs Vaughan Davies
##Mrs Davis
##Miss Dawnay (2)
##Mrs de Arcos
##Misses De Brienen (2)
##[Miss] La Baronne de Friesen
##Mrs R. C. de Grey Vyner
##[Miss] La Baronne Sirtema de Grovestins [?]
##Mr and Mrs J. de la Cour
##Mr and Mrs Edwin de Lisle
##Mr W. E. Denison
##Mrs Denny
##Miss De Perpigna
##Mrs de Salis
##Mr de Soria
##Mr De Trafford, Miss De Trafford
##Mr Deverell, Miss Deverell
##Mr and Mrs W. de Winton, Miss De Winton
##Mr and Mrs Gerard Dicconson
##Mr and Mrs Dicken
##Mr and Mrs C. S. Dickson, Mrs Dickson
##Mr J. K. Digby, Kenelm E. Digby, Mrs Digby, Misses Digby (2), Miss Digby
##Mr and Mrs J. Diggle
##Mr Lee Dillon, Misses Dillon (2)
##Mr and Mrs Coningsby Disraeli, Mr and Mrs R. Disraeli, Miss Disraeli
##Mrs Domvile, Miss Domvile
##Mr Greville Douglas, Mrs A. L. Douglas, Misses Douglas (2)
##Mrs Akers-Douglas
##Miss Dowell
##Mr and Mrs Doxford, Miss Doxford
##Mrs Geoffrey Drage
##Mr A. Drummond, Mr and Mrs G. Drummond, Mrs A. Hay Drummond, Mrs Lawrence Drummond, Mrs Drummond, Miss Edith Drummond,  Misses Drummond (2), Miss Mary Drummond, Miss Adelizs [?] Drummond, Misses Drummond (2) [sic 2x]
##Misses Du Cane (2)
##Miss Du Chair
##Mr W. H. Dudley-Ward, Miss Sybil Dudley-Ward
##Mr F. Dugdale
##Misses Duncombe (2)
##Mrs Dundas, Miss May Dundas
##Miss Dunn
##Mrs Dunne, Miss Marion Dunne
##Mr Du Plat Taylor, Mrs G. Du Plat Taylor
##Mrs Durnford
##Mr and Mrs Thiselton Dyer
##Mrs East, Misses East (2)
##Mr F. Eaton
##Mr R. Edgcumb
##Mrs Edis, Misses Edis (2)
##Mr Bevan Edwards, Miss Bevan Edwards (2), Mr C. C. Edwards, Mrs Edwards
##Mrs Egerton, Miss Egerton (2), Miss Egerton
##Miss Grey Egerton
##Mr and Mrs M. Eliot, Misses Eliot (2)
##Miss Ellaby
##Mrs Ellicott, Miss Ellicott
##Mr and Mrs F. Elliot, Mr T. H. Elliott, Miss Gertrude Elliot
##Mr T. E. Ellis, Miss Ellis (2), Miss Evelyn Ellis
##Mrs Ellison, Miss Ellison
##Misses Elphinstone (2)
##Mr Cary-Elwes
##Mr Erskine, Miss Rachel Erskine
##Mr Maurice Euphrussi
##Mr W. H. Evans, Misses Evans (2)
##Mr H. P. Ewart, Mrs C. B. Ewart
##Mr Eyre
##Mr Cecil Fane, Mr G. H. Fane, Mr Fane
##Mr Dyafer Fakhry
##Misses Keith Falconer (2)
##Mrs Fane
##Mrs Fanshawe, Miss Fanshawe
##Mr and Mrs Fardell, Misses Fardell (2)
##Mr and Mrs Farmer, Mrs Lancelot Farmer, Miss Farmer
##Mrs Farnham
##Mr Alfred Farquhar, Mr W. Farquhar, Mr and Mrs E. Farquhar, Mrs G. M. Farquhar
##Mr J. N. Farquharson, Miss Amelia Farquharson, Miss Henrietta Farquharson
##Mr and Mrs Farquharson of Invercauld, Misses Farquharson of Invercauld (2)
##Misses Feilding (2)
##Mrs Fellowes
##Mrs Fenn
##Mrs Fenwick, Misses Fenwick (2)
##Mr and Mrs Johnson-Ferguson
##Mr Munro-Ferguson
##Misses Ferguson of Pitfour (2)
##Miss Fergusson
##Miss Dorothy Ffolkes
##Mrs Field
##Mr and Mrs Fielden, Misses Fielden (2)
##Mrs G. H. Finch, Mrs Wynne Finch, Misses Finch (2)
##Mr and Mrs Firbank
##Mr Herbert Fisher, Mr and Mrs Hayes Fisher, Misses Fisher (2)
##Mr and Mrs Fison, Miss Fison
##Miss FitzClarence (2)
##Mrs FitzGeorge, Miss Olga FitzGeorge
##Mr Fitzgerald, Mr F. G. Fitzgerald, Miss Fitz Gerald
##Mr and Mrs Almeric Fitzroy, Miss Ethel Fitz-Roy
##Mrs R. Fitzwilliam, Misses Fitzwilliam (2)
##Mr Flannery
##Mr E. Flower, Miss Flower, Miss Flower [sic 2x]
##Mrs Floyd
##Mrs H. Fludyer
##Mr H. St. George Foley
##Mrs Barrington Foote
##Mr J. S. Forbes, Mr Forbes
##Mr John Ford
##Mr H. W. Forster
##Mr and Mrs Arnold-Forster
##Mr and Mrs Bevill Fortescue
##Misses Forwood (2)
##Mr W. S. Foster, Mrs W. H. Foster, Mrs H. S. Foster, Miss Foster
##Misses Fowler (2)
##Mr Franklin
##Mrs Houston French
##Misses Frere (2)
##Mr L. Fry
##Mrs Fullerton, Misses Fullerton (2)
##Mr Gadson
##Mr Wilhelm Ganz
##Miss Gardiner, Miss Gardiner [sic 2x]
##Mrs Gardner
##Mr and Mrs Garfit [?]
##Mrs Gathorne-Hardy, Miss Gathorne-Hardy
##Mr Hamilton Gatliff
##Mr and Mrs Scott Gatty
##Mr and Mrs Sydney Gedge
##Mr Geoffrey Drage [sic; does this belong here?]
##Mr F. W. Gibbs, Misses Gibbs (2)
##Mr and Mrs Walter Gibson
##Miss Gilbey
##Mr and Mrs Tyrell Giles
##Mr W. Gillett
##Mr and Mrs Gilliat, Misses Gilliat (2)
##Mr Henry Gladstone, Mrs Gladstone, Miss Helen Gladstone
##Miss Glyn
##Misses Godley (2)
##Mrs Godson
##Mr and Mrs Goelet, Miss Goelet
##Mr Charles Gold, Miss Gold
##Mr G. P. Goldney
##Mr and Mrs S. Hoffnung Goldnung Goldsmid
##Mrs A. Goldsmid, Miss Goldsmid
##Mr Otto Goldsmidt
##Mrs Goldsworthy
##Mrs Goodden, Miss Gurrney Goodden
##Mrs Goodenough
##Mr and Mrs John Gordon, Mr and Mrs J. E. Gordon, Mrs Gordon, Mrs G. G. Gordon, Mrs S. Gordon, Mrs Gordon [sic 2x], Miss Hamilton Gordon, Misses Gordon (2)
##Mr and Mrs Frank Gore, V. Gore, Mr and Mrs S. W. Gore, Mrs Gore, Miss Gore
##Mr and Mrs Goschen, Misses Goschen (2)
##Mr and Mrs A. Gosling, Miss Gosling
##Mr and Mrs F. R. Gosset
##Misses Gough-Calthorpe (2)
##Mr E. A. Goulding
##Mr G. Leveson-Gower
##Mr F. Graham, Mr Graham, Mr H. R. Graham, Mr and Mrs C. C. Graham
##Mrs Grant, Miss Grant
##Miss Victona Grant-Duff
##Mr and Mrs Henry Graves, Miss Graves
##Mr Ernest Gray
##Mrs Green
##Mr H. D. Greene, Mr W. R. Greene
##Mrs Gregory, Miss Gregory
##Mr and Mrs W. H. Grenfell, Mrs H. Grenfell, Miss Maud Grenfell
##Mr J. A. Gretton
##Mr Howard of Greystoke
##Mr Grifflth-Boscawen
##Mr and Mrs W. H. Kendal Grimston
##Mr George Grossmith [&quot;George Grossmith was not a little lionised by titled ladies&quot;&lt;ref name=&quot;:3&quot;&gt;“The Queen’s Garden Party. Buckingham Palace Grounds. A Brilliant Scene. The Queen’s Cup of Tea.” ''Daily News'' (London) 29 June 1897, Tuesday: 5 [of 10], Col. 6a [of 7] – 6, Col. 2a. ''British Newspaper Archive''  https://www.britishnewspaperarchive.co.uk/viewer/bl/0000051/18970629/021/0005. Print pp. 5–6.&lt;/ref&gt; (5, Col. 6b)]
##Mr Montagu Guest
##Mrs Gunter, Misses Gunter (2)
##Mr Gurdon
##Mrs Gurney
##Mrs Guy-Pym
##Mr and Mrs Gye
##Mr and Mrs Carl Haag, Miss Carl Haag
##The Munshi Abdul Hafiz Karim
##Mr and Mrs Haggard
##Miss Haig
##Mr R. B. Haldane
##Mr Halford, Misses Halford (2)
##Mr and Mrs Lewis Hall, Mrs Hall, Miss Hall, Miss (Lewis) Hall
##Mr and Mrs Thomas Halsey, Misses Halsey (2)
##Mr Francis Hamilton, Mrs R. W. Hamilton, Mrs Ian Hamilton, Misses Hamilton (2), Miss Hamilton
##Mrs Hammet
##Mr and Mrs Hanbury, Miss Dora Hanbury, Mrs Hanbury
##Miss V. Hanson
##Mr L. V. Harcourt
##Mr and Mrs Hardcastle, Misses Hardcastle (2)
##Mr and Mrs Hardy, Misses Hardy (2)
##Mr Cozens-Hardy
##Mr T. Hare, Mr Augustus Hare, Mr and Mrs John Hare, Mrs Marcus Hare, Mrs Marcus Hare [sic 2x], Miss Hare, Misses Hare (2), Misses Hare (2) [sic 2x]
##Mrs Harford
##Mrs Hargreaves-Rogers
##Mr C. Harrison, Miss Harrison
##Miss Hart
##Misses Hart-Dyke (2)
##Mr and Mrs Hartmann
##Mr George Harwood, Misses Harwood (2)
##Mr Hatch
##Mrs Hatton
##Mrs Haweis
##Mr and Mrs Claude Hay, Misses Hay (2), Misses Hay (2) [sic 2x]
##Mrs Arthur Heath, Mrs Heath
##Miss Louisa Heathcote
##Mr J. Henniker Heaton
##Misses Hemming (2)
##Mr Philip Henriques
##Mrs Heneage, Miss Heneage
##Mrs Henderson
##Miss (Brydges) Henniker
##Mrs Philip Henriques
##Mrs Herbert, Miss Herbert
##Mr and Mrs Hermon-Hodge
##Miss Heron Maxwell (2)
##Mr G. T. Hertslet
##Mrs Hervey, Miss Hervey
##Misses Hervey-Bathurst (2)
##Mr and Mrs Heseltine, Miss Heseltine
##Miss Hickman
##Mr H. Higgins, Mr Cecil Higgins, Mrs Higgins
##Mrs Platt-Higgins
##Miss Gladys Higginson
##Mrs Hildyard
##Mrs Staveley Hill, Miss Hill
##Misses (Stock) Hill (2)
##Mrs Hills
##Mrs Hippisley
##Mr and Mrs E. Brodie Hoare, Misses (Brodie) Hoare (2), Mr G. Hoare, Mrs S. Hoare, Misses Hoare (2)
##Mr and Mrs H. Hobhouse
##Mr R. K. Hodgson
##Mr and Mrs C. D. Hohler
##Mr R. R. Holmes, Mrs Holmes
##Mr R. Hallett Holt
##Mr Maurice Holzman
##Misses Hood (2)
##Miss Margaret Acland Hood
##Miss Hooker
##Mr and Mrs E. Hope, Mr and Mrs Adrian Hope, Misses Adrian Hope (2), Mr and Mrs James Hope, Mr Hope, Miss Mary Hope, Miss Hope
##Mr and Mrs Beresford-Hope, Miss Agnes Beresford Hope
##Mr and Mrs W. H. Hornby, Misses Hornby (2)
##Mr and Mrs Horner
##Mr and Mrs Hornyold [sic]
##Mr J. C. Horsley
##Miss Jean Hotham
##Misses Houldsworth (2)
##Mr R. P. Houston
##Mr E. S. Howard, Mr and Mrs A. C. Howard, Mr Joseph and Mrs J. Howard, Mr and Mrs H. Howard, Mrs Howard, Misses Howard (2), Miss Howard, Misses Howard (2) [sic 2x]
##Mr J. Hozier
##Mr and Mrs G. B. Hudson
##Hughes, Misses Hughes (2)
##Mr and Mrs A. C. Humphreys-Owen
##Mr and Mrs Hungerford
##Miss M. Carew Hunt
##Mrs W. G. G. Hutchinson, Miss Hutchinson
##Mr and Mrs G. M. Hutton, Mr John Hutton, Mr A. E. Hutton, Mrs G. Hutton
##Mrs Inglefield
##Mr and Mrs Wootton Isaacson
##Mrs Jackson, Miss Grace Jackson
##Mr Jacobs
##Mrs Jacoby
##Mr and Mrs W. James, Mr Arthur and Mrs A. James, Miss Helena James
##Mrs J. E. Jameson, Misses Jameson (2)
##Mr and Mrs Jebb
##Mr and Mrs A. F. Jeffreys
##Mr E. R. Jenkins, Missess Jenkins (2)
##Mrs Jenkinson
##Miss Jenner
##Mr and Mrs H. C. Jervoise, Miss Jervoise
##Mrs Jessel, Miss Jessel
##Mrs Cotton-Jodrell, Miss Cotton-Jodrell
##Mr and Mrs J. H. Johnstone, Mrs G. Johnstone, Miss Johnstone, Miss S. L. Johnstone
##Mrs Joicey, Miss Joicey
##Misses Jolliffe (2)
##Mr and Mrs Atherley-Jones
##Mr and Mrs Brynmor-Jones
##Mrs Inigo Jones
##Mr Philip Burne Jones
##Mrs Pryce Jones
##Mr Henry Joslin
##Mr and Mrs Kearley
##Mrs Keeley
##Misses Keith-Falconer (2)
##Miss Kemball
##Mr George Kemp
##Mr C. Kempe
##Mr and Mrs A. Kennard, Mrs Hegan Kennard, Miss Kennard, Misses Kennard (2)
##Misses Kennaway (2)
##Mrs Kennedy, Miss Kennedy
##Mrs Kennion [?]
##Mrs Kennison
##Mr and Mrs W. Kenny, Miss Ethel Kenny
##Mr J. Kenyon
##Mrs Colin Keppel, Miss Keppel
##Misses Ker (2)
##Misses Kerr (2), Miss Nona Kerr
##Mrs Kilkelly
##Mr and Mrs Kimber, Miss Kimber
##Mr King King, Miss King King
##Mr Nigel Kingscote, Mr T. Kingscote
##Mrs Kingston
##Mrs Kitching
##Misses Kitson (2)
##Mr Lees Knowles and Mrs Knowles
##Mr Knowles
##Mr and Mrs Kuhe
##Mrs A. P. Lake
##Misses Lambart (2)
##Miss Aline Lambton
##Mr Landon
##Mrs Lane
##Mrs Langenbach
##Miss Larking
##Mrs Lascelles
##Mr W. F. Laurence
##Mr Edwin Laurence
##Misses Laurie (2)
##Mrs Laurier
##Mr and Mrs E. Law
##Mrs E. Lawrence, Misses Lawrence (2)
##Mrs Lawrie
##Mr J. Grant Lawson, Miss J. Lawson
##Mr and Mrs Lecky
##Mrs Hanning Lee, Misses Hanning Lee (2)
##Miss C. Lees
##Miss Leese
##Miss Violet Leigh
##Mr and Mrs S. Leighton, Misses Leighton (2)
##Mrs Leslie
##Mr L’Estrange, Miss l’Estrange
##Mr Letchworth
##Mrs Lewis, Misses Lewis (2)
##Mrs Naylor Leyland
##Mrs Liddell, Miss Liddell, Misses Liddell (2)
##Mrs Lidderdale, Misses Lidderdale (2)
##Misses Lindley (2)
##Mr Henry Gore Lindsay, Miss Gore Lindsay, Mr H. B. Lindsay, Mr W. A. Lindsay, Miss Lindsay, Miss Lindsay [sic 2x]
##Mr Leonard Lindsey
##Misses Linton (2)
##Miss Lister
##Mr Cecil Lister-Kaye
##Miss Llewelyn
##Mr E. Lloyd, Mrs Lloyd, Misses Lloyd (2)
##Miss Alice Loch, Miss Emily Loch
##Mrs Lockhart
##Mrs Lockwood, Miss Lockwood
##Mr Loder
##Miss Loftus
##Mr Heathcote Long and Mrs Long
##Mr H. T. Lopes, Misses Lopes (2)
##Mr Drury Lowe and Mrs Lowe, Miss Drury Lowe
##Mr and Mrs J. W. Lowther, Miss Aimee Lowther
##Mr E. H. Loyd, Mr and Mrs A. K. Loyd
##Mr and Mrs H Lubbock, Miss Lubbock
##Mr Reginald Lucas, Mrs Lucas
##Mrs Lucas-Shadwell, Miss Lucas-Shadwell
##Mrs Luck
##Mr and Mrs Fairfax Lucy
##Mr H. Luttrell, Mr W. C. F. Luttrell, Mrs Luttrell, Miss Luttrell
##Miss Lyall
##Misses Lyell (2)
##Mr and Mrs Lyon
##Miss Lyson
##Misses Lyte (2)
##, , , , , , , , , , , , , , , , , , , , ,  T. C. March, C. J. Murray, Mount, Morrell, R. J. More, Moon, E. P. Monckton, Monk, F. Bingham Mildmay, Beresford Melville, M’Laren, M'Ewan, Martin, H. H. Marks, lan Z. Malcolm, H. L. B. MCalmont, J. W. Maclure, Campbell Munro, J. Maclean, J. C. Macdona, W. G. E. Macartney, Muir Mackenzie, Hugh Morrison, G. H. Murray, P. C. Milbank, Bingham Mildmay, Alpin Macgregor, M. Myther, V. Montagu, Frederick Macmillan, C. M’Neill, Arundel St. John Mildmay, C. Maud, Fuller Maitland, A. Milman, W. A M’Arthur, Marjoribanks, W. H. Myers, F. W. Maude, Muntz, Charles Morley, Murdoch, A. B. F. Mitford, B. Mallet, Mure, Madden, W. J. Mure, R. Maguire, Mackinnon, Montgomerie, Maxwell-Lyte, Mason, Ronald Moncrieffe, Milvain, T. G. Menzies, G. Manners, Nicol, F. A. Newdigate, G. Noel, T. W. Nussey, Charles Orde, R. A. Oswald, M. Oldroyd, J. C. O'Dowd, Oswald, Oppenheim, Arthur Oliphant, C. L. Orr-Ewing, J. L. Pattison, J. Balfour Paul, Paton, A. Peckover, Archibald Peel, Perks, J. Pender, J. Penn, Price, Powell, Paoli, Constantine Phipps[,] Charles Phipps, Leslie Probyn, B. Faudel-Phillips, Wilton Phipps, L. Faudel-Phillips, Joseph Pease, Pollock, Arthur Pease, Roland Protheroe, Walter Peace, J. M. Paulton, Platt-Higgins, Pennefather, Provand, Guy Pym , A. E. Pease, Godfrey Pearse, Algernon Peel, A. V. Pryor, Montagu Price, Phelips, John Ponsonby, Hussey Packe, Wyndham Portal, Henry Petre, Lort Phillips, H. W. Primrose, E. Parkes, Herbert Praed, Heber Percy, Quilter, J. Rankin, Renshaw, J. A. Rentoul, H. C. Richards, Read, T. Richardson, A. T. Phillips Roberts, Hugo von Ruffer, Alexander Ramsay, Alderman and Sheriff Ritchie, Richardson, Rebow, G. L. Ryder, G. A. Redford, G. W. E. Russell, H. J. H. Russell, Pandeli Ralli, John Rutherford, J. Rennell Rodd, Leopold Rothschild, T. W. RusseII, Forbes Robertson, Alfred Rothschild, Brooke Robinson, Edmund Robertson, Repton, James Round, Royds, Henry Raikes, Bowen Rowlands, J. D. Ryder, Sheriff Hargreaves Rogers, Skeffington Smyth, Augustus Spalding, H. H. Shaw, E. Strachey, J. Murray Scott, J. Stern, P. L. Sclater, R. Sassoon, W. Sidebottom, Abel Smith, Louis Sinclair, C. H. Seely, Lucas Shadwell, W. E. T. Sharpe, C. E. Shaw, E. B. Sparke, T. H. Sidebottom, Steward, Stibbert, H. Somerset, H. S. Samuel, J. P. Smith, Horace Seymour, A. H. Smith, H. M. Stanley, J. A. Swettenham, A. Spicer, Stevenson, J. H. Stock, J. Sturgis, H. C. Smith, C. J. Stewart, Leslie Stephen, T. Smith, Senhouse, Eames Storey, Christopher Sykes, H. Seton-Karr, Philip Somers-Cocks, T. Skewes-Cox, Shelley-Bontein, Salting, Leo Schuster, Smith, Arthur Sassoon, G. D. Smith, Shaw, Michael Shaw-Stewart, E. J. Stanley, Albert Sandeman, Scaramanga, Sant, F. Sutton, Dudley Smith, C. E. Tritton. W. E. M. Tomlinson, H. F. Tollemache, A. M. Torrance, Tarleton, Edward Tighe, Alma-Tadema, W. H. Wilson-Todd, P. Thornton, F. Taylor, Beerbohm Tree, Dan Tupper, Montagu Tharp, Abel Thomas, Algernon Turnor, Tudway, C. W. Trotter, H. J. Tennant, J. C. Thynne, H. D. Trelawny, C. E. Thynne, F. J. Thynne, Montagu Thorold, Tremayne, H. Graham Toler, John Taylor, A. J. R. Trendell, Tosti, Christopher Tower, T. Usher, A. Ure, T. Usborne, Chas van Raalte, Graham Vivian, R. C. de Grey Vyner, Hope Vere, F. E. Villiers, Von André, Venning, L. Van Loon, Van De Weyer, Val Prinsep, Walter, Thomas Wayman, Hwfa Williams, Cornwallis West, R. G. Webster, Sackville West, Wanklyn, A. S. Wiison, G. Fleetwood Wilson, A. F. Warr, F. W. Wilson, Piers Egerton Warburton, S. Wombwell, Weigall, Powell Williams, John Welby, Wingfleld, Whitbread, J. W. Wilson, Walton, D’Arcy Wyvill, Wodehouse, Wylie, A. Wilson, John Wilson, C. H. Wilson, Herbert Whiteley, Wynne, Lee Warner, W. West, G. Whiteley, Spencer Walpole, H. C. Woods, M.D., Deputy Inspector-General, Charles Wyndham, J. Humphrey Ward, F. Walker, Whateley, W. Woodall, Wyndham, Godfrey Webb, J. Welby, Charles Waldstein, H. Yorke and Yerburgh
#Mesdames&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 7a–b) —  , , , , , , , , , , , , , , , , , , , , , , , , , Maxwell-Lyte, F. A. Lucas, G. Manners, Beresford Melville, Morrell, Victor Milward, Marshall, Maclure, J. Maclean, M'Laren, M'Ewan, R. B. Martin, Marks, Markham, J. M'Calmont, F. W. Maude, Napier Miles, M’Neill, Max Muller, Meeking, Manvers Moorson, Arundel St. John Mildmay, Frederick Macmillan, Mount, Muntz, Murdoch, Wyndham Murray, W. J. Mure, Graham Murray, Montefiore, W. C. F. Molyneux, Newton Mant, Millett, Malet, Ashurst Morris, May, Maurice, Milvain, Marjoribanks, J. C. Macdona, Moorhouse, Muir Mackenzie, G. Moncrieff, J. Murray, Montgomery, Milbank, Bingham Mildmay, Mellor, C. Maude, T. G. Menzies, J. M'Donald, W. A. M'Arthur, M'Neile, M'N'eill, Campbell Munro, Mostyn, A. Milman, Majoribanks, Noel, H. F. Nicholson, F. Neville, Nicol, Nevul [?], Nugent, Newhouse, Oppenheim, M. Oldroyd, Charles Orde, H. H. Oldham, R. A. Oswald, Oswald. A. Oliphant, Oakley, J. L Pattison, Price, Perowne, Perks, Constantine Phipps, Peacocke [?], R. Prothero, Powell, Leslie Probyn, Pitman, Upton Prior, Lort Phillips, Primrose, Powlett, Pakenham, Peyton, Parkes, Wyndham Portal, Pipon, Pender, Phillpotts, Pollock, Montagu Price, Phellps, John Ponsonby, Fox Pitt, A. Peel, Aldrich Pelham, J. Pease, Poe, G. Pearse, A. Paget, A. Pease. N. G. Philips, Pirie [?], Dampier Palmer, F. Post, Pakenham, Paget, H. Parr, Wilton[?]-Phipps, Quilter, Rebow, J. C. Russell, Rolfe, Rutherford, I[?]. Richardson. James Ronand, Robins, Rennell Rodd, W. W. Russon[?], Alexander Ramsay, Robinson, J. Rennell Rodd, Redford, Harcourt Rose, Royds, H. Raikes, Carl Rosa, Ronalds, Arrnold Royle [? Royce?], Rice, Leopold Rothschild, Raikes, J. Rankin, Renshaw, F. Russell, Ricardo, Riddel, Robertson, G. Royle, Teignmouth [?] Shore, Sandeman, Stopford, Graham Smith, Salting, Brinsley Sheridan, Salmon [?], Salmond, Edgar Shephard, Sant. A. Sandeman, H. Seymour, H. S. Samuel, St. Clair, AbeI Smith, J. P. Smith, H. M. Stanley, A. Spicer, Stevenson, Swaine, Sullivan, J. H. Stock. E. B. Sparke, J. Sturgis, Louis Sinclair, H. Seton-Karr, Slade, J. Stern, Skefflngton Smyth, P. L. Slater, A. C. Stewart, R. Sassoon, C. Smith, E. Strachey, Napier Sturt, Steward, Eames Storey, Starkie, Senhouse, Bridgman Simpson, Seddon, T. Smith, Leslie Stephen, Settle, Scaramanga [?], Arthur Sassoon, L. Seymour, Shaw. R. F. Synge, T. Skewes-Cox, Stevenson, H. C. Smith, Sterling, T. H. Sidebottom, C. H. Seely, Shelley-Bontem [?], Sandford, Hawley Smart, Sergison [?], Frederick Slade, Scobell, Graves Sawle, Scott, Settle, Smith-Barry, Stewart, J. A. Swettenham, Surtees, Synge, Dudley Smith, Thomson, M. Thorold, H. Graham Toler [?], J. W. Taylor, Christopher Tower, Tosti, Temple, Beerbohm Tree, Dan Tupper, R. T. Thynne, Montagu Tharp, Trotter, Anstruther Thomson, Tupper, Taylor, C. E. Tritton, C. F. Anstruther Thomson, Edward Tighe, F. Taylor, Tillard, Tillbrook, Brook Taylor, Tudway, C. E. Thynne, J. C. Thynne, H. Thomas, Thwaites, Tarleton, A. Ure, Usher, R. Vivian, Val Prinsep, Edmund Vaughan, E. Villiers, C. van Raalte, Von André, Verschoyle, F. E. Villiers, Vance, Hope Vere, Villiers, Venning, Sackviile West, Whatman, Williams Wynn, Watson, Wharton, John Wilson, Williams, Stuart Wortley, Wood, C. H. Wilson, S. J. Way, Walton, H. Whiteley, G. Whiteley, Ellis Williams, Wilson, Weywan, E. F. Wodehouse, John Welby, Wray, Wickham, Whatley, Spencer Walpole, Hwfa Williams. J. Woodford, Charles Wyndham, Wingfield, Charles Wood, Lee Warner, Warre, Humphrey Ward, Wallis, Wilberforce, Wynne, J. Welby, Eardley Wilmot, A. S. Wilson, C. [?] E. Ward, Walter, Warner, R. G. Webster, Wells, Cornwallis-West, F. Charteris Wemyss, Yerburgh
#Misses&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 7c – 5, Col. 1a) —  , , , , , , , ,  , , , , Satyendra Bala '''Tagore''', , , Graham Murray, Mellor (2), Milward, Monk (2), Maxwell (2), Massey-Mainwaring, Mackworth (2), Markham (2), Macdonald, More-Molyneux, Cicely Monson, Maclure, Lena Milman, Morris (2), Macnaghten, Mowatt (2), Margaret Muir MacKenzie, Murray, Mundella, Mowbray, Ethel Morris, Beatrice Mildmay, May Milbank, Evelyn Moreton, Magniac (2), Mackenzie, M'Clintock, Madden, MacGregor(2) Mount (2), Muntz, Murdoch, Mitford, Montagu (2), Mure, Menzies, Macpherson-Grant, Malet, Moseley (2), Meeking, Macgregor, Mary Moore, Montgomery, St. John Mildmay, Madden (2), Milman (2), Constance Maude, Martyn, Campbell Munro, Nevill (2),Noel (2), Nevill, Nicol, Neville, Nelson (2). Olpherts [?] (2), Oakley, Ogilvy, Humphreys Owen, V. A. Okeover, O’Brien (2), Linda Oppenheim, Phoebe Otway, Alina O'Shee, Anderson Pelham, Pole, Pereira, Peyton, Pattison, Orde Powlett, Powlett, Pelly (2), Perowne, Charlotte Probyn, Julia Ponsonby, Peekover, Peel (2), Penn, Peace, Baden Powell, Powell (2), Pease, Priestley, Palgrave, Post, Parker, Pease, Palmer, Packe (2), Alice Paget, Paget, Paget of Cranmore (2), [?] Phillips, Phipps, Cecilia Peel, Chandos Pole, Pollock Phellps, Parry, Ponsonby, Wilton Phipps, Quain, Quilter (2), Russell of Killowen(2), Ritchie (2), Robins, Sibyl Robertson, Round (2), Royds (2), Russell, Rebow, Jane Ryan, Ramsay, Ricardo (2), Rigby, Russell (2), Lucy Raikes, Rankin, Frances Rod, Beatrix Rice, Russell (2), K. Reiss, Ricardo, Smith, Stafford, Stevenson, Stopford (2), M. Seymour, Kay Shuttleworth [?] (2), Seymour (2), Shaw, Shaw-Stewart, Evelyn Starling[?], Maxwell Scott, Abel Smith, Sartorius (2), Maud S[?]hey, Stewart, Magaret Stanley, Dorrien Smith, Smith (2), [?]-Smith, Saurin, Salmond (2), Sandeman (2), Sant, Dudley Smith[?], Swaine, Stephenson (2), Stewart (2), Dora Stone, Sparkes, [Stanley?], Nita Houston Stewart, Lily Severn, Evelyn Stanley, [Sheppart?], Saumarez Smith, Truda Saunderson, Swinburne, [Sullivan?], Mabel Seymour, Shute, R. Sterling, Stern (2), Sar[?] (2), Sassoon (2), P. L. Sclater, Sparke, Smith (Clement), [Sanderson?], Hilda Stewart, Seddon (2), Shelley, Sprigg (2), [?] Stephen, Ruby Spencer Churchill, Rachel Smith, [?], Tremayne (2), Ellen Terry, Ethel Thomas, Muriel [?], Taylor, Mary Talbot, Tomlinson, G. le M. Tupper, [?], Ella Taylor, Thorold, Taylor (2), E. Tuson, Trelawny [?], Adela[?] Trefusis, Rachel Thynne, Tritton (2), Thomson (2), [?], Thesiger, Thynne, I. C. (2), Thynne (2), Thornton (2), [Temple?], Turner, Talbot, Thynne, Usher, Van de Weyer (2), [Vivian?] (2), Dorothy Villiers, Freda Villiers, Verschoyle, Van [der Byl?], Villiers, Venning, Hilda von Deichmann, Wood[ford?], Fleetwood Wilson, Eardley-Wilmot, Maud Walpole, [?hend?] Wilson, Wilson, Wilberforce, Warren (2), [W?vil?] (2), Wills (2), Warrender (2), Walrond (2), Wynd[ham?] (2), Webster (2), Watson, Wombwell, Whitehead (2), [W?Ieyer?] (2), Evelyn Wellesley, Cornwallis West, Whatman {2), [?] (2), Rachel Weigall, F. Walker, Smart Walker, Wood (2), de la Wood[?], Ward, Wilbraham, Wilberforce (2), Walker, Williams, [Workham?] (2), Yeatman
#Admirals of the Fleet [initial large caps, rest sm caps] — Earl of Clanwilliam, Lord John [Hay?], the Hon. Sir H. Keppel
#Admirals — H. G. Andoe, C. E. Buckle, Sir F. Bedford, Britten, the Hon. W. Carpenter, H. F. Cleveland, Sir H. Chads, Close, [?], Carr, E. J. Church, Sir W. Dowell, R. G. Douglas, A. L. [?], C. E. Domvile, A. T. Dale, D’Eyncourt, Field, Sir A. [Farquhar?], Fitzgerald, Fellowes, Fanshawe, Sir H. Fairfax, Sir [?] Fisher, C. J. Fane, Fullerton, the Hon. Sir E. Fremantle, [?] FitzGeorge, Woods Pasha, Sir W. Hunt-Grubbe, Sir Anthony [?] Hoskin, Lord Hood of Avalon, Sir Leopold Heath, Sir [?] [F.?] Hotham, Sir Algernon Heneage, R. H. Hamond, the Right Hon. Sir [J.?] Hay, St. G. C. D’Arcy Irvine, Jones, Kennedy, Sir A. [?s], A. P. Lake, R. M. Lloyd, Sir L. Loraine, A. H. Markham, [Sir?] R. More-Molyneux, Sir F. L. M'Clintock, Sir R. Macdonald, [the?] Hon. V. Montagu, Nicholson, Noel, Marquis of Northampton, Sir E. Ommaney [?], Sir Augustus Phillimore, A. T. Powlett, [?], [?. ?.] Rowley, Sir F. Richards, Lord Charles Scott, [? St.? John?], W. H. C. St. Clair, Bowden Smith, Sulivan, E. H. Sey[mour?], H. Stephenson, Sir Nowell Salmon, Sir W. Houston [Stewart?], Sir M. [Cuhne?]-Seymour, E. W. Turnour, E. W. Van[?] Wharton, Sir G. Willes, the Hon. W. J. Ward
#Captain, R.N. — W. A. D. Acland, C. J. Barlow, F. R. Board[?], H. Bainbridge, Hon. T. Brand, Bickford, Lord Charles [B?ford?], B. F. Clark, Colville, Carter, Hon. S. Cecil Colville, [?ford?], A. G. Douglas, Sir C. Domville, Hon. A. Hay Dru[?], [?] [W.?] [?] Gordon, Hammet, Hon. Curzon Howe, Hender[?], [?] Ingles, Jellicoe, Jephson, Johnstone, Jeffreys, H. C. [?], Hon. A. Littleton, Hon. Hedworth Lambton, Moore, May, [? Net?], Poe, Pipon [?], Aldrich Pelham, Alfred Paget, [Bi.idcl?], Rolleston, John Sinclair, Bridgeman Simpson, [?], Van Koughnet [?], Burges Watson, Eardley-Wilmot, [?ham, Winsloe, Hon. J. Yorke
#[Lieutenants???] — Anson, G. R. Bethell, Blair, Bayley, Cave[?], [?] Cave,Hon. Cecil Cadogan, de Salis, Fraser, Floyd, Hon. [?] [F?], Alaric Grant, Morgan, Moore, Marescaux, [?] Stuart, Tupper, Wells, Williams, G. J. S. Warrender
#[Lieutenants?] R.N. — Alton, Murray Aynsley, Boyle, Bather, [?], [R. F.?] Boyle, Chaytor, Sir Charles Cust, G. W. Davy, [?] Wyndham-Fiennes, Fair, Godfrey Faussett, Garforth, [L?]ord Clifford, Hopkinson, Henderson, Keyes, Keppel, [?] Lloyd, Majendie, Mitchell, Morant, Kerr-Pearse, [?] Richmond, Rae, Stewart, Hon. Victor Stanley, [?] [Calta?]-Seymoar, Trye, Thring, Hon. Cyril Ward, W[?], R. E. Wemyss, Woolcombe
#[Captain?] Trinity House, Sir J. Sydney Webbe
#[Field?] Marshall — Sir F. P. Haines, Sir Lintorn Simmons, Sir [?] Stewart, Lord Roberts of Kandahar, Viscount Wolseley
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===Entertainment===
&quot;The Bands of the 1st Life Guards, Grenadier Guards, and Royal Artillery played a selection of music during the afternoon.&quot;&lt;ref name=&quot;:1&quot; /&gt; (4, Col. 2c)

==Anthology==

====Quote Intro====
&lt;quote&gt;&lt;/quote&gt; ()

== Notes and Questions ==
# 

==References==
*
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==Main Section==
Hello there! I am a school student currently living in the United Kingdom, in London. I have helped out on other Wikimedia projects such as Wikivoyage and Wikimedia Commons, and I hope to make good learning resources for everyone here to enjoy, which is my main goal here.
[[File:Permit To Travel machine at Salfords station.jpg|alt=PERTIS machine|thumb|A photo of mine]]

I have always been interested in transportation, mainly trains and mechanisms. I have also enjoyed helping out on:

Wikivoyage(my main home wiki)

Wikimedia Commons

Wikiversity (this website)

Please also make sure to ask any questions on my talk page, or if you just want to chat about certain topics.

Have a great day! :)

=== Pages that I am currently working on: ===
[[Basic Scratch Coding]]

[[Food Tests]]</text>
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== Introduction ==
This page on the topic &quot;Path Integral&quot; can be displayed as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Pathintegral&amp;author=Course:Functiontheory&amp;language=de&amp;audioslide=yes&amp;shorttitle=Pathintegral&amp;coursetitle=Course:Functiontheory Wiki2Reveal Slides]'''.
Individual sections are considered as slides, and changes to the slides immediately affect the content of the slides.
The following subtopics are treated in detail:

(1) Paths as continuous mappings from an interval &lt;math&gt;[a,b]&lt;/math&gt; into the complex numbers &lt;math&gt;\mathbb{C}&lt;/math&gt; over which integration is performed,

(2) Derivatives of curves/paths as a prerequisite for the definition of path integrals,

(3) Definition of path integrals


== Learning requirements ==
The learning resource on the topic &quot;Path Integral&quot; has the following learning prerequisites, which are helpful or necessary for understanding the subsequent explanations:

*Concept of[[w:en:Paths in a topological Space|Paths in a topological  Space]],

*Differentiability in real analysis,

*Integration in real analysis.


== Basic Geometric  Idea of the Path Integral == 
The following curve &lt;math&gt;\gamma&lt;/math&gt; loops around a point &lt;math&gt;z_0\in \mathbb{C}&lt;/math&gt; twice.

[[File:Windungszahl5.png|150px|center|Path around a point]]

== Integral over an Interval ==
 Let &lt;math&gt;G\subseteq \mathbb{C}&lt;/math&gt; be a [[w:en:Domain (mathematics)|domain]] and &lt;math&gt;g\colon [a,b] \to\mathbb{C}&lt;/math&gt; a [[w:en:Complex function|complex-valued function]]. The function &lt;math&gt;g&lt;/math&gt; is called integrable if
::&lt;math&gt;\operatorname{Re}(g):G \to\mathbb{R}&lt;/math&gt; and &lt;math&gt;\operatorname{Im}(g):G \to\mathbb{R}&lt;/math&gt; with &lt;math&gt;g=\operatorname{Re}(g) + i \cdot \operatorname{Im}(g)&lt;/math&gt; are integrable functions.
It is defined as
:&lt;math&gt;\int\limits_a^b g(x)\mathrm{d} x := \int\limits_a^b\operatorname{Re}(g)(x)\mathrm{d}x +\mathrm{i}\int\limits_a^b\operatorname{Im}(g)(x)\mathrm{d}x&lt;/math&gt;.
Thus, the integral is &lt;math&gt;\mathbb{C}&lt;/math&gt;-linear. If &lt;math&gt;g&lt;/math&gt; is continuous and &lt;math&gt;G&lt;/math&gt; is an antiderivative of &lt;math&gt;g&lt;/math&gt;, then as in the real case,
:&lt;math&gt;\int\limits_a^b g(x)\mathrm{d}x = G(b)-G(a)&lt;/math&gt;.

== Extension of the Integral Concept == 
The integral concept is extended through the definition of an integration path in the complex plane as follows: If &lt;math&gt;f\colon G\to\mathbb{C}&lt;/math&gt; is a complex-valued function on a [[w:en:Domain (mathematics)|domain]] &lt;math&gt;G\subseteq\mathbb{C}&lt;/math&gt;, and &lt;math&gt;\gamma\colon[a,b]\to G&lt;/math&gt; is a piecewise continuously differentiable [[w:en:Path (mathematics)|path]] in &lt;math&gt;G&lt;/math&gt;, then the ''path integral'' of &lt;math&gt;f&lt;/math&gt; along the path &lt;math&gt;\gamma&lt;/math&gt; is defined as
: &lt;math&gt;\int\limits_\gamma f:=\int\limits_\gamma f(z),\mathrm dz:=\int\limits_a^b f(\gamma(t))\cdot \gamma'(t),\mathrm dt.&lt;/math&gt;
Here, the multiplication sign refers to complex multiplication.&lt;ref&gt;„Curve Integral“. In: Wikipedia, The Free Encyclopedia. Editing status: November 24, 2017, 16:22 UTC. URL: https://en.wikipedia.org/w/index.php?title=Curve_integral&amp;oldid=171345033 (Accessed: December 8, 2017, 14:27 UTC) &lt;/ref&gt;

== Cauchy's Integral Theorem ==
The central statement about path integrals of complex functions is the [[w:en:Cauchy's_integral_theorem|Cauchy Integral Theorem]]: For a [[w:en:Holomorphic function|holomorphic]] function &lt;math&gt;f&lt;/math&gt;, the path integral depends only on the [[w:en:Homotopy|homotopy]] class of &lt;math&gt;\gamma&lt;/math&gt;. If &lt;math&gt;U&lt;/math&gt; is [[w:en:Simply_connected_space|simply connected]], then the integral depends not on &lt;math&gt;\gamma&lt;/math&gt;, but only on the starting and ending points.

Analogous to the real case, the ''length'' of the path &lt;math&gt;\gamma:[a,b]\rightarrow \mathbb{C}&lt;/math&gt; is defined as
:&lt;math&gt;\mathcal{L}(\gamma):=\int\limits_a^b \left| \gamma'(t) \right| \mathrm{d}t&lt;/math&gt;.

For theoretical purposes, the following inequality, called the ''standard estimate'', is of particular interest:
:&lt;math&gt;\left| \int_\gamma f(z) , \mathrm dz \right| \leq \mathcal{L}(\gamma)\cdot C&lt;/math&gt;, if &lt;math&gt;\left| f(z) \right|\leq C&lt;/math&gt; for all &lt;math&gt;z\in\gamma([0,1])&lt;/math&gt;.

As in the real case, the path integral is independent of the parametrization of the path &lt;math&gt;\gamma&lt;/math&gt;, i.e., it is not strictly necessary to choose &lt;math&gt;[0,1]&lt;/math&gt; as the parameter domain, as can be shown by substitution. This allows the definition of complex curve integrals by replacing the above formulas with a curve &lt;math&gt;\mathcal{C}&lt;/math&gt; in &lt;math&gt;\mathbb{C}&lt;/math&gt;.

== Exercises ==

*Be &lt;math&gt;\gamma\colon[a,b]\to G&lt;/math&gt; with &lt;math&gt;t\mapsto \gamma(t)= \sin(t)+i\cdot t^2&lt;/math&gt;. Determine &lt;math&gt;\gamma'(t)&lt;/math&gt;!
*Compute the path integral &lt;math&gt;\int\limits_\gamma \frac{1} {z},\mathrm dz&lt;/math&gt; for the path &lt;math&gt;\gamma\colon[0,2\pi] \to \mathbb{C}&lt;/math&gt; with &lt;math&gt;t\mapsto \gamma(t)= r\cdot e^{i\cdot t}&lt;/math&gt;.

*Calculate the length of the path &lt;math&gt;\mathcal(\gamma)&lt;/math&gt; with &lt;math&gt;t\mapsto \gamma(t)= r\cdot e^{i\cdot t}&lt;/math&gt;.

== See also ==

*[[w:en: Function theory (course)|Function Theory Course]]

*[[w:en: Line integral|Line integral]]


== Literature ==
 &lt;references/&gt;


== Page Information ==
You can display this page as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=%20Path%20Integral&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=%20Path%20Integral&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]'''

=== Wiki2Reveal ===
The '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=%20Path%20Integral&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=%20Path%20Integral&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator].
&lt;!--
* Contents of the page are based on:
** [https://en.wikipedia.org/wiki/%20Path%20Integral https://en.wikiversity.org/wiki/%20Path%20Integral]
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=== Translation and Version Control ===

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* Source: [[v:de:Wegintegral|Wegintegral]] - URL: https://en.wikiversity.org/wiki/Wegintegral
* Date: 11/20/2024

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== Introduction ==
This page on the topic &quot;Path Integral&quot; can be displayed as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Pathintegral&amp;author=Course:Functiontheory&amp;language=de&amp;audioslide=yes&amp;shorttitle=Pathintegral&amp;coursetitle=Course:Functiontheory Wiki2Reveal Slides]'''.
Individual sections are considered as slides, and changes to the slides immediately affect the content of the slides.
The following subtopics are treated in detail:

(1) Paths as continuous mappings from an interval &lt;math&gt;[a,b]&lt;/math&gt; into the complex numbers &lt;math&gt;\mathbb{C}&lt;/math&gt; over which integration is performed,

(2) Derivatives of curves/paths as a prerequisite for the definition of path integrals,

(3) Definition of path integrals


== Learning requirements ==
The learning resource on the topic &quot;Path Integral&quot; has the following learning prerequisites, which are helpful or necessary for understanding the subsequent explanations:

*Concept of[[w:en:Paths in a topological Space|Paths in a topological  Space]],

*Differentiability in real analysis,

*Integration in real analysis.


== Basic Geometric  Idea of the Path Integral == 
The following curve &lt;math&gt;\gamma&lt;/math&gt; loops around a point &lt;math&gt;z_0\in \mathbb{C}&lt;/math&gt; twice.

[[File:Windungszahl5.png|150px|center|Path around a point]]

== Integral over an Interval ==
 Let &lt;math&gt;G\subseteq \mathbb{C}&lt;/math&gt; be a [[w:en:Domain (mathematics)|domain]] and &lt;math&gt;g\colon [a,b] \to\mathbb{C}&lt;/math&gt; a [[w:en:Complex function|complex-valued function]]. The function &lt;math&gt;g&lt;/math&gt; is called integrable if
::&lt;math&gt;\operatorname{Re}(g):G \to\mathbb{R}&lt;/math&gt; and &lt;math&gt;\operatorname{Im}(g):G \to\mathbb{R}&lt;/math&gt; with &lt;math&gt;g=\operatorname{Re}(g) + i \cdot \operatorname{Im}(g)&lt;/math&gt; are integrable functions.
It is defined as
:&lt;math&gt;\int\limits_a^b g(x)\mathrm{d} x := \int\limits_a^b\operatorname{Re}(g)(x)\mathrm{d}x +\mathrm{i}\int\limits_a^b\operatorname{Im}(g)(x)\mathrm{d}x&lt;/math&gt;.
Thus, the integral is &lt;math&gt;\mathbb{C}&lt;/math&gt;-linear. If &lt;math&gt;g&lt;/math&gt; is continuous and &lt;math&gt;G&lt;/math&gt; is an antiderivative of &lt;math&gt;g&lt;/math&gt;, then as in the real case,
:&lt;math&gt;\int\limits_a^b g(x)\mathrm{d}x = G(b)-G(a)&lt;/math&gt;.

== Extension of the Integral Concept == 
The integral concept is extended through the definition of an integration path in the complex plane as follows: If &lt;math&gt;f\colon G\to\mathbb{C}&lt;/math&gt; is a complex-valued function on a [[w:en:Domain (mathematics)|domain]] &lt;math&gt;G\subseteq\mathbb{C}&lt;/math&gt;, and &lt;math&gt;\gamma\colon[a,b]\to G&lt;/math&gt; is a piecewise continuously differentiable [[w:en:Path (mathematics)|path]] in &lt;math&gt;G&lt;/math&gt;, then the ''path integral'' of &lt;math&gt;f&lt;/math&gt; along the path &lt;math&gt;\gamma&lt;/math&gt; is defined as
: &lt;math&gt;\int\limits_\gamma f:=\int\limits_\gamma f(z),\mathrm dz:=\int\limits_a^b f(\gamma(t))\cdot \gamma'(t),\mathrm dt.&lt;/math&gt;
Here, the multiplication sign refers to complex multiplication.&lt;ref&gt;„Curve Integral“. In: Wikipedia, The Free Encyclopedia. Editing status: November 24, 2017, 16:22 UTC. URL: https://en.wikipedia.org/w/index.php?title=Curve_integral&amp;oldid=171345033 (Accessed: December 8, 2017, 14:27 UTC) &lt;/ref&gt;

== Cauchy's Integral Theorem ==
The central statement about path integrals of complex functions is the [[w:en:Cauchy's_integral_theorem|Cauchy Integral Theorem]]: For a [[w:en:Holomorphic function|holomorphic]] function &lt;math&gt;f&lt;/math&gt;, the path integral depends only on the [[w:en:Homotopy|homotopy]] class of &lt;math&gt;\gamma&lt;/math&gt;. If &lt;math&gt;U&lt;/math&gt; is [[w:en:Simply_connected_space|simply connected]], then the integral depends not on &lt;math&gt;\gamma&lt;/math&gt;, but only on the starting and ending points.

Analogous to the real case, the ''length'' of the path &lt;math&gt;\gamma:[a,b]\rightarrow \mathbb{C}&lt;/math&gt; is defined as
:&lt;math&gt;\mathcal{L}(\gamma):=\int\limits_a^b \left| \gamma'(t) \right| \mathrm{d}t&lt;/math&gt;.

For theoretical purposes, the following inequality, called the ''standard estimate'', is of particular interest:
:&lt;math&gt;\left| \int_\gamma f(z) , \mathrm dz \right| \leq \mathcal{L}(\gamma)\cdot C&lt;/math&gt;, if &lt;math&gt;\left| f(z) \right|\leq C&lt;/math&gt; for all &lt;math&gt;z\in\gamma([0,1])&lt;/math&gt;.

As in the real case, the path integral is independent of the parametrization of the path &lt;math&gt;\gamma&lt;/math&gt;, i.e., it is not strictly necessary to choose &lt;math&gt;[0,1]&lt;/math&gt; as the parameter domain, as can be shown by substitution. This allows the definition of complex curve integrals by replacing the above formulas with a curve &lt;math&gt;\mathcal{C}&lt;/math&gt; in &lt;math&gt;\mathbb{C}&lt;/math&gt;.

== Exercises ==

*Be &lt;math&gt;\gamma\colon[a,b]\to G&lt;/math&gt; with &lt;math&gt;t\mapsto \gamma(t)= \sin(t)+i\cdot t^2&lt;/math&gt;. Determine &lt;math&gt;\gamma'(t)&lt;/math&gt;!
*Compute the path integral &lt;math&gt;\int\limits_\gamma \frac{1} {z},\mathrm dz&lt;/math&gt; for the path &lt;math&gt;\gamma\colon[0,2\pi] \to \mathbb{C}&lt;/math&gt; with &lt;math&gt;t\mapsto \gamma(t)= r\cdot e^{i\cdot t}&lt;/math&gt;.

*Calculate the length of the path &lt;math&gt;\mathcal(\gamma)&lt;/math&gt; with &lt;math&gt;t\mapsto \gamma(t)= r\cdot e^{i\cdot t}&lt;/math&gt;.

== See also ==
* [[Complex Analysis]]

* [[w:en: Line integral|Line integral]]

== Literature ==
 &lt;references/&gt;


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      <text bytes="6022" sha1="ejl15u2j1ihzb4ptpc80lfojjjovry2" xml:space="preserve">== Smooth paths and path subdivision ==
The following definitions were abbreviated with acronyms and are used as justifications for transformations or conclusions in proofs.

* '''(WG1) Definition (Smooth path):''' A path &lt;math&gt;\gamma: [a,b] \to\mathbb{C}&lt;/math&gt; is smooth if it is continuously differentiable.
* '''(UT) Definition (Subdivision):''' Let &lt;math&gt;[a,b]&lt;/math&gt; be an interval, &lt;math&gt;n \in\mathbb{N}&lt;/math&gt; and &lt;math&gt;{a}={u}_{{0}} &lt; {\ldots} &lt; {{u}}_{n}={b}&lt;/math&gt;. &lt;math&gt;{\left({u}_{{0}},\ldots,{u}_{{{n}}}\right)}\in\mathbb{R}^{n+1}&lt;/math&gt; is called a subdivision of &lt;math&gt;{\left[{a},{b}\right]}&lt;/math&gt;.
* '''(WG2) Definition (Path subdivision):''' Let &lt;math&gt;\gamma: [a,b] \to\mathbb{C}&lt;/math&gt; be a path in &lt;math&gt;{U}\subseteq\mathbb{C}&lt;/math&gt;, &lt;math&gt;{n}\in\mathbb{N}&lt;/math&gt;, &lt;math&gt;{\left({u}_{{0}},\ldots,{u}_{{{n}}}\right)}&lt;/math&gt; a subdivision of &lt;math&gt;[a,b]&lt;/math&gt;, &lt;math&gt;\gamma_{{k}}:{\left[{u}_{{{k}-{1}}},{u}_{{k}}\right]}\to\mathbb{C}&lt;/math&gt; for all &lt;math&gt;{k}\in{\left\lbrace{1},\ldots,{n}\right\rbrace}&lt;/math&gt; a path in &lt;math&gt;{U}&lt;/math&gt;. &lt;math&gt;{\left(\gamma_{{{1}}},\ldots,\gamma_{{{n}}}\right)}&lt;/math&gt; is called a path subdivision of &lt;math&gt;\gamma&lt;/math&gt; if &lt;math&gt;\gamma_{{n}}{\left({b}\right)}=\gamma{\left({b}\right)}&lt;/math&gt; and for all &lt;math&gt;{k}\in{\left\lbrace{1},\ldots,{n}\right\rbrace}&lt;/math&gt; and &lt;math&gt;{t}\in{\left[{u}_{{{k}-{1}}},{u}_{{k}}\right]}&lt;/math&gt; we have &lt;math&gt;\gamma_{{k}}{\left({t}\right)}=\gamma{\left({t}\right)}\wedge\gamma_{{k}}{\left({u}_{{{k}-{1}}}\right)}=\gamma_{{{k}-{1}}}{\left({u}_{{k}}\right)}&lt;/math&gt;.
* '''(WG3) Definition (Piecewise smooth path):''' A path &lt;math&gt;\gamma:{\left[{a},{b}\right]}\to\mathbb{C}&lt;/math&gt; is piecewise smooth if there exists a path subdivision &lt;math&gt;{\left(\gamma_{{1}},\ldots\gamma_{{n}}\right)}&lt;/math&gt; of &lt;math&gt;\gamma&lt;/math&gt; consisting of smooth paths &lt;math&gt;\gamma_{{k}}&lt;/math&gt; for all &lt;math&gt;{k}\in{\left\lbrace{1},\ldots,{n}\right\rbrace}&lt;/math&gt;.

== Integration path ==
* '''(WG4) Definition (Path integral):''' Let &lt;math&gt;f : U \to\mathbb{C}&lt;/math&gt; be a continuous function and &lt;math&gt;\gamma: [a,b] \to U &lt;/math&gt; a smooth path, then the path integral is defined as: &lt;math&gt;\int_{\gamma} f := \int_{\gamma} f(z) \, dz := \int_a^b f(\gamma(t)) \cdot\gamma'(t)\, dt &lt;/math&gt;. If &lt;math&gt;\gamma&lt;/math&gt; is only piecewise smooth with respect to a path subdivision &lt;math&gt;( \gamma_1 ,\ldots,\gamma_n ) &lt;/math&gt;, then we define &lt;math&gt;\int_{\gamma} f(z) \, dz :=\sum_{k=1}^{n} \int_{\gamma_k} f(z) \, dz&lt;/math&gt;.
* '''Definition (Integration path):''' An integration path is a piecewise smooth (piecewise continuously differentiable) path.

== Example ==
[[File:Dreiecksweg.svg|mini|Integration path on the triangle edge]]

The following path is piecewise continuously differentiable (smooth) and for the vertices &lt;math&gt;z_1,z_2,z_3\in \text{Spur}(\gamma)&lt;/math&gt; the closed triangle path &lt;math&gt;\gamma : [0,3] \to \mathbb{C}&lt;/math&gt; is not differentiable. The triangle path is defined on the interval &lt;math&gt;[0,3]&lt;/math&gt; as follows:
:&lt;math&gt;
\gamma(t) := \left\langle z_1 ,z_2 ,z_3 \right\rangle (t) := 
\begin{cases}
 (1-t)\cdot z_1 + t\cdot z_2 &amp; \text{for } t \in [0,1] \\
 (2-t)\cdot z_2 + (t-1)\cdot z_3 &amp; \text{for } t \in (1,2] \\ 
 (3-t)\cdot z_3 + (t-2) \cdot z_1 &amp; \text{for } t \in (2,3] \\ 
\end{cases}
&lt;/math&gt;

=== Paths from convex combinations ===
The piecewise continuously differentiable path is formed from [[convex combination]].The sub-paths
* &lt;math&gt;\gamma_1 := \left\langle z_1 ,z_2 \right\rangle &lt;/math&gt; with &lt;math&gt;\gamma_1 : [0,1] \to \mathbb{C}, \ (1-t)\cdot z_1 + t\cdot z_2&lt;/math&gt; 
* &lt;math&gt;\gamma_2 := \left\langle z_2 ,z_3 \right\rangle &lt;/math&gt; with &lt;math&gt;\gamma_2 : [1,2] \to \mathbb{C}, \ (2-t)\cdot z_2 + (t-1)\cdot z_3 &lt;/math&gt;
* &lt;math&gt;\gamma_3 := \left\langle z_3 ,z_1 \right\rangle &lt;/math&gt; with &lt;math&gt;\gamma_3 : [2,3] \to \mathbb{C}, \ (3-t)\cdot z_3 + (t-2)\cdot z_1 &lt;/math&gt;
are continuously differentiable.

== See also ==
* [[Complex Analysis/Goursat's Lemma (Details)|Goursat's Lemma (Details)]]
* [[convex combination]]
* [https://www.geogebra.org/m/rwwjymrv Convex combinations and interpolation on triangles in the plane]


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      <text bytes="5994" sha1="8amiq4892v86e9ka0sxjq527uja6t1q" xml:space="preserve">== Smooth paths and path subdivision ==
The following definitions were abbreviated with acronyms and are used as justifications for transformations or conclusions in proofs.

* '''(WG1) Definition (Smooth path):''' A path &lt;math&gt;\gamma: [a,b] \to\mathbb{C}&lt;/math&gt; is smooth if it is continuously differentiable.
* '''(UT) Definition (Subdivision):''' Let &lt;math&gt;[a,b]&lt;/math&gt; be an interval, &lt;math&gt;n \in\mathbb{N}&lt;/math&gt; and &lt;math&gt;{a}={u}_{{0}} &lt; {\ldots} &lt; {{u}}_{n}={b}&lt;/math&gt;. &lt;math&gt;{\left({u}_{{0}},\ldots,{u}_{{{n}}}\right)}\in\mathbb{R}^{n+1}&lt;/math&gt; is called a subdivision of &lt;math&gt;{\left[{a},{b}\right]}&lt;/math&gt;.
* '''(WG2) Definition (Path subdivision):''' Let &lt;math&gt;\gamma: [a,b] \to\mathbb{C}&lt;/math&gt; be a path in &lt;math&gt;{U}\subseteq\mathbb{C}&lt;/math&gt;, &lt;math&gt;{n}\in\mathbb{N}&lt;/math&gt;, &lt;math&gt;{\left({u}_{{0}},\ldots,{u}_{{{n}}}\right)}&lt;/math&gt; a subdivision of &lt;math&gt;[a,b]&lt;/math&gt;, &lt;math&gt;\gamma_{{k}}:{\left[{u}_{{{k}-{1}}},{u}_{{k}}\right]}\to\mathbb{C}&lt;/math&gt; for all &lt;math&gt;{k}\in{\left\lbrace{1},\ldots,{n}\right\rbrace}&lt;/math&gt; a path in &lt;math&gt;{U}&lt;/math&gt;. &lt;math&gt;{\left(\gamma_{{{1}}},\ldots,\gamma_{{{n}}}\right)}&lt;/math&gt; is called a path subdivision of &lt;math&gt;\gamma&lt;/math&gt; if &lt;math&gt;\gamma_{{n}}{\left({b}\right)}=\gamma{\left({b}\right)}&lt;/math&gt; and for all &lt;math&gt;{k}\in{\left\lbrace{1},\ldots,{n}\right\rbrace}&lt;/math&gt; and &lt;math&gt;{t}\in{\left[{u}_{{{k}-{1}}},{u}_{{k}}\right]}&lt;/math&gt; we have &lt;math&gt;\gamma_{{k}}{\left({t}\right)}=\gamma{\left({t}\right)}\wedge\gamma_{{k}}{\left({u}_{{{k}-{1}}}\right)}=\gamma_{{{k}-{1}}}{\left({u}_{{k}}\right)}&lt;/math&gt;.
* '''(WG3) Definition (Piecewise smooth path):''' A path &lt;math&gt;\gamma:{\left[{a},{b}\right]}\to\mathbb{C}&lt;/math&gt; is piecewise smooth if there exists a path subdivision &lt;math&gt;{\left(\gamma_{{1}},\ldots\gamma_{{n}}\right)}&lt;/math&gt; of &lt;math&gt;\gamma&lt;/math&gt; consisting of smooth paths &lt;math&gt;\gamma_{{k}}&lt;/math&gt; for all &lt;math&gt;{k}\in{\left\lbrace{1},\ldots,{n}\right\rbrace}&lt;/math&gt;.

== Integration path ==
* '''(WG4) Definition (Path integral):''' Let &lt;math&gt;f : U \to\mathbb{C}&lt;/math&gt; be a continuous function and &lt;math&gt;\gamma: [a,b] \to U &lt;/math&gt; a smooth path, then the path integral is defined as: &lt;math&gt;\int_{\gamma} f := \int_{\gamma} f(z) \, dz := \int_a^b f(\gamma(t)) \cdot\gamma'(t)\, dt &lt;/math&gt;. If &lt;math&gt;\gamma&lt;/math&gt; is only piecewise smooth with respect to a path subdivision &lt;math&gt;( \gamma_1 ,\ldots,\gamma_n ) &lt;/math&gt;, then we define &lt;math&gt;\int_{\gamma} f(z) \, dz :=\sum_{k=1}^{n} \int_{\gamma_k} f(z) \, dz&lt;/math&gt;.
* '''Definition (Integration path):''' An integration path is a piecewise smooth (piecewise continuously differentiable) path.

== Example ==
[[File:Dreiecksweg.svg|mini|Integration path on the triangle edge]]

The following path is piecewise continuously differentiable (smooth) and for the vertices &lt;math&gt;z_1,z_2,z_3\in \text{Spur}(\gamma)&lt;/math&gt; the closed triangle path &lt;math&gt;\gamma : [0,3] \to \mathbb{C}&lt;/math&gt; is not differentiable. The triangle path is defined on the interval &lt;math&gt;[0,3]&lt;/math&gt; as follows:
:&lt;math&gt;
\gamma(t) := \left\langle z_1 ,z_2 ,z_3 \right\rangle (t) := 
\begin{cases}
 (1-t)\cdot z_1 + t\cdot z_2 &amp; \text{for } t \in [0,1] \\
 (2-t)\cdot z_2 + (t-1)\cdot z_3 &amp; \text{for } t \in (1,2] \\ 
 (3-t)\cdot z_3 + (t-2) \cdot z_1 &amp; \text{for } t \in (2,3] \\ 
\end{cases}
&lt;/math&gt;

=== Paths from convex combinations ===
The piecewise continuously differentiable path is formed from [[convex combination]].The sub-paths
* &lt;math&gt;\gamma_1 := \left\langle z_1 ,z_2 \right\rangle &lt;/math&gt; with &lt;math&gt;\gamma_1 : [0,1] \to \mathbb{C}, \ (1-t)\cdot z_1 + t\cdot z_2&lt;/math&gt; 
* &lt;math&gt;\gamma_2 := \left\langle z_2 ,z_3 \right\rangle &lt;/math&gt; with &lt;math&gt;\gamma_2 : [1,2] \to \mathbb{C}, \ (2-t)\cdot z_2 + (t-1)\cdot z_3 &lt;/math&gt;
* &lt;math&gt;\gamma_3 := \left\langle z_3 ,z_1 \right\rangle &lt;/math&gt; with &lt;math&gt;\gamma_3 : [2,3] \to \mathbb{C}, \ (3-t)\cdot z_3 + (t-2)\cdot z_1 &lt;/math&gt;
are continuously differentiable.

== See also ==
* [[Complex Analysis/Goursat's Lemma (Details)|Goursat's Lemma (Details)]]
* [[convex combination]]
* [https://www.geogebra.org/m/rwwjymrv Convex combinations and interpolation on triangles in the plane]


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&lt;noinclude&gt;[[de:Integrationsweg]]&lt;/noinclude&gt;

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      <text bytes="6030" sha1="cy6iyw5nrzk61j8mlphclz6xcxasuph" xml:space="preserve">== Smooth paths and path subdivision ==
The following definitions were abbreviated with acronyms and are used as justifications for transformations or conclusions in proofs.

* '''(WG1) Definition (Smooth path):''' A path &lt;math&gt;\gamma: [a,b] \to\mathbb{C}&lt;/math&gt; is smooth if it is continuously differentiable.
* '''(UT) Definition (Subdivision):''' Let &lt;math&gt;[a,b]&lt;/math&gt; be an interval, &lt;math&gt;n \in\mathbb{N}&lt;/math&gt; and &lt;math&gt;{a}={u}_{{0}} &lt; {\ldots} &lt; {{u}}_{n}={b}&lt;/math&gt;. &lt;math&gt;{\left({u}_{{0}},\ldots,{u}_{{{n}}}\right)}\in\mathbb{R}^{n+1}&lt;/math&gt; is called a subdivision of &lt;math&gt;{\left[{a},{b}\right]}&lt;/math&gt;.
* '''(WG2) Definition (Path subdivision):''' Let &lt;math&gt;\gamma: [a,b] \to\mathbb{C}&lt;/math&gt; be a path in &lt;math&gt;{U}\subseteq\mathbb{C}&lt;/math&gt;, &lt;math&gt;{n}\in\mathbb{N}&lt;/math&gt;, &lt;math&gt;{\left({u}_{{0}},\ldots,{u}_{{{n}}}\right)}&lt;/math&gt; a subdivision of &lt;math&gt;[a,b]&lt;/math&gt;, &lt;math&gt;\gamma_{{k}}:{\left[{u}_{{{k}-{1}}},{u}_{{k}}\right]}\to\mathbb{C}&lt;/math&gt; for all &lt;math&gt;{k}\in{\left\lbrace{1},\ldots,{n}\right\rbrace}&lt;/math&gt; a path in &lt;math&gt;{U}&lt;/math&gt;. &lt;math&gt;{\left(\gamma_{{{1}}},\ldots,\gamma_{{{n}}}\right)}&lt;/math&gt; is called a path subdivision of &lt;math&gt;\gamma&lt;/math&gt; if &lt;math&gt;\gamma_{{n}}{\left({b}\right)}=\gamma{\left({b}\right)}&lt;/math&gt; and for all &lt;math&gt;{k}\in{\left\lbrace{1},\ldots,{n}\right\rbrace}&lt;/math&gt; and &lt;math&gt;{t}\in{\left[{u}_{{{k}-{1}}},{u}_{{k}}\right]}&lt;/math&gt; we have &lt;math&gt;\gamma_{{k}}{\left({t}\right)}=\gamma{\left({t}\right)}\wedge\gamma_{{k}}{\left({u}_{{{k}-{1}}}\right)}=\gamma_{{{k}-{1}}}{\left({u}_{{k}}\right)}&lt;/math&gt;.
* '''(WG3) Definition (Piecewise smooth path):''' A path &lt;math&gt;\gamma:{\left[{a},{b}\right]}\to\mathbb{C}&lt;/math&gt; is piecewise smooth if there exists a path subdivision &lt;math&gt;{\left(\gamma_{{1}},\ldots\gamma_{{n}}\right)}&lt;/math&gt; of &lt;math&gt;\gamma&lt;/math&gt; consisting of smooth paths &lt;math&gt;\gamma_{{k}}&lt;/math&gt; for all &lt;math&gt;{k}\in{\left\lbrace{1},\ldots,{n}\right\rbrace}&lt;/math&gt;.

== Integration path ==
* '''(WG4) Definition (Path integral):''' Let &lt;math&gt;f : U \to\mathbb{C}&lt;/math&gt; be a continuous function and &lt;math&gt;\gamma: [a,b] \to U &lt;/math&gt; a smooth path, then the path integral is defined as: &lt;math&gt;\int_{\gamma} f := \int_{\gamma} f(z) \, dz := \int_a^b f(\gamma(t)) \cdot\gamma'(t)\, dt &lt;/math&gt;. If &lt;math&gt;\gamma&lt;/math&gt; is only piecewise smooth with respect to a path subdivision &lt;math&gt;( \gamma_1 ,\ldots,\gamma_n ) &lt;/math&gt;, then we define &lt;math&gt;\int_{\gamma} f(z) \, dz :=\sum_{k=1}^{n} \int_{\gamma_k} f(z) \, dz&lt;/math&gt;.
* '''Definition (Integration path):''' An integration path is a piecewise smooth (piecewise continuously differentiable) path.

== Example ==
[[File:Dreiecksweg.svg|mini|Integration path on the triangle edge]]

The following path is piecewise continuously differentiable (smooth) and for the vertices &lt;math&gt;z_1,z_2,z_3\in \text{Spur}(\gamma)&lt;/math&gt; the closed triangle path &lt;math&gt;\gamma : [0,3] \to \mathbb{C}&lt;/math&gt; is not differentiable. The triangle path is defined on the interval &lt;math&gt;[0,3]&lt;/math&gt; as follows:
:&lt;math&gt;
\gamma(t) := \left\langle z_1 ,z_2 ,z_3 \right\rangle (t) := 
\begin{cases}
 (1-t)\cdot z_1 + t\cdot z_2 &amp; \text{for } t \in [0,1] \\
 (2-t)\cdot z_2 + (t-1)\cdot z_3 &amp; \text{for } t \in (1,2] \\ 
 (3-t)\cdot z_3 + (t-2) \cdot z_1 &amp; \text{for } t \in (2,3] \\ 
\end{cases}
&lt;/math&gt;

=== Paths from convex combinations ===
The piecewise continuously differentiable path is formed from [[convex combination]].The sub-paths
* &lt;math&gt;\gamma_1 := \left\langle z_1 ,z_2 \right\rangle &lt;/math&gt; with &lt;math&gt;\gamma_1 : [0,1] \to \mathbb{C}, \ (1-t)\cdot z_1 + t\cdot z_2&lt;/math&gt; 
* &lt;math&gt;\gamma_2 := \left\langle z_2 ,z_3 \right\rangle &lt;/math&gt; with &lt;math&gt;\gamma_2 : [1,2] \to \mathbb{C}, \ (2-t)\cdot z_2 + (t-1)\cdot z_3 &lt;/math&gt;
* &lt;math&gt;\gamma_3 := \left\langle z_3 ,z_1 \right\rangle &lt;/math&gt; with &lt;math&gt;\gamma_3 : [2,3] \to \mathbb{C}, \ (3-t)\cdot z_3 + (t-2)\cdot z_1 &lt;/math&gt;
are continuously differentiable.

== See also ==
* [[Complex Analysis/Goursat's Lemma (Details)|Goursat's Lemma (Details)]]
* [[convex combination]]
* [https://www.geogebra.org/m/rwwjymrv Convex combinations and interpolation on triangles in the plane]


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* Source: [[v:de:Funktionentheorie/Integrationsweg|Funktionentheorie/Integrationsweg]] - URL: https://en.wikiversity.org/wiki/Wegintegral
* Date: 11/20/2024

&lt;span type=&quot;translate&quot; src=&quot;Integrationsweg&quot; srclang=&quot;de&quot; date=&quot;12/17/2024&quot; time=&quot;11:42&quot; status=&quot;inprogress&quot;&gt;&lt;/span&gt;

&lt;noinclude&gt;[[de:Integrationsweg]]&lt;/noinclude&gt;

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        <id>2387134</id>
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      <text bytes="6114" sha1="dk2mxpuigyhzcfhf8lbb7ylekcxc1ar" xml:space="preserve">== Smooth paths and path subdivision ==
The following definitions were abbreviated with acronyms and are used as justifications for transformations or conclusions in proofs.

* '''(WG1) Definition (Smooth path):''' A path &lt;math&gt;\gamma: [a,b] \to\mathbb{C}&lt;/math&gt; is smooth if it is continuously differentiable.
* '''(UT) Definition (Subdivision):''' Let &lt;math&gt;[a,b]&lt;/math&gt; be an interval, &lt;math&gt;n \in\mathbb{N}&lt;/math&gt; and &lt;math&gt;{a}={u}_{{0}} &lt; {\ldots} &lt; {{u}}_{n}={b}&lt;/math&gt;. &lt;math&gt;{\left({u}_{{0}},\ldots,{u}_{{{n}}}\right)}\in\mathbb{R}^{n+1}&lt;/math&gt; is called a subdivision of &lt;math&gt;{\left[{a},{b}\right]}&lt;/math&gt;.
* '''(WG2) Definition (Path subdivision):''' Let &lt;math&gt;\gamma: [a,b] \to\mathbb{C}&lt;/math&gt; be a path in &lt;math&gt;{U}\subseteq\mathbb{C}&lt;/math&gt;, &lt;math&gt;{n}\in\mathbb{N}&lt;/math&gt;, &lt;math&gt;{\left({u}_{{0}},\ldots,{u}_{{{n}}}\right)}&lt;/math&gt; a subdivision of &lt;math&gt;[a,b]&lt;/math&gt;, &lt;math&gt;\gamma_{{k}}:{\left[{u}_{{{k}-{1}}},{u}_{{k}}\right]}\to\mathbb{C}&lt;/math&gt; for all &lt;math&gt;{k}\in{\left\lbrace{1},\ldots,{n}\right\rbrace}&lt;/math&gt; a path in &lt;math&gt;{U}&lt;/math&gt;. &lt;math&gt;{\left(\gamma_{{{1}}},\ldots,\gamma_{{{n}}}\right)}&lt;/math&gt; is called a path subdivision of &lt;math&gt;\gamma&lt;/math&gt; if &lt;math&gt;\gamma_{{n}}{\left({b}\right)}=\gamma{\left({b}\right)}&lt;/math&gt; and for all &lt;math&gt;{k}\in{\left\lbrace{1},\ldots,{n}\right\rbrace}&lt;/math&gt; and &lt;math&gt;{t}\in{\left[{u}_{{{k}-{1}}},{u}_{{k}}\right]}&lt;/math&gt; we have &lt;math&gt;\gamma_{{k}}{\left({t}\right)}=\gamma{\left({t}\right)}\wedge\gamma_{{k}}{\left({u}_{{{k}-{1}}}\right)}=\gamma_{{{k}-{1}}}{\left({u}_{{k}}\right)}&lt;/math&gt;.
* '''(WG3) Definition (Piecewise smooth path):''' A path &lt;math&gt;\gamma:{\left[{a},{b}\right]}\to\mathbb{C}&lt;/math&gt; is piecewise smooth if there exists a path subdivision &lt;math&gt;{\left(\gamma_{{1}},\ldots\gamma_{{n}}\right)}&lt;/math&gt; of &lt;math&gt;\gamma&lt;/math&gt; consisting of smooth paths &lt;math&gt;\gamma_{{k}}&lt;/math&gt; for all &lt;math&gt;{k}\in{\left\lbrace{1},\ldots,{n}\right\rbrace}&lt;/math&gt;.

== Integration path ==
* '''(WG4) Definition (Path integral):''' Let &lt;math&gt;f : U \to\mathbb{C}&lt;/math&gt; be a continuous function and &lt;math&gt;\gamma: [a,b] \to U &lt;/math&gt; a smooth path, then the path integral is defined as: &lt;math&gt;\int_{\gamma} f := \int_{\gamma} f(z) \, dz := \int_a^b f(\gamma(t)) \cdot\gamma'(t)\, dt &lt;/math&gt;. If &lt;math&gt;\gamma&lt;/math&gt; is only piecewise smooth with respect to a path subdivision &lt;math&gt;( \gamma_1 ,\ldots,\gamma_n ) &lt;/math&gt;, then we define &lt;math&gt;\int_{\gamma} f(z) \, dz :=\sum_{k=1}^{n} \int_{\gamma_k} f(z) \, dz&lt;/math&gt;.
* '''Definition (Integration path):''' An integration path is a piecewise smooth (piecewise continuously differentiable) path.

== Example ==
[[File:Dreiecksweg.svg|mini|Integration path on the triangle edge]]

The following path is piecewise continuously differentiable (smooth) and for the vertices &lt;math&gt;z_1,z_2,z_3\in \text{Spur}(\gamma)&lt;/math&gt; the closed triangle path &lt;math&gt;\gamma : [0,3] \to \mathbb{C}&lt;/math&gt; is not differentiable. The triangle path is defined on the interval &lt;math&gt;[0,3]&lt;/math&gt; as follows:
:&lt;math&gt;
\gamma(t) := \left\langle z_1 ,z_2 ,z_3 \right\rangle (t) := 
\begin{cases}
 (1-t)\cdot z_1 + t\cdot z_2 &amp; \text{for } t \in [0,1] \\
 (2-t)\cdot z_2 + (t-1)\cdot z_3 &amp; \text{for } t \in (1,2] \\ 
 (3-t)\cdot z_3 + (t-2) \cdot z_1 &amp; \text{for } t \in (2,3] \\ 
\end{cases}
&lt;/math&gt;

=== Paths from convex combinations ===
The piecewise continuously differentiable path is formed from [[convex combination]].The sub-paths
* &lt;math&gt;\gamma_1 := \left\langle z_1 ,z_2 \right\rangle &lt;/math&gt; with &lt;math&gt;\gamma_1 : [0,1] \to \mathbb{C}, \ (1-t)\cdot z_1 + t\cdot z_2&lt;/math&gt; 
* &lt;math&gt;\gamma_2 := \left\langle z_2 ,z_3 \right\rangle &lt;/math&gt; with &lt;math&gt;\gamma_2 : [1,2] \to \mathbb{C}, \ (2-t)\cdot z_2 + (t-1)\cdot z_3 &lt;/math&gt;
* &lt;math&gt;\gamma_3 := \left\langle z_3 ,z_1 \right\rangle &lt;/math&gt; with &lt;math&gt;\gamma_3 : [2,3] \to \mathbb{C}, \ (3-t)\cdot z_3 + (t-2)\cdot z_1 &lt;/math&gt;
are continuously differentiable.

== See also ==
* [[Complex Analysis/Goursat's Lemma (Details)|Goursat's Lemma (Details)]]
* [[convex combination]]
* [https://www.geogebra.org/m/rwwjymrv Convex combinations and interpolation on triangles in the plane]


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* Source: [[v:de:Kurs:Funktionentheorie/Integrationsweg|Kurs:Funktionentheorie/Integrationsweg]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Integrationsweg
* Date: 11/20/2024

&lt;span type=&quot;translate&quot; src=&quot;Kurs:Funktionentheorie/Integrationsweg&quot; srclang=&quot;de&quot; date=&quot;12/17/2024&quot; time=&quot;11:42&quot; status=&quot;inprogress&quot;&gt;&lt;/span&gt;

&lt;noinclude&gt;[[de:Kurs:Funktionentheorie/Integrationsweg]]&lt;/noinclude&gt;

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      <text bytes="6137" sha1="4p62gxy4n59ou20c6kaachktn55rj8z" xml:space="preserve">== Smooth paths and path subdivision ==
The following definitions were abbreviated with acronyms and are used as justifications for transformations or conclusions in proofs.

* '''(WG1) Definition (Smooth path):''' A path &lt;math&gt;\gamma: [a,b] \to\mathbb{C}&lt;/math&gt; is smooth if it is continuously differentiable.
* '''(UT) Definition (Subdivision):''' Let &lt;math&gt;[a,b]&lt;/math&gt; be an interval, &lt;math&gt;n \in\mathbb{N}&lt;/math&gt; and &lt;math&gt;{a}={u}_{{0}} &lt; {\ldots} &lt; {{u}}_{n}={b}&lt;/math&gt;. &lt;math&gt;{\left({u}_{{0}},\ldots,{u}_{{{n}}}\right)}\in\mathbb{R}^{n+1}&lt;/math&gt; is called a subdivision of &lt;math&gt;{\left[{a},{b}\right]}&lt;/math&gt;.
* '''(WG2) Definition (Path subdivision):''' Let &lt;math&gt;\gamma: [a,b] \to\mathbb{C}&lt;/math&gt; be a path in &lt;math&gt;{U}\subseteq\mathbb{C}&lt;/math&gt;, &lt;math&gt;{n}\in\mathbb{N}&lt;/math&gt;, &lt;math&gt;{\left({u}_{{0}},\ldots,{u}_{{{n}}}\right)}&lt;/math&gt; a subdivision of &lt;math&gt;[a,b]&lt;/math&gt;, &lt;math&gt;\gamma_{{k}}:{\left[{u}_{{{k}-{1}}},{u}_{{k}}\right]}\to\mathbb{C}&lt;/math&gt; for all &lt;math&gt;{k}\in{\left\lbrace{1},\ldots,{n}\right\rbrace}&lt;/math&gt; a path in &lt;math&gt;{U}&lt;/math&gt;. &lt;math&gt;{\left(\gamma_{{{1}}},\ldots,\gamma_{{{n}}}\right)}&lt;/math&gt; is called a path subdivision of &lt;math&gt;\gamma&lt;/math&gt; if &lt;math&gt;\gamma_{{n}}{\left({b}\right)}=\gamma{\left({b}\right)}&lt;/math&gt; and for all &lt;math&gt;{k}\in{\left\lbrace{1},\ldots,{n}\right\rbrace}&lt;/math&gt; and &lt;math&gt;{t}\in{\left[{u}_{{{k}-{1}}},{u}_{{k}}\right]}&lt;/math&gt; we have &lt;math&gt;\gamma_{{k}}{\left({t}\right)}=\gamma{\left({t}\right)}\wedge\gamma_{{k}}{\left({u}_{{{k}-{1}}}\right)}=\gamma_{{{k}-{1}}}{\left({u}_{{k}}\right)}&lt;/math&gt;.
* '''(WG3) Definition (Piecewise smooth path):''' A path &lt;math&gt;\gamma:{\left[{a},{b}\right]}\to\mathbb{C}&lt;/math&gt; is piecewise smooth if there exists a path subdivision &lt;math&gt;{\left(\gamma_{{1}},\ldots\gamma_{{n}}\right)}&lt;/math&gt; of &lt;math&gt;\gamma&lt;/math&gt; consisting of smooth paths &lt;math&gt;\gamma_{{k}}&lt;/math&gt; for all &lt;math&gt;{k}\in{\left\lbrace{1},\ldots,{n}\right\rbrace}&lt;/math&gt;.

== Integration path ==
* '''(WG4) Definition (Path integral):''' Let &lt;math&gt;f : U \to\mathbb{C}&lt;/math&gt; be a continuous function and &lt;math&gt;\gamma: [a,b] \to U &lt;/math&gt; a smooth path, then the path integral is defined as: &lt;math&gt;\int_{\gamma} f := \int_{\gamma} f(z) \, dz := \int_a^b f(\gamma(t)) \cdot\gamma'(t)\, dt &lt;/math&gt;. If &lt;math&gt;\gamma&lt;/math&gt; is only piecewise smooth with respect to a path subdivision &lt;math&gt;( \gamma_1 ,\ldots,\gamma_n ) &lt;/math&gt;, then we define &lt;math&gt;\int_{\gamma} f(z) \, dz :=\sum_{k=1}^{n} \int_{\gamma_k} f(z) \, dz&lt;/math&gt;.
* '''Definition (Integration path):''' An integration path is a piecewise smooth (piecewise continuously differentiable) path.

== Example ==
[[File:Dreiecksweg.svg|mini|Integration path on the triangle edge]]

The following path is piecewise continuously differentiable (smooth) and for the vertices &lt;math&gt;z_1,z_2,z_3\in \text{Spur}(\gamma)&lt;/math&gt; the closed triangle path &lt;math&gt;\gamma : [0,3] \to \mathbb{C}&lt;/math&gt; is not differentiable. The triangle path is defined on the interval &lt;math&gt;[0,3]&lt;/math&gt; as follows:
:&lt;math&gt;
\gamma(t) := \left\langle z_1 ,z_2 ,z_3 \right\rangle (t) := 
\begin{cases}
 (1-t)\cdot z_1 + t\cdot z_2 &amp; \text{for } t \in [0,1] \\
 (2-t)\cdot z_2 + (t-1)\cdot z_3 &amp; \text{for } t \in (1,2] \\ 
 (3-t)\cdot z_3 + (t-2) \cdot z_1 &amp; \text{for } t \in (2,3] \\ 
\end{cases}
&lt;/math&gt;

=== Paths from convex combinations ===
The piecewise continuously differentiable path is formed from [[convex combination]].The sub-paths
* &lt;math&gt;\gamma_1 := \left\langle z_1 ,z_2 \right\rangle &lt;/math&gt; with &lt;math&gt;\gamma_1 : [0,1] \to \mathbb{C}, \ (1-t)\cdot z_1 + t\cdot z_2&lt;/math&gt; 
* &lt;math&gt;\gamma_2 := \left\langle z_2 ,z_3 \right\rangle &lt;/math&gt; with &lt;math&gt;\gamma_2 : [1,2] \to \mathbb{C}, \ (2-t)\cdot z_2 + (t-1)\cdot z_3 &lt;/math&gt;
* &lt;math&gt;\gamma_3 := \left\langle z_3 ,z_1 \right\rangle &lt;/math&gt; with &lt;math&gt;\gamma_3 : [2,3] \to \mathbb{C}, \ (3-t)\cdot z_3 + (t-2)\cdot z_1 &lt;/math&gt;
are continuously differentiable.

== See also ==
* [[Complex Analysis/Goursat's Lemma (Details)|Goursat's Lemma (Details)]]
* [[convex combination]]
* [https://www.geogebra.org/m/rwwjymrv Convex combinations and interpolation on triangles in the plane]


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* Date: 11/20/2024

&lt;span type=&quot;translate&quot; src=&quot;Kurs:Funktionentheorie/Integrationsweg&quot; srclang=&quot;de&quot; date=&quot;12/17/2024&quot; time=&quot;11:42&quot; status=&quot;inprogress&quot;&gt;&lt;/span&gt;

&lt;noinclude&gt;[[de:Kurs:Funktionentheorie/Integrationsweg]]&lt;/noinclude&gt;

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    <title>Complex Analysis/Goursat's Lemma (Details)</title>
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      <text bytes="16453" sha1="hv89safumapezbr411ps6v6md6weve0" xml:space="preserve">Goursat's Lemma, also known as the Goursat's Theorem, is a theorem in [[Complex analysis]].
Goursat's lemma is a precursor to the [[w:en:Cauchy's integral theorem|Cauchy's integral theorem]] and is often used in its proof. It plays an important role in the development of complex analysis. Remarkably, the lemma only requires  [[Holomorphic function|Complex differentiability]] but not [[w:en:Continuity|continuous]] differentiability. The lemma was proved in its rectangular form by [[w:en:Édouard Goursat|Édouard Goursat]] ([[w:en:1858|1858]]–[[w:en:1936|1936]]) and published in [[w:en:1884|1884]]. The triangular form commonly used today was introduced by [[w:en:Alfred Pringsheim|Alfred Pringsheim]].
== Goursat's Lemma ==
Given the following assumptions:

* (P1) Let &lt;math&gt;{U}\subseteq\mathbb{C}&lt;/math&gt; be an open subset,
* (P2) Let &lt;math&gt;{z}_{{1}},{z}_{{2}},{z}_{{3}}\in\mathbb{C}&lt;/math&gt; be three non-collinear points that define the triangle
:&lt;math&gt;\Delta{\left({z}_{{{1}}},{z}_{{{2}}},{z}_{{{3}}}\right)} := \left\{ \sum_{k=1}^{3} \lambda_{k} \cdot{z}_{k} {\mid} {\left({\sum_{{{k} {1}}}^{{3}}}\lambda_{{k}}={1}\right)}\wedge\forall{k}\in{\left\lbrace{1},{2},{3} \right\rbrace} \lambda_{{k}}\in [{0},{1}]\right\} \subset{U} &lt;/math&gt;
* (P3) Let &lt;math&gt;{f}:{U}\to\mathbb{C}&lt;/math&gt; be a holomorphic function,
* (P4) Let &lt;math&gt;{\left\langle{z}_{{1}},{z}_{{2}},{z}_{{3}}\right\rangle}:{\left[{0},{3}\right]}\to\mathbb{C}&lt;/math&gt; be the closed path over the triangle edge of &lt;math&gt;\Delta{\left({z}_{{{1}}},{z}_{{{2}}},{z}_{{{3}}}\right)}&lt;/math&gt; with starting point &lt;math&gt;{z}_{{1}}&lt;/math&gt;,
then the following statements hold:

* (C1) &lt;math&gt;\int_{{{\left\langle{z}_{{1}},{z}_{{2}},{z}_{{3}}\right\rangle}}}{f{{\left({z}\right)}}}{d}{z}={0}&lt;/math&gt;
== Proof ==
[[File:Dreiecksweg.svg|thumb|Integration path along the triangle boundary]]
[[File:Lemma goursat2 seitenmitten m1m2m3.svg|thumb|Subdivision of the outer paths and insertion of additional paths between the midpoints of the sides, which cancel out in the line integral due to the reversed direction of the integration path, resulting in a sum of 0 and leaving the total integral unchanged.]]
[[File:Lemma goursat3 wege.svg|thumb|Inductive definition of the paths. The subtriangles are [[w:de:Ähnlichkeit_(Geometrie)|similar]] to the original triangle. By using the midpoints of the sides, the perimeter of a triangle &lt;math&gt;\Delta^{(n)}&lt;/math&gt; is halved with each step to &lt;math&gt;\Delta^{(n+1)}&lt;/math&gt;.]]
(S1) We define a sequence of triangular paths recursively as &lt;math&gt;\gamma^{(n)} := {\left\langle z_{1}^{(n)} , z_{2}^{(n)} , z_{3}^{(n)} \right\rangle}&lt;/math&gt;.
=== Proof part 1: Definition of the triangle paths ===
* (S2) (DEF) For &lt;math&gt;{n}={0}&lt;/math&gt; let the closed triangle path &lt;math&gt;\gamma^{(0)} : [0,3] \to \mathbb{C}&lt;/math&gt; be defined as:
:&lt;math&gt;
\gamma^{(0)}(t) := \left\langle z_1 ,z_2 ,z_3 \right\rangle (t) := 
\begin{cases}
 (1-t)\cdot z_1 + t\cdot z_2 &amp; \text{for } t \in [0,1] \\
 (2-t)\cdot z_2 + (t-1) \cdot z_3 &amp; \text{for } t \in (1,2] \\ 
 (3-t)\cdot z_3 + (t-2) \cdot z_1 &amp; \text{for } t \in (2,3] \\ 
\end{cases}
&lt;/math&gt; 
Furthermore, let &lt;math&gt;\gamma^{(n)}&lt;/math&gt; be already defined. We define &lt;math&gt;\gamma^{(n+1)}&lt;/math&gt; inductively.
:: Justification: (P4,UT)
* (S3) (DEF) Definition: Triangle path &lt;math&gt;{\gamma_{1}^{(n)}} := {\left\langle\frac{{ z_{1}^{(n)} + z_{2}^{(n)} }}{{2}}, z_{2}^{(n)} ,\frac{{ z_{2}^{(n)} + z_{3}^{(n)} }}{{2}}\right\rangle}&lt;/math&gt;,
:: Justification: (S3,S4,S5)
* (S4) (DEF) Definition: Triangle path &lt;math&gt;{\gamma_{2}^{(n)}} := {\left\langle\frac{{ z_{2}^{(n)} + z_{3}^{(n)} }}{{2}}, z_{3}^{(n)} ,\frac{{ z_{1}^{(n)} + z_{3}^{(n)} }}{{2}}\right\rangle}&lt;/math&gt;,
* (S5) (DEF) Definition: Triangle path &lt;math&gt;{\gamma_{3}^{(n)}} := {\left\langle\frac{{ z_{1}^{(n)} + z_{3}^{(n)} }}{{2}}, z_{1}^{(n)} , 
\frac{{ z_{1}^{(n)} + z_{2}^{(n)} }}{{2}}\right\rangle}&lt;/math&gt;,
* (S6) (DEF) Definition: Triangle path &lt;math&gt;{\gamma_{4}^{(n)}} := {\left\langle\frac{{ z_{1}^{(n)} + z_{2}^{(n)} }}{{2}},\frac{{ z_{2}^{(n)} + z_{3}^{(n)} }}{{2}},\frac{{ z_{1}^{(n)} + z_{3}^{(n)} }}{{2}}\right\rangle}&lt;/math&gt;
* (S7) (DEF) Let &lt;math&gt;{i}\in{\left\lbrace{1},{2},{3},{4}\right\rbrace}&lt;/math&gt; be the smallest index with &lt;math&gt;\forall_{{{k}\in{\left\lbrace{1},,{2},{3},{4}\right\rbrace}}}:{\left|\int_{{{\gamma_{{k}}^{(n)}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}\le{\left|\int_{{{\gamma_{{i}}^{(n)}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}&lt;/math&gt; and &lt;math&gt;\gamma^{{{\left({n}+{1}\right)}}} := {\gamma_{{i}}^{(n)}}&lt;/math&gt;

=== Proof part 2: Estimates ===

* (S8) &lt;math&gt;\Rightarrow&lt;/math&gt; &lt;math&gt; \int_{\gamma^{(n)}} f(z) \, dz = \sum_{k=1}^{4} \int_{\gamma_k}^{(n)} f(z) \, dz&lt;/math&gt;
* (S9) &lt;math&gt;\Rightarrow&lt;/math&gt; &lt;math&gt;{\left|\int_{{\gamma^{{{n}}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}={\left|{\sum_{{{k}={1}}}^{{4}}}\int_{{{\gamma_{{k}}^{(n)}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}\le{\sum_{{{k}={1}}}^{{4}}}{\left|\int_{{{\gamma_{{k}}^{(n)}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}\le{4}\cdot{\left|\int_{{{\gamma_{{k}}^{(n)}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}&lt;/math&gt; for all &lt;math&gt;{n}\in\mathbb{N}&lt;/math&gt;
:: Justification: (S7,WG4,DU)
* (S10) &lt;math&gt;\Rightarrow&lt;/math&gt; &lt;math&gt;{0}\le{\left|\int_{{{\left\langle{z}_{{1}},{z}_{{2}},{z}_{{3}}\right\rangle}}}{f{{\left({z}\right)}}}{d}{z}\right|}={\left|\int_{\gamma^{(0)}} f(z) {\left.{d}{z}\right.}\right|}\le{4}\cdot{\left|\int_{{\gamma^{{{\left({1}\right)}}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}\le\ldots\le{4}^{{n}}\cdot{\left|\int_{{{\gamma_{{i}}^{(n)}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}={4}^{{n}}\cdot{\left|\int_{{\gamma^{{{\left({n}+{1}\right)}}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}&lt;/math&gt;

=== Proof part 3: Diameter of the sub-triangles === 
* (S11) The nested definition of the sub-triangles yields for all &lt;math&gt;{n}\in\mathbb{N}&lt;/math&gt;: &lt;math&gt;\Delta{\left({{z}_{1}^{(n)}},{{z}_{2}^{(n)}},{{z}_{3}^{(n)}}\right)}\supset\Delta{\left({{z}_{1}^{{{\left({n}+{1}\right)}}}},{{z}_{2}^{{{\left({n}+{1}\right)}}}},{{z}_{3}^{{{\left({n}+{1}\right)}}}}\right)}&lt;/math&gt; and 
:: &lt;math&gt;\lim_{n\to\infty} \, \text{diam} \left(\Delta{\left({{z}_{1}^{(n)}},{{z}_{2}^{(n)}},{{z}_{3}^{(n)}}\right)}\right) = 0 &lt;/math&gt;
* (S12) &lt;math&gt;\Rightarrow&lt;/math&gt; &lt;math&gt;\exists_{{{z}_{{0}}\in{U}}}\forall_{{{n}\in\mathbb{N}}}:{z}_{{0}}\in\Delta^{(n)} := \Delta{\left({{z}_{1}^{(n)}},{{z}_{2}^{(n)}},{{z}_{3}^{(n)}}\right)}&lt;/math&gt; and &lt;math&gt;{\left\lbrace{z}_{{0}}\right\rbrace}=\bigcap_{{{n}\in\mathbb{N}}}\Delta^{(n)}&lt;/math&gt;

=== Proof part 4: Use of holomorphism (P3) ===
* (S13) We use the holomorphism of &lt;math&gt;f&lt;/math&gt; in &lt;math&gt;z_0 \in U &lt;/math&gt; for further steps with 
: &lt;math&gt;f(z) := f(z_0) +f'(z_0) \cdot ( z - z_0 ) + r(z)&lt;/math&gt; and &lt;math&gt;\lim_{z \to z_0} \frac{r(z)}{z - z_0} = 0 &lt;/math&gt;
:: Justification: (P3)
* (S14) &lt;math&gt;\Rightarrow&lt;/math&gt; The function &lt;math&gt;{h}:{U}\to\mathbb{C}&lt;/math&gt; with &lt;math&gt;h(z) := f(z_0) +f'(z_0) \cdot ( z - z_0 )&lt;/math&gt; has a primitive &lt;math&gt;H(z) := f(z_0) +f'(z_0) \cdot \frac{1}{2}\cdot (z - z_0)^2 &lt;/math&gt;
:: Justification: since &lt;math&gt;h(z)&lt;/math&gt; is a polynomial of degree 1.
* (S15) &lt;math&gt;\Rightarrow&lt;/math&gt; The path integral over the closed paths &lt;math&gt;\gamma^{(n)}&lt;/math&gt; of the function &lt;math&gt;{h}:{U}\to\mathbb{C}&lt;/math&gt; is thus &lt;math&gt;\int_{{{\gamma_{{k}}^{(n)}}}}{h}{\left({z}\right)}={0}&lt;/math&gt;
:: Justification: (SF)
* (S16) &lt;math&gt;\Rightarrow&lt;/math&gt; For the path integral over the closed paths &lt;math&gt;\gamma^{(n)}&lt;/math&gt; of the function &lt;math&gt;{f}:{U}\to\mathbb{C}&lt;/math&gt; we have &lt;math&gt;\int_{{{\gamma_{{k}}^{(n)}}}}{f{{\left({z}\right)}{\left.{d}{z}\right.}}}=\int_{{{\gamma_{{k}}^{(n)}}}}{h}{\left({z}\right)}+{r}{\left({z}\right)}{\left.{d}{z}\right.}=\int_{{{\gamma_{{k}}^{(n)}}}}{r}{\left({z}\right)}{\left.{d}{z}\right.}&lt;/math&gt;

=== Proof part 4: Estimate of the remainder term &lt;math&gt;r(z)&lt;/math&gt; ===
* (S17) &lt;math&gt;\Rightarrow&lt;/math&gt; With &lt;math&gt;\lim_{z \to z_0} \frac{r(z)}{ z - z_0 }= 0 &lt;/math&gt; we have: For all &lt;math&gt;\epsilon&gt;{0}&lt;/math&gt; there exists a &lt;math&gt;\delta&gt;{0}&lt;/math&gt;
:: &lt;math&gt; | z - z_0 | &lt; \delta \Rightarrow \left| \frac{ r(z) }{ z - z_0 } \right| &lt; \epsilon&lt;/math&gt;
: Justification: &lt;math&gt;\epsilon&lt;/math&gt;-&lt;math&gt;\delta&lt;/math&gt;-criterion applied to &lt;math&gt;g(z):=\frac{r(z)}{ z - z_0 }&lt;/math&gt; and continuity of &lt;math&gt;g&lt;/math&gt; in &lt;math&gt;z_0&lt;/math&gt; 
* (S18) &lt;math&gt;\Rightarrow&lt;/math&gt; For all &lt;math&gt;\epsilon&gt;{0}&lt;/math&gt; there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt;: &lt;math&gt;| z - z_0 | &lt; \delta\Rightarrow | r(z) | &lt; \epsilon \cdot | z - z_0 |&lt;/math&gt; 
:: &lt;math&gt;0 \leq \left|\int_{\left\langle z_1 , z_2 , z_3 \right\rangle} f(z) \, dz \right| \leq 4^n \cdot \left| \int_{ \gamma_{k}^{(n)} } f(z)\, dz \right| = 4^n \cdot \left| \int_{ \gamma_{k}^{(n)} } r(z) \, dz \right| \leq 4^n \cdot \int_{ \gamma_{k}^{(n)} } |r(z)| \, dz \leq 4^n \cdot \int_{ \gamma_{k}^{(n)} } \epsilon \cdot | z - z_0 | \, dz&lt;/math&gt;
:: Justification: (S2)
* (S20) From the condition &lt;math&gt;\lim_{n\to\infty}\text{diam} \left( \Delta^{(n)} \right) = 0&lt;/math&gt; there exists for all &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; an &lt;math&gt;n_{\delta} \in\mathbb{N}&lt;/math&gt; with &lt;math&gt;\Delta^{(n)}\subseteq {D}_{\delta}(z_0)&lt;/math&gt; for all &lt;math&gt;n &gt; n_{\delta}&lt;/math&gt;.
* (S21) &lt;math&gt;\Rightarrow&lt;/math&gt; &lt;math&gt;| z - z_0 | &lt; L \left(\gamma^{(n)}\right) = \frac{1}{2^n} \cdot L \left(\gamma\right) &lt;/math&gt; for all &lt;math&gt; n \in\mathbb{N}&lt;/math&gt; and all &lt;math&gt; z \in \Delta^{(n)}&lt;/math&gt;
:: Justification: The factor &lt;math&gt;\frac{1}{2^n}&lt;/math&gt; arises from the continued halving of the sides of the triangles &lt;math&gt;\Delta^{(n)}&lt;/math&gt;
* (S22) This implies:
:&lt;math&gt; 0 \leq \left|\int_{ \langle z_1 , z_2 , z_3 \rangle } f(z) \, dz \right| \leq 4^n \cdot \int_{ \gamma_{k}^{(n)} } \epsilon \cdot | z - z_0 | \, dz \leq 4^n \cdot \epsilon \cdot \int_{ \gamma_{k}^{(n)} } \frac{1}{2^n} \cdot\mathcal{L}(\gamma) \, dz = 4^n \cdot \epsilon \cdot \frac{1}{2^n} \cdot \mathcal{L}(\gamma) \underbrace{\leq \int_{ \gamma_{k}^{(n)} } 1 \, dz}_{\mathcal{L}(\gamma_{k}^{(n)}) } &lt;/math&gt;
::&lt;math&gt;\leq 
4^n \cdot \epsilon \cdot \frac{1}{2^n} \cdot L(\gamma) \cdot \mathcal{L}(\gamma_{k}^{(n)} ) \leq 
4^n \cdot \epsilon \cdot \frac{\mathcal{L}(\gamma)}{4^n} = \epsilon \cdot \mathcal{L}(\gamma) &lt;/math&gt; for all &lt;math&gt;\epsilon &gt;0&lt;/math&gt;
:: Justification: (S19,LIW,IAL)
* (C1) &lt;math&gt;\Rightarrow&lt;/math&gt; &lt;math&gt;\int_{{{\left\langle{z}_{{1}},{z}_{{2}},{z}_{{3}}\right\rangle}}}{f{{\left({z}\right)}}}{d}{z}={0}&lt;/math&gt;


== Abbreviations for justifications ==
* (DU) &lt;math&gt;\forall_{a,b \in\mathbb{C}} : | a+b | \leq |a| + |b| &lt;/math&gt;
&lt;!--
* (DG) &lt;math&gt;\forall_{a,b,c \in\mathbb{M}} : a \cdot (b+c) = a \cdot b + a \cdot c &lt;/math&gt;
* (AG&lt;math&gt;+&lt;/math&gt;) &lt;math&gt;\forall_{a,b,c \in\mathbb{M}} : a+ (b + c) = (a + b) + c &lt;/math&gt;
* (AG&lt;math&gt;\cdot&lt;/math&gt;) &lt;math&gt;\forall_{a,b,c \in\mathbb{M}} : a\cdot (b \cdot c) = (a \cdot b) \cdot c &lt;/math&gt;
--&gt;
* (DI) Definition: Let &lt;math&gt; M\subset{C}&lt;/math&gt; be a set &lt;math&gt;\text{diam}(M) :=\text{sup} \lbrace |b-a| \, :\, a,b \in M \rbrace &lt;/math&gt;
* (WE) Definition (Path): Let &lt;math&gt;{U}\subseteq\mathbb{C}&lt;/math&gt; be a subset and &lt;math&gt; a,b \in\mathbb{R}&lt;/math&gt; with &lt;math&gt;a &lt; b &lt;/math&gt;. A path &lt;math&gt;\gamma&lt;/math&gt; in &lt;math&gt;U \subseteq \mathbb{C}&lt;/math&gt; is a continuous mapping &lt;math&gt;\gamma: [a,b] \to U &lt;/math&gt;.
* (SPU) Definition (Trace): Let &lt;math&gt;\gamma: [a,b] \to U&lt;/math&gt; be a path in &lt;math&gt;{U}\subseteq\mathbb{C}&lt;/math&gt;. The trace of &lt;math&gt;\gamma&lt;/math&gt; is defined as: &lt;math&gt;\text{Spur}(\gamma) := \lbrace \gamma(t)\in\mathbb{C} \, {\mid} \, t \in [a,b] \rbrace &lt;/math&gt;.
* (WZ) Definition (Path-connected): Let &lt;math&gt;{U}\subseteq\mathbb{C}&lt;/math&gt; be a subset. &lt;math&gt;{U}&lt;/math&gt; is called path-connected if there exists a path &lt;math&gt;\gamma:[a,b] \to U&lt;/math&gt; in &lt;math&gt;U \subseteq \mathbb{C}&lt;/math&gt; with &lt;math&gt; \gamma(a) = z_1 &lt;/math&gt;, &lt;math&gt;\gamma(b) = z_2 &lt;/math&gt; and &lt;math&gt;\text{Spur}(\gamma) \subseteq U&lt;/math&gt;.
* (GE) Definition (Domain): A subset &lt;math&gt;{G}\subseteq\mathbb{C}&lt;/math&gt; is called a domain if (1) &lt;math&gt;{G}&lt;/math&gt; is open, (2) &lt;math&gt;{G}\ne\emptyset&lt;/math&gt; and (3) &lt;math&gt;{G}&lt;/math&gt; is path-connected.
* (WG1) Definition (Smooth path): A path &lt;math&gt;\gamma: [a,b] \to\mathbb{C}&lt;/math&gt; is smooth if it is continuously differentiable.
* (UT) Definition (Subdivision): Let &lt;math&gt;[a,b]&lt;/math&gt; be an interval, &lt;math&gt;n \in\mathbb{N}&lt;/math&gt; and &lt;math&gt;{a}={u}_{{0}} &lt; {\ldots} &lt; {{u}}_{n}={b}&lt;/math&gt;. &lt;math&gt;{\left({u}_{{0}},\ldots,{u}_{{{n}}}\right)}\in\mathbb{R}^{n+1}&lt;/math&gt; is called a subdivision of &lt;math&gt;{\left[{a},{b}\right]}&lt;/math&gt;.
* (WG2) Definition (Path subdivision): Let &lt;math&gt;\gamma: [a,b] \to\mathbb{C}&lt;/math&gt; be a path in &lt;math&gt;{U}\subseteq\mathbb{C}&lt;/math&gt;, &lt;math&gt;{n}\in\mathbb{N}&lt;/math&gt;, &lt;math&gt;{\left({u}_{{0}},\ldots,{u}_{{{n}}}\right)}&lt;/math&gt; a subdivision of &lt;math&gt;[a,b]&lt;/math&gt;, &lt;math&gt;\gamma_{{k}}:{\left[{u}_{{{k}-{1}}},{u}_{{k}}\right]}\to\mathbb{C}&lt;/math&gt; for all &lt;math&gt;{k}\in{\left\lbrace{1},\ldots,{n}\right\rbrace}&lt;/math&gt; a path in &lt;math&gt;{U}&lt;/math&gt;. &lt;math&gt;{\left(\gamma_{{{1}}},\ldots,\gamma_{{{n}}}\right)}&lt;/math&gt; is called a path subdivision of &lt;math&gt;\gamma&lt;/math&gt; if &lt;math&gt;\gamma_{{n}}{\left({b}\right)}=\gamma{\left({b}\right)}&lt;/math&gt; and &lt;math&gt;\forall_{{{k}\in{\left\lbrace{1},\ldots,{n}\right\rbrace}}}\forall_{{{t}\in{\left[{u}_{{{k}-{1}}},{u}_{{k}}\right)}}}:\gamma_{{k}}{\left({t}\right)}=\gamma{\left({t}\right)}\wedge\gamma_{{k}}{\left({u}_{{{k}-{1}}}\right)}=\gamma_{{{k}-{1}}}{\left({u}_{{k}}\right)}&lt;/math&gt;.
* (WG3) Definition (Piecewise smooth path): A path &lt;math&gt;\gamma:{\left[{a},{b}\right]}\to\mathbb{C}&lt;/math&gt; is piecewise smooth if there exists a path subdivision &lt;math&gt;{\left(\gamma_{{1}},\ldots\gamma_{{n}}\right)}&lt;/math&gt; of &lt;math&gt;\gamma&lt;/math&gt; consisting of smooth paths &lt;math&gt;\gamma_{{k}}&lt;/math&gt; for all &lt;math&gt;{k}\in{\left\lbrace{1},\ldots,{n}\right\rbrace}&lt;/math&gt;.
* (WG4) Definition (Path integral): Let &lt;math&gt;f : U \to\mathbb{C}&lt;/math&gt; be a continuous function and &lt;math&gt;\gamma: [a,b] \to U &lt;/math&gt; a smooth path, then the path integral is defined as: &lt;math&gt;\int_{\gamma} f := \int_{\gamma} f(z) \, dz := \int_a^b f(\gamma(t)) \cdot\gamma'(t)\, dt &lt;/math&gt;. If &lt;math&gt;\gamma&lt;/math&gt; is only piecewise smooth with respect to a path subdivision &lt;math&gt;( \gamma_1 ,\ldots,\gamma_n ) &lt;/math&gt;, then we define &lt;math&gt;\int_{\gamma} f(z) \, dz :=\sum_{k=1}^{n} \int_{\gamma_k} f(z) \, dz&lt;/math&gt;.
* (SF) Theorem (Primitive with closed paths): If a continuous function &lt;math&gt;f : U \to\mathbb{C}&lt;/math&gt; has a primitive &lt;math&gt;F : U \to\mathbb{C}&lt;/math&gt;, then for a piecewise smooth path &lt;math&gt;\gamma: [a,b] \to U&lt;/math&gt; we have &lt;math&gt;\int_{\gamma} f(z) \, dz =F(b) - F(a)&lt;/math&gt;.
* (LIW) Length of the integration path: Let &lt;math&gt;\gamma:{\left[{a},{b}\right]}\to\mathbb{C}&lt;/math&gt; be a smooth path, then the &lt;math&gt;\mathcal{L}(\gamma)&lt;/math&gt; is defined as:
:: &lt;math&gt;\mathcal{L}(\gamma) := \int_{a}^{b} |\gamma'(t)| \, dt&lt;/math&gt;. 
: If &lt;math&gt;\gamma:{\left[{a},{b}\right]}\to\mathbb{C}&lt;/math&gt; is a general integration path with the path subdivision &lt;math&gt;{\left(\gamma_{{1}},\ldots\gamma_{{n}}\right)}&lt;/math&gt; of smooth paths &lt;math&gt;\gamma_{{k}}&lt;/math&gt;, then &lt;math&gt;\mathcal{L}(\gamma)&lt;/math&gt; is defined as the sum of the lengths of the smooth paths &lt;math&gt;\gamma_{{k}}&lt;/math&gt;, i.e.:
:: &lt;math&gt;\mathcal{L}(\gamma) := \sum_{k=1}^{n} \mathcal{L}(\gamma_{k})&lt;/math&gt;
* (IAL) Integral estimate over the length of the integration path: Let &lt;math&gt;\gamma:{\left[{a},{b}\right]}\to\mathbb{G}&lt;/math&gt; be an integration path on the domain &lt;math&gt;G \subseteq \mathbb{C}&lt;/math&gt;, then for a continuous function &lt;math&gt;f&lt;/math&gt; on &lt;math&gt;\text{Spur}(\gamma)&lt;/math&gt; we have the estimate:
::&lt;math&gt;\left| \int_{\gamma} f(z) \, dz \right| \leq \max_{z \in \text{Spur}(\gamma)} |f(z)| \cdot \mathcal{L}(\gamma)&lt;/math&gt;

== Literature ==
* Eberhard Freitag &amp; Rolf Busam: ''Funktionentheorie 1'', Springer-Verlag, Berlin

== See also ==
* [[Complex Analysis/Goursat's Lemma]]-Short form
* [[Complex Analysis/Path of Integration|Path of Integration]]

[[Category:Complex Analysis]]
[[Category:Theorem (Mathematics)|Goursat, Lemma of]]

== Page information ==
This page was created based on the following Wikipedia source:
* [https://de.wikipedia.org/wiki/Lemma%20von%20Goursat Lemma von Goursat] https://de.wikipedia.org/wiki/Lemma%20von%20Goursat
* Date: 14.12.2018
* [https://niebert.github.com/Wikipedia2Wikiversity Wikipedia2Wikiversity-Converter]: https://niebert.github.com/Wikipedia2Wikiversity

&lt;noninclude&gt;[[de:Kurs:Funktionentheorie/Lemma von Goursat (Details)]]&lt;/noinclude&gt;</text>
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      <text bytes="18230" sha1="bmw96jpt0s9v018jed66p1n0srvn7tb" xml:space="preserve">Goursat's Lemma, also known as the Goursat's Theorem, is a theorem in [[Complex analysis]].
Goursat's lemma is a precursor to the [[w:en:Cauchy's integral theorem|Cauchy's integral theorem]] and is often used in its proof. It plays an important role in the development of complex analysis. Remarkably, the lemma only requires  [[Holomorphic function|Complex differentiability]] but not [[w:en:Continuity|continuous]] differentiability. The lemma was proved in its rectangular form by [[w:en:Édouard Goursat|Édouard Goursat]] ([[w:en:1858|1858]]–[[w:en:1936|1936]]) and published in [[w:en:1884|1884]]. The triangular form commonly used today was introduced by [[w:en:Alfred Pringsheim|Alfred Pringsheim]].
== Goursat's Lemma ==
Given the following assumptions:

* (P1) Let &lt;math&gt;{U}\subseteq\mathbb{C}&lt;/math&gt; be an open subset,
* (P2) Let &lt;math&gt;{z}_{{1}},{z}_{{2}},{z}_{{3}}\in\mathbb{C}&lt;/math&gt; be three non-collinear points that define the triangle
:&lt;math&gt;\Delta{\left({z}_{{{1}}},{z}_{{{2}}},{z}_{{{3}}}\right)} := \left\{ \sum_{k=1}^{3} \lambda_{k} \cdot{z}_{k} {\mid} {\left({\sum_{{{k} {1}}}^{{3}}}\lambda_{{k}}={1}\right)}\wedge\forall{k}\in{\left\lbrace{1},{2},{3} \right\rbrace} \lambda_{{k}}\in [{0},{1}]\right\} \subset{U} &lt;/math&gt;
* (P3) Let &lt;math&gt;{f}:{U}\to\mathbb{C}&lt;/math&gt; be a holomorphic function,
* (P4) Let &lt;math&gt;{\left\langle{z}_{{1}},{z}_{{2}},{z}_{{3}}\right\rangle}:{\left[{0},{3}\right]}\to\mathbb{C}&lt;/math&gt; be the closed path over the triangle edge of &lt;math&gt;\Delta{\left({z}_{{{1}}},{z}_{{{2}}},{z}_{{{3}}}\right)}&lt;/math&gt; with starting point &lt;math&gt;{z}_{{1}}&lt;/math&gt;,
then the following statements hold:

* (C1) &lt;math&gt;\int_{{{\left\langle{z}_{{1}},{z}_{{2}},{z}_{{3}}\right\rangle}}}{f{{\left({z}\right)}}}{d}{z}={0}&lt;/math&gt;
== Proof ==
[[File:Dreiecksweg.svg|thumb|Integration path along the triangle boundary]]
[[File:Lemma goursat2 seitenmitten m1m2m3.svg|thumb|Subdivision of the outer paths and insertion of additional paths between the midpoints of the sides, which cancel out in the line integral due to the reversed direction of the integration path, resulting in a sum of 0 and leaving the total integral unchanged.]]
[[File:Lemma goursat3 wege.svg|thumb|Inductive definition of the paths. The subtriangles are [[w:de:Ähnlichkeit_(Geometrie)|similar]] to the original triangle. By using the midpoints of the sides, the perimeter of a triangle &lt;math&gt;\Delta^{(n)}&lt;/math&gt; is halved with each step to &lt;math&gt;\Delta^{(n+1)}&lt;/math&gt;.]]
(S1) We define a sequence of triangular paths recursively as &lt;math&gt;\gamma^{(n)} := {\left\langle z_{1}^{(n)} , z_{2}^{(n)} , z_{3}^{(n)} \right\rangle}&lt;/math&gt;.
=== Proof part 1: Definition of the triangle paths ===
* (S2) (DEF) For &lt;math&gt;{n}={0}&lt;/math&gt; let the closed triangle path &lt;math&gt;\gamma^{(0)} : [0,3] \to \mathbb{C}&lt;/math&gt; be defined as:
:&lt;math&gt;
\gamma^{(0)}(t) := \left\langle z_1 ,z_2 ,z_3 \right\rangle (t) := 
\begin{cases}
 (1-t)\cdot z_1 + t\cdot z_2 &amp; \text{for } t \in [0,1] \\
 (2-t)\cdot z_2 + (t-1) \cdot z_3 &amp; \text{for } t \in (1,2] \\ 
 (3-t)\cdot z_3 + (t-2) \cdot z_1 &amp; \text{for } t \in (2,3] \\ 
\end{cases}
&lt;/math&gt; 
Furthermore, let &lt;math&gt;\gamma^{(n)}&lt;/math&gt; be already defined. We define &lt;math&gt;\gamma^{(n+1)}&lt;/math&gt; inductively.
:: Justification: (P4,UT)
* (S3) (DEF) Definition: Triangle path &lt;math&gt;{\gamma_{1}^{(n)}} := {\left\langle\frac{{ z_{1}^{(n)} + z_{2}^{(n)} }}{{2}}, z_{2}^{(n)} ,\frac{{ z_{2}^{(n)} + z_{3}^{(n)} }}{{2}}\right\rangle}&lt;/math&gt;,
:: Justification: (S3,S4,S5)
* (S4) (DEF) Definition: Triangle path &lt;math&gt;{\gamma_{2}^{(n)}} := {\left\langle\frac{{ z_{2}^{(n)} + z_{3}^{(n)} }}{{2}}, z_{3}^{(n)} ,\frac{{ z_{1}^{(n)} + z_{3}^{(n)} }}{{2}}\right\rangle}&lt;/math&gt;,
* (S5) (DEF) Definition: Triangle path &lt;math&gt;{\gamma_{3}^{(n)}} := {\left\langle\frac{{ z_{1}^{(n)} + z_{3}^{(n)} }}{{2}}, z_{1}^{(n)} , 
\frac{{ z_{1}^{(n)} + z_{2}^{(n)} }}{{2}}\right\rangle}&lt;/math&gt;,
* (S6) (DEF) Definition: Triangle path &lt;math&gt;{\gamma_{4}^{(n)}} := {\left\langle\frac{{ z_{1}^{(n)} + z_{2}^{(n)} }}{{2}},\frac{{ z_{2}^{(n)} + z_{3}^{(n)} }}{{2}},\frac{{ z_{1}^{(n)} + z_{3}^{(n)} }}{{2}}\right\rangle}&lt;/math&gt;
* (S7) (DEF) Let &lt;math&gt;{i}\in{\left\lbrace{1},{2},{3},{4}\right\rbrace}&lt;/math&gt; be the smallest index with &lt;math&gt;\forall_{{{k}\in{\left\lbrace{1},,{2},{3},{4}\right\rbrace}}}:{\left|\int_{{{\gamma_{{k}}^{(n)}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}\le{\left|\int_{{{\gamma_{{i}}^{(n)}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}&lt;/math&gt; and &lt;math&gt;\gamma^{{{\left({n}+{1}\right)}}} := {\gamma_{{i}}^{(n)}}&lt;/math&gt;

=== Proof part 2: Estimates ===

* (S8) &lt;math&gt;\Rightarrow&lt;/math&gt; &lt;math&gt; \int_{\gamma^{(n)}} f(z) \, dz = \sum_{k=1}^{4} \int_{\gamma_k}^{(n)} f(z) \, dz&lt;/math&gt;
* (S9) &lt;math&gt;\Rightarrow&lt;/math&gt; &lt;math&gt;{\left|\int_{{\gamma^{{{n}}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}={\left|{\sum_{{{k}={1}}}^{{4}}}\int_{{{\gamma_{{k}}^{(n)}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}\le{\sum_{{{k}={1}}}^{{4}}}{\left|\int_{{{\gamma_{{k}}^{(n)}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}\le{4}\cdot{\left|\int_{{{\gamma_{{k}}^{(n)}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}&lt;/math&gt; for all &lt;math&gt;{n}\in\mathbb{N}&lt;/math&gt;
:: Justification: (S7,WG4,DU)
* (S10) &lt;math&gt;\Rightarrow&lt;/math&gt; &lt;math&gt;{0}\le{\left|\int_{{{\left\langle{z}_{{1}},{z}_{{2}},{z}_{{3}}\right\rangle}}}{f{{\left({z}\right)}}}{d}{z}\right|}={\left|\int_{\gamma^{(0)}} f(z) {\left.{d}{z}\right.}\right|}\le{4}\cdot{\left|\int_{{\gamma^{{{\left({1}\right)}}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}\le\ldots\le{4}^{{n}}\cdot{\left|\int_{{{\gamma_{{i}}^{(n)}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}={4}^{{n}}\cdot{\left|\int_{{\gamma^{{{\left({n}+{1}\right)}}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}&lt;/math&gt;

=== Proof part 3: Diameter of the sub-triangles === 
* (S11) The nested definition of the sub-triangles yields for all &lt;math&gt;{n}\in\mathbb{N}&lt;/math&gt;: &lt;math&gt;\Delta{\left({{z}_{1}^{(n)}},{{z}_{2}^{(n)}},{{z}_{3}^{(n)}}\right)}\supset\Delta{\left({{z}_{1}^{{{\left({n}+{1}\right)}}}},{{z}_{2}^{{{\left({n}+{1}\right)}}}},{{z}_{3}^{{{\left({n}+{1}\right)}}}}\right)}&lt;/math&gt; and 
:: &lt;math&gt;\lim_{n\to\infty} \, \text{diam} \left(\Delta{\left({{z}_{1}^{(n)}},{{z}_{2}^{(n)}},{{z}_{3}^{(n)}}\right)}\right) = 0 &lt;/math&gt;
* (S12) &lt;math&gt;\Rightarrow&lt;/math&gt; &lt;math&gt;\exists_{{{z}_{{0}}\in{U}}}\forall_{{{n}\in\mathbb{N}}}:{z}_{{0}}\in\Delta^{(n)} := \Delta{\left({{z}_{1}^{(n)}},{{z}_{2}^{(n)}},{{z}_{3}^{(n)}}\right)}&lt;/math&gt; and &lt;math&gt;{\left\lbrace{z}_{{0}}\right\rbrace}=\bigcap_{{{n}\in\mathbb{N}}}\Delta^{(n)}&lt;/math&gt;

=== Proof part 4: Use of holomorphism (P3) ===
* (S13) We use the holomorphism of &lt;math&gt;f&lt;/math&gt; in &lt;math&gt;z_0 \in U &lt;/math&gt; for further steps with 
: &lt;math&gt;f(z) := f(z_0) +f'(z_0) \cdot ( z - z_0 ) + r(z)&lt;/math&gt; and &lt;math&gt;\lim_{z \to z_0} \frac{r(z)}{z - z_0} = 0 &lt;/math&gt;
:: Justification: (P3)
* (S14) &lt;math&gt;\Rightarrow&lt;/math&gt; The function &lt;math&gt;{h}:{U}\to\mathbb{C}&lt;/math&gt; with &lt;math&gt;h(z) := f(z_0) +f'(z_0) \cdot ( z - z_0 )&lt;/math&gt; has a primitive &lt;math&gt;H(z) := f(z_0) +f'(z_0) \cdot \frac{1}{2}\cdot (z - z_0)^2 &lt;/math&gt;
:: Justification: since &lt;math&gt;h(z)&lt;/math&gt; is a polynomial of degree 1.
* (S15) &lt;math&gt;\Rightarrow&lt;/math&gt; The path integral over the closed paths &lt;math&gt;\gamma^{(n)}&lt;/math&gt; of the function &lt;math&gt;{h}:{U}\to\mathbb{C}&lt;/math&gt; is thus &lt;math&gt;\int_{{{\gamma_{{k}}^{(n)}}}}{h}{\left({z}\right)}={0}&lt;/math&gt;
:: Justification: (SF)
* (S16) &lt;math&gt;\Rightarrow&lt;/math&gt; For the path integral over the closed paths &lt;math&gt;\gamma^{(n)}&lt;/math&gt; of the function &lt;math&gt;{f}:{U}\to\mathbb{C}&lt;/math&gt; we have &lt;math&gt;\int_{{{\gamma_{{k}}^{(n)}}}}{f{{\left({z}\right)}{\left.{d}{z}\right.}}}=\int_{{{\gamma_{{k}}^{(n)}}}}{h}{\left({z}\right)}+{r}{\left({z}\right)}{\left.{d}{z}\right.}=\int_{{{\gamma_{{k}}^{(n)}}}}{r}{\left({z}\right)}{\left.{d}{z}\right.}&lt;/math&gt;

=== Proof part 4: Estimate of the remainder term &lt;math&gt;r(z)&lt;/math&gt; ===
* (S17) &lt;math&gt;\Rightarrow&lt;/math&gt; With &lt;math&gt;\lim_{z \to z_0} \frac{r(z)}{ z - z_0 }= 0 &lt;/math&gt; we have: For all &lt;math&gt;\epsilon&gt;{0}&lt;/math&gt; there exists a &lt;math&gt;\delta&gt;{0}&lt;/math&gt;
:: &lt;math&gt; | z - z_0 | &lt; \delta \Rightarrow \left| \frac{ r(z) }{ z - z_0 } \right| &lt; \epsilon&lt;/math&gt;
: Justification: &lt;math&gt;\epsilon&lt;/math&gt;-&lt;math&gt;\delta&lt;/math&gt;-criterion applied to &lt;math&gt;g(z):=\frac{r(z)}{ z - z_0 }&lt;/math&gt; and continuity of &lt;math&gt;g&lt;/math&gt; in &lt;math&gt;z_0&lt;/math&gt; 
* (S18) &lt;math&gt;\Rightarrow&lt;/math&gt; For all &lt;math&gt;\epsilon&gt;{0}&lt;/math&gt; there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt;: &lt;math&gt;| z - z_0 | &lt; \delta\Rightarrow | r(z) | &lt; \epsilon \cdot | z - z_0 |&lt;/math&gt; 
:: &lt;math&gt;0 \leq \left|\int_{\left\langle z_1 , z_2 , z_3 \right\rangle} f(z) \, dz \right| \leq 4^n \cdot \left| \int_{ \gamma_{k}^{(n)} } f(z)\, dz \right| = 4^n \cdot \left| \int_{ \gamma_{k}^{(n)} } r(z) \, dz \right| \leq 4^n \cdot \int_{ \gamma_{k}^{(n)} } |r(z)| \, dz \leq 4^n \cdot \int_{ \gamma_{k}^{(n)} } \epsilon \cdot | z - z_0 | \, dz&lt;/math&gt;
:: Justification: (S2)
* (S20) From the condition &lt;math&gt;\lim_{n\to\infty}\text{diam} \left( \Delta^{(n)} \right) = 0&lt;/math&gt; there exists for all &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; an &lt;math&gt;n_{\delta} \in\mathbb{N}&lt;/math&gt; with &lt;math&gt;\Delta^{(n)}\subseteq {D}_{\delta}(z_0)&lt;/math&gt; for all &lt;math&gt;n &gt; n_{\delta}&lt;/math&gt;.
* (S21) &lt;math&gt;\Rightarrow&lt;/math&gt; &lt;math&gt;| z - z_0 | &lt; L \left(\gamma^{(n)}\right) = \frac{1}{2^n} \cdot L \left(\gamma\right) &lt;/math&gt; for all &lt;math&gt; n \in\mathbb{N}&lt;/math&gt; and all &lt;math&gt; z \in \Delta^{(n)}&lt;/math&gt;
:: Justification: The factor &lt;math&gt;\frac{1}{2^n}&lt;/math&gt; arises from the continued halving of the sides of the triangles &lt;math&gt;\Delta^{(n)}&lt;/math&gt;
* (S22) This implies:
:&lt;math&gt; 0 \leq \left|\int_{ \langle z_1 , z_2 , z_3 \rangle } f(z) \, dz \right| \leq 4^n \cdot \int_{ \gamma_{k}^{(n)} } \epsilon \cdot | z - z_0 | \, dz \leq 4^n \cdot \epsilon \cdot \int_{ \gamma_{k}^{(n)} } \frac{1}{2^n} \cdot\mathcal{L}(\gamma) \, dz = 4^n \cdot \epsilon \cdot \frac{1}{2^n} \cdot \mathcal{L}(\gamma) \underbrace{\leq \int_{ \gamma_{k}^{(n)} } 1 \, dz}_{\mathcal{L}(\gamma_{k}^{(n)}) } &lt;/math&gt;
::&lt;math&gt;\leq 
4^n \cdot \epsilon \cdot \frac{1}{2^n} \cdot L(\gamma) \cdot \mathcal{L}(\gamma_{k}^{(n)} ) \leq 
4^n \cdot \epsilon \cdot \frac{\mathcal{L}(\gamma)}{4^n} = \epsilon \cdot \mathcal{L}(\gamma) &lt;/math&gt; for all &lt;math&gt;\epsilon &gt;0&lt;/math&gt;
:: Justification: (S19,LIW,IAL)
* (C1) &lt;math&gt;\Rightarrow&lt;/math&gt; &lt;math&gt;\int_{{{\left\langle{z}_{{1}},{z}_{{2}},{z}_{{3}}\right\rangle}}}{f{{\left({z}\right)}}}{d}{z}={0}&lt;/math&gt;


== Abbreviations for justifications ==
* (DU) &lt;math&gt;\forall_{a,b \in\mathbb{C}} : | a+b | \leq |a| + |b| &lt;/math&gt;
&lt;!--
* (DG) &lt;math&gt;\forall_{a,b,c \in\mathbb{M}} : a \cdot (b+c) = a \cdot b + a \cdot c &lt;/math&gt;
* (AG&lt;math&gt;+&lt;/math&gt;) &lt;math&gt;\forall_{a,b,c \in\mathbb{M}} : a+ (b + c) = (a + b) + c &lt;/math&gt;
* (AG&lt;math&gt;\cdot&lt;/math&gt;) &lt;math&gt;\forall_{a,b,c \in\mathbb{M}} : a\cdot (b \cdot c) = (a \cdot b) \cdot c &lt;/math&gt;
--&gt;
* (DI) Definition: Let &lt;math&gt; M\subset{C}&lt;/math&gt; be a set &lt;math&gt;\text{diam}(M) :=\text{sup} \lbrace |b-a| \, :\, a,b \in M \rbrace &lt;/math&gt;
* (WE) Definition (Path): Let &lt;math&gt;{U}\subseteq\mathbb{C}&lt;/math&gt; be a subset and &lt;math&gt; a,b \in\mathbb{R}&lt;/math&gt; with &lt;math&gt;a &lt; b &lt;/math&gt;. A path &lt;math&gt;\gamma&lt;/math&gt; in &lt;math&gt;U \subseteq \mathbb{C}&lt;/math&gt; is a continuous mapping &lt;math&gt;\gamma: [a,b] \to U &lt;/math&gt;.
* (SPU) Definition (Trace): Let &lt;math&gt;\gamma: [a,b] \to U&lt;/math&gt; be a path in &lt;math&gt;{U}\subseteq\mathbb{C}&lt;/math&gt;. The trace of &lt;math&gt;\gamma&lt;/math&gt; is defined as: &lt;math&gt;\text{Spur}(\gamma) := \lbrace \gamma(t)\in\mathbb{C} \, {\mid} \, t \in [a,b] \rbrace &lt;/math&gt;.
* (WZ) Definition (Path-connected): Let &lt;math&gt;{U}\subseteq\mathbb{C}&lt;/math&gt; be a subset. &lt;math&gt;{U}&lt;/math&gt; is called path-connected if there exists a path &lt;math&gt;\gamma:[a,b] \to U&lt;/math&gt; in &lt;math&gt;U \subseteq \mathbb{C}&lt;/math&gt; with &lt;math&gt; \gamma(a) = z_1 &lt;/math&gt;, &lt;math&gt;\gamma(b) = z_2 &lt;/math&gt; and &lt;math&gt;\text{Spur}(\gamma) \subseteq U&lt;/math&gt;.
* (GE) Definition (Domain): A subset &lt;math&gt;{G}\subseteq\mathbb{C}&lt;/math&gt; is called a domain if (1) &lt;math&gt;{G}&lt;/math&gt; is open, (2) &lt;math&gt;{G}\ne\emptyset&lt;/math&gt; and (3) &lt;math&gt;{G}&lt;/math&gt; is path-connected.
* (WG1) Definition (Smooth path): A path &lt;math&gt;\gamma: [a,b] \to\mathbb{C}&lt;/math&gt; is smooth if it is continuously differentiable.
* (UT) Definition (Subdivision): Let &lt;math&gt;[a,b]&lt;/math&gt; be an interval, &lt;math&gt;n \in\mathbb{N}&lt;/math&gt; and &lt;math&gt;{a}={u}_{{0}} &lt; {\ldots} &lt; {{u}}_{n}={b}&lt;/math&gt;. &lt;math&gt;{\left({u}_{{0}},\ldots,{u}_{{{n}}}\right)}\in\mathbb{R}^{n+1}&lt;/math&gt; is called a subdivision of &lt;math&gt;{\left[{a},{b}\right]}&lt;/math&gt;.
* (WG2) Definition (Path subdivision): Let &lt;math&gt;\gamma: [a,b] \to\mathbb{C}&lt;/math&gt; be a path in &lt;math&gt;{U}\subseteq\mathbb{C}&lt;/math&gt;, &lt;math&gt;{n}\in\mathbb{N}&lt;/math&gt;, &lt;math&gt;{\left({u}_{{0}},\ldots,{u}_{{{n}}}\right)}&lt;/math&gt; a subdivision of &lt;math&gt;[a,b]&lt;/math&gt;, &lt;math&gt;\gamma_{{k}}:{\left[{u}_{{{k}-{1}}},{u}_{{k}}\right]}\to\mathbb{C}&lt;/math&gt; for all &lt;math&gt;{k}\in{\left\lbrace{1},\ldots,{n}\right\rbrace}&lt;/math&gt; a path in &lt;math&gt;{U}&lt;/math&gt;. &lt;math&gt;{\left(\gamma_{{{1}}},\ldots,\gamma_{{{n}}}\right)}&lt;/math&gt; is called a path subdivision of &lt;math&gt;\gamma&lt;/math&gt; if &lt;math&gt;\gamma_{{n}}{\left({b}\right)}=\gamma{\left({b}\right)}&lt;/math&gt; and &lt;math&gt;\forall_{{{k}\in{\left\lbrace{1},\ldots,{n}\right\rbrace}}}\forall_{{{t}\in{\left[{u}_{{{k}-{1}}},{u}_{{k}}\right)}}}:\gamma_{{k}}{\left({t}\right)}=\gamma{\left({t}\right)}\wedge\gamma_{{k}}{\left({u}_{{{k}-{1}}}\right)}=\gamma_{{{k}-{1}}}{\left({u}_{{k}}\right)}&lt;/math&gt;.
* (WG3) Definition (Piecewise smooth path): A path &lt;math&gt;\gamma:{\left[{a},{b}\right]}\to\mathbb{C}&lt;/math&gt; is piecewise smooth if there exists a path subdivision &lt;math&gt;{\left(\gamma_{{1}},\ldots\gamma_{{n}}\right)}&lt;/math&gt; of &lt;math&gt;\gamma&lt;/math&gt; consisting of smooth paths &lt;math&gt;\gamma_{{k}}&lt;/math&gt; for all &lt;math&gt;{k}\in{\left\lbrace{1},\ldots,{n}\right\rbrace}&lt;/math&gt;.
* (WG4) Definition (Path integral): Let &lt;math&gt;f : U \to\mathbb{C}&lt;/math&gt; be a continuous function and &lt;math&gt;\gamma: [a,b] \to U &lt;/math&gt; a smooth path, then the path integral is defined as: &lt;math&gt;\int_{\gamma} f := \int_{\gamma} f(z) \, dz := \int_a^b f(\gamma(t)) \cdot\gamma'(t)\, dt &lt;/math&gt;. If &lt;math&gt;\gamma&lt;/math&gt; is only piecewise smooth with respect to a path subdivision &lt;math&gt;( \gamma_1 ,\ldots,\gamma_n ) &lt;/math&gt;, then we define &lt;math&gt;\int_{\gamma} f(z) \, dz :=\sum_{k=1}^{n} \int_{\gamma_k} f(z) \, dz&lt;/math&gt;.
* (SF) Theorem (Primitive with closed paths): If a continuous function &lt;math&gt;f : U \to\mathbb{C}&lt;/math&gt; has a primitive &lt;math&gt;F : U \to\mathbb{C}&lt;/math&gt;, then for a piecewise smooth path &lt;math&gt;\gamma: [a,b] \to U&lt;/math&gt; we have &lt;math&gt;\int_{\gamma} f(z) \, dz =F(b) - F(a)&lt;/math&gt;.
* (LIW) Length of the integration path: Let &lt;math&gt;\gamma:{\left[{a},{b}\right]}\to\mathbb{C}&lt;/math&gt; be a smooth path, then the &lt;math&gt;\mathcal{L}(\gamma)&lt;/math&gt; is defined as:
:: &lt;math&gt;\mathcal{L}(\gamma) := \int_{a}^{b} |\gamma'(t)| \, dt&lt;/math&gt;. 
: If &lt;math&gt;\gamma:{\left[{a},{b}\right]}\to\mathbb{C}&lt;/math&gt; is a general integration path with the path subdivision &lt;math&gt;{\left(\gamma_{{1}},\ldots\gamma_{{n}}\right)}&lt;/math&gt; of smooth paths &lt;math&gt;\gamma_{{k}}&lt;/math&gt;, then &lt;math&gt;\mathcal{L}(\gamma)&lt;/math&gt; is defined as the sum of the lengths of the smooth paths &lt;math&gt;\gamma_{{k}}&lt;/math&gt;, i.e.:
:: &lt;math&gt;\mathcal{L}(\gamma) := \sum_{k=1}^{n} \mathcal{L}(\gamma_{k})&lt;/math&gt;
* (IAL) Integral estimate over the length of the integration path: Let &lt;math&gt;\gamma:{\left[{a},{b}\right]}\to\mathbb{G}&lt;/math&gt; be an integration path on the domain &lt;math&gt;G \subseteq \mathbb{C}&lt;/math&gt;, then for a continuous function &lt;math&gt;f&lt;/math&gt; on &lt;math&gt;\text{Spur}(\gamma)&lt;/math&gt; we have the estimate:
::&lt;math&gt;\left| \int_{\gamma} f(z) \, dz \right| \leq \max_{z \in \text{Spur}(\gamma)} |f(z)| \cdot \mathcal{L}(\gamma)&lt;/math&gt;

== Literature ==
* Eberhard Freitag &amp; Rolf Busam: ''Funktionentheorie 1'', Springer-Verlag, Berlin

== See also ==
* [[Complex Analysis/Goursat's Lemma]]-Short form
* [[Complex Analysis/Path of Integration|Path of Integration]]

[[Category:Complex Analysis]]
[[Category:Theorem (Mathematics)|Goursat, Lemma of]]


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* [https://de.wikipedia.org/wiki/Lemma%20von%20Goursat Lemma von Goursat] https://de.wikipedia.org/wiki/Lemma%20von%20Goursat
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&lt;noninclude&gt;[[de:Kurs:Funktionentheorie/Lemma von Goursat (Details)]]&lt;/noinclude&gt;</text>
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      <text bytes="18517" sha1="r690d7vfp5zgulbapghatkbd0pzerwq" xml:space="preserve">Goursat's Lemma, also known as the Goursat's Theorem, is a theorem in [[Complex analysis]].
Goursat's lemma is a precursor to the [[w:en:Cauchy's integral theorem|Cauchy's integral theorem]] and is often used in its proof. It plays an important role in the development of complex analysis. Remarkably, the lemma only requires  [[Holomorphic function|Complex differentiability]] but not [[w:en:Continuity|continuous]] differentiability. The lemma was proved in its rectangular form by [[w:en:Édouard Goursat|Édouard Goursat]] ([[w:en:1858|1858]]–[[w:en:1936|1936]]) and published in [[w:en:1884|1884]]. The triangular form commonly used today was introduced by [[w:en:Alfred Pringsheim|Alfred Pringsheim]].
== Goursat's Lemma ==
Given the following assumptions:

* (P1) Let &lt;math&gt;{U}\subseteq\mathbb{C}&lt;/math&gt; be an open subset,
* (P2) Let &lt;math&gt;{z}_{{1}},{z}_{{2}},{z}_{{3}}\in\mathbb{C}&lt;/math&gt; be three non-collinear points that define the triangle
:&lt;math&gt;\Delta{\left({z}_{{{1}}},{z}_{{{2}}},{z}_{{{3}}}\right)} := \left\{ \sum_{k=1}^{3} \lambda_{k} \cdot{z}_{k} {\mid} {\left({\sum_{{{k} {1}}}^{{3}}}\lambda_{{k}}={1}\right)}\wedge\forall{k}\in{\left\lbrace{1},{2},{3} \right\rbrace} \lambda_{{k}}\in [{0},{1}]\right\} \subset{U} &lt;/math&gt;
* (P3) Let &lt;math&gt;{f}:{U}\to\mathbb{C}&lt;/math&gt; be a holomorphic function,
* (P4) Let &lt;math&gt;{\left\langle{z}_{{1}},{z}_{{2}},{z}_{{3}}\right\rangle}:{\left[{0},{3}\right]}\to\mathbb{C}&lt;/math&gt; be the closed path over the triangle edge of &lt;math&gt;\Delta{\left({z}_{{{1}}},{z}_{{{2}}},{z}_{{{3}}}\right)}&lt;/math&gt; with starting point &lt;math&gt;{z}_{{1}}&lt;/math&gt;,
then the following statements hold:

* (C1) &lt;math&gt;\int_{{{\left\langle{z}_{{1}},{z}_{{2}},{z}_{{3}}\right\rangle}}}{f{{\left({z}\right)}}}{d}{z}={0}&lt;/math&gt;
== Proof ==
[[File:Dreiecksweg.svg|thumb|Integration path along the triangle boundary]]
[[File:Lemma goursat2 seitenmitten m1m2m3.svg|thumb|Subdivision of the outer paths and insertion of additional paths between the midpoints of the sides, which cancel out in the line integral due to the reversed direction of the integration path, resulting in a sum of 0 and leaving the total integral unchanged.]]
[[File:Lemma goursat3 wege.svg|thumb|Inductive definition of the paths. The subtriangles are [[w:de:Ähnlichkeit_(Geometrie)|similar]] to the original triangle. By using the midpoints of the sides, the perimeter of a triangle &lt;math&gt;\Delta^{(n)}&lt;/math&gt; is halved with each step to &lt;math&gt;\Delta^{(n+1)}&lt;/math&gt;.]]
(S1) We define a sequence of triangular paths recursively as &lt;math&gt;\gamma^{(n)} := {\left\langle z_{1}^{(n)} , z_{2}^{(n)} , z_{3}^{(n)} \right\rangle}&lt;/math&gt;.
=== Proof part 1: Definition of the triangle paths ===
* (S2) (DEF) For &lt;math&gt;{n}={0}&lt;/math&gt; let the closed triangle path &lt;math&gt;\gamma^{(0)} : [0,3] \to \mathbb{C}&lt;/math&gt; be defined as:
:&lt;math&gt;
\gamma^{(0)}(t) := \left\langle z_1 ,z_2 ,z_3 \right\rangle (t) := 
\begin{cases}
 (1-t)\cdot z_1 + t\cdot z_2 &amp; \text{for } t \in [0,1] \\
 (2-t)\cdot z_2 + (t-1) \cdot z_3 &amp; \text{for } t \in (1,2] \\ 
 (3-t)\cdot z_3 + (t-2) \cdot z_1 &amp; \text{for } t \in (2,3] \\ 
\end{cases}
&lt;/math&gt; 
Furthermore, let &lt;math&gt;\gamma^{(n)}&lt;/math&gt; be already defined. We define &lt;math&gt;\gamma^{(n+1)}&lt;/math&gt; inductively.
:: Justification: (P4,UT)
* (S3) (DEF) Definition: Triangle path &lt;math&gt;{\gamma_{1}^{(n)}} := {\left\langle\frac{{ z_{1}^{(n)} + z_{2}^{(n)} }}{{2}}, z_{2}^{(n)} ,\frac{{ z_{2}^{(n)} + z_{3}^{(n)} }}{{2}}\right\rangle}&lt;/math&gt;,
:: Justification: (S3,S4,S5)
* (S4) (DEF) Definition: Triangle path &lt;math&gt;{\gamma_{2}^{(n)}} := {\left\langle\frac{{ z_{2}^{(n)} + z_{3}^{(n)} }}{{2}}, z_{3}^{(n)} ,\frac{{ z_{1}^{(n)} + z_{3}^{(n)} }}{{2}}\right\rangle}&lt;/math&gt;,
* (S5) (DEF) Definition: Triangle path &lt;math&gt;{\gamma_{3}^{(n)}} := {\left\langle\frac{{ z_{1}^{(n)} + z_{3}^{(n)} }}{{2}}, z_{1}^{(n)} , 
\frac{{ z_{1}^{(n)} + z_{2}^{(n)} }}{{2}}\right\rangle}&lt;/math&gt;,
* (S6) (DEF) Definition: Triangle path &lt;math&gt;{\gamma_{4}^{(n)}} := {\left\langle\frac{{ z_{1}^{(n)} + z_{2}^{(n)} }}{{2}},\frac{{ z_{2}^{(n)} + z_{3}^{(n)} }}{{2}},\frac{{ z_{1}^{(n)} + z_{3}^{(n)} }}{{2}}\right\rangle}&lt;/math&gt;
* (S7) (DEF) Let &lt;math&gt;{i}\in{\left\lbrace{1},{2},{3},{4}\right\rbrace}&lt;/math&gt; be the smallest index with &lt;math&gt;\forall_{{{k}\in{\left\lbrace{1},,{2},{3},{4}\right\rbrace}}}:{\left|\int_{{{\gamma_{{k}}^{(n)}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}\le{\left|\int_{{{\gamma_{{i}}^{(n)}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}&lt;/math&gt; and &lt;math&gt;\gamma^{{{\left({n}+{1}\right)}}} := {\gamma_{{i}}^{(n)}}&lt;/math&gt;

=== Proof part 2: Estimates ===

* (S8) &lt;math&gt;\Rightarrow&lt;/math&gt; &lt;math&gt; \int_{\gamma^{(n)}} f(z) \, dz = \sum_{k=1}^{4} \int_{\gamma_k}^{(n)} f(z) \, dz&lt;/math&gt;
* (S9) &lt;math&gt;\Rightarrow&lt;/math&gt; &lt;math&gt;{\left|\int_{{\gamma^{{{n}}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}={\left|{\sum_{{{k}={1}}}^{{4}}}\int_{{{\gamma_{{k}}^{(n)}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}\le{\sum_{{{k}={1}}}^{{4}}}{\left|\int_{{{\gamma_{{k}}^{(n)}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}\le{4}\cdot{\left|\int_{{{\gamma_{{k}}^{(n)}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}&lt;/math&gt; for all &lt;math&gt;{n}\in\mathbb{N}&lt;/math&gt;
:: Justification: (S7,WG4,DU)
* (S10) &lt;math&gt;\Rightarrow&lt;/math&gt; &lt;math&gt;{0}\le{\left|\int_{{{\left\langle{z}_{{1}},{z}_{{2}},{z}_{{3}}\right\rangle}}}{f{{\left({z}\right)}}}{d}{z}\right|}={\left|\int_{\gamma^{(0)}} f(z) {\left.{d}{z}\right.}\right|}\le{4}\cdot{\left|\int_{{\gamma^{{{\left({1}\right)}}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}\le\ldots\le{4}^{{n}}\cdot{\left|\int_{{{\gamma_{{i}}^{(n)}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}={4}^{{n}}\cdot{\left|\int_{{\gamma^{{{\left({n}+{1}\right)}}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}&lt;/math&gt;

=== Proof part 3: Diameter of the sub-triangles === 
* (S11) The nested definition of the sub-triangles yields for all &lt;math&gt;{n}\in\mathbb{N}&lt;/math&gt;: &lt;math&gt;\Delta{\left({{z}_{1}^{(n)}},{{z}_{2}^{(n)}},{{z}_{3}^{(n)}}\right)}\supset\Delta{\left({{z}_{1}^{{{\left({n}+{1}\right)}}}},{{z}_{2}^{{{\left({n}+{1}\right)}}}},{{z}_{3}^{{{\left({n}+{1}\right)}}}}\right)}&lt;/math&gt; and 
:: &lt;math&gt;\lim_{n\to\infty} \, \text{diam} \left(\Delta{\left({{z}_{1}^{(n)}},{{z}_{2}^{(n)}},{{z}_{3}^{(n)}}\right)}\right) = 0 &lt;/math&gt;
* (S12) &lt;math&gt;\Rightarrow&lt;/math&gt; &lt;math&gt;\exists_{{{z}_{{0}}\in{U}}}\forall_{{{n}\in\mathbb{N}}}:{z}_{{0}}\in\Delta^{(n)} := \Delta{\left({{z}_{1}^{(n)}},{{z}_{2}^{(n)}},{{z}_{3}^{(n)}}\right)}&lt;/math&gt; and &lt;math&gt;{\left\lbrace{z}_{{0}}\right\rbrace}=\bigcap_{{{n}\in\mathbb{N}}}\Delta^{(n)}&lt;/math&gt;

=== Proof part 4: Use of holomorphism (P3) ===
* (S13) We use the holomorphism of &lt;math&gt;f&lt;/math&gt; in &lt;math&gt;z_0 \in U &lt;/math&gt; for further steps with 
: &lt;math&gt;f(z) := f(z_0) +f'(z_0) \cdot ( z - z_0 ) + r(z)&lt;/math&gt; and &lt;math&gt;\lim_{z \to z_0} \frac{r(z)}{z - z_0} = 0 &lt;/math&gt;
:: Justification: (P3)
* (S14) &lt;math&gt;\Rightarrow&lt;/math&gt; The function &lt;math&gt;{h}:{U}\to\mathbb{C}&lt;/math&gt; with &lt;math&gt;h(z) := f(z_0) +f'(z_0) \cdot ( z - z_0 )&lt;/math&gt; has a primitive &lt;math&gt;H(z) := f(z_0) +f'(z_0) \cdot \frac{1}{2}\cdot (z - z_0)^2 &lt;/math&gt;
:: Justification: since &lt;math&gt;h(z)&lt;/math&gt; is a polynomial of degree 1.
* (S15) &lt;math&gt;\Rightarrow&lt;/math&gt; The path integral over the closed paths &lt;math&gt;\gamma^{(n)}&lt;/math&gt; of the function &lt;math&gt;{h}:{U}\to\mathbb{C}&lt;/math&gt; is thus &lt;math&gt;\int_{{{\gamma_{{k}}^{(n)}}}}{h}{\left({z}\right)}={0}&lt;/math&gt;
:: Justification: (SF)
* (S16) &lt;math&gt;\Rightarrow&lt;/math&gt; For the path integral over the closed paths &lt;math&gt;\gamma^{(n)}&lt;/math&gt; of the function &lt;math&gt;{f}:{U}\to\mathbb{C}&lt;/math&gt; we have &lt;math&gt;\int_{{{\gamma_{{k}}^{(n)}}}}{f{{\left({z}\right)}{\left.{d}{z}\right.}}}=\int_{{{\gamma_{{k}}^{(n)}}}}{h}{\left({z}\right)}+{r}{\left({z}\right)}{\left.{d}{z}\right.}=\int_{{{\gamma_{{k}}^{(n)}}}}{r}{\left({z}\right)}{\left.{d}{z}\right.}&lt;/math&gt;

=== Proof part 4: Estimate of the remainder term &lt;math&gt;r(z)&lt;/math&gt; ===
* (S17) &lt;math&gt;\Rightarrow&lt;/math&gt; With &lt;math&gt;\lim_{z \to z_0} \frac{r(z)}{ z - z_0 }= 0 &lt;/math&gt; we have: For all &lt;math&gt;\epsilon&gt;{0}&lt;/math&gt; there exists a &lt;math&gt;\delta&gt;{0}&lt;/math&gt;
:: &lt;math&gt; | z - z_0 | &lt; \delta \Rightarrow \left| \frac{ r(z) }{ z - z_0 } \right| &lt; \epsilon&lt;/math&gt;
: Justification: &lt;math&gt;\epsilon&lt;/math&gt;-&lt;math&gt;\delta&lt;/math&gt;-criterion applied to &lt;math&gt;g(z):=\frac{r(z)}{ z - z_0 }&lt;/math&gt; and continuity of &lt;math&gt;g&lt;/math&gt; in &lt;math&gt;z_0&lt;/math&gt; 
* (S18) &lt;math&gt;\Rightarrow&lt;/math&gt; For all &lt;math&gt;\epsilon&gt;{0}&lt;/math&gt; there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt;: &lt;math&gt;| z - z_0 | &lt; \delta\Rightarrow | r(z) | &lt; \epsilon \cdot | z - z_0 |&lt;/math&gt; 
:: &lt;math&gt;0 \leq \left|\int_{\left\langle z_1 , z_2 , z_3 \right\rangle} f(z) \, dz \right| \leq 4^n \cdot \left| \int_{ \gamma_{k}^{(n)} } f(z)\, dz \right| = 4^n \cdot \left| \int_{ \gamma_{k}^{(n)} } r(z) \, dz \right| \leq 4^n \cdot \int_{ \gamma_{k}^{(n)} } |r(z)| \, dz \leq 4^n \cdot \int_{ \gamma_{k}^{(n)} } \epsilon \cdot | z - z_0 | \, dz&lt;/math&gt;
:: Justification: (S2)
* (S20) From the condition &lt;math&gt;\lim_{n\to\infty}\text{diam} \left( \Delta^{(n)} \right) = 0&lt;/math&gt; there exists for all &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; an &lt;math&gt;n_{\delta} \in\mathbb{N}&lt;/math&gt; with &lt;math&gt;\Delta^{(n)}\subseteq {D}_{\delta}(z_0)&lt;/math&gt; for all &lt;math&gt;n &gt; n_{\delta}&lt;/math&gt;.
* (S21) &lt;math&gt;\Rightarrow&lt;/math&gt; &lt;math&gt;| z - z_0 | &lt; L \left(\gamma^{(n)}\right) = \frac{1}{2^n} \cdot L \left(\gamma\right) &lt;/math&gt; for all &lt;math&gt; n \in\mathbb{N}&lt;/math&gt; and all &lt;math&gt; z \in \Delta^{(n)}&lt;/math&gt;
:: Justification: The factor &lt;math&gt;\frac{1}{2^n}&lt;/math&gt; arises from the continued halving of the sides of the triangles &lt;math&gt;\Delta^{(n)}&lt;/math&gt;
* (S22) This implies:
:&lt;math&gt; 0 \leq \left|\int_{ \langle z_1 , z_2 , z_3 \rangle } f(z) \, dz \right| \leq 4^n \cdot \int_{ \gamma_{k}^{(n)} } \epsilon \cdot | z - z_0 | \, dz \leq 4^n \cdot \epsilon \cdot \int_{ \gamma_{k}^{(n)} } \frac{1}{2^n} \cdot\mathcal{L}(\gamma) \, dz = 4^n \cdot \epsilon \cdot \frac{1}{2^n} \cdot \mathcal{L}(\gamma) \underbrace{\leq \int_{ \gamma_{k}^{(n)} } 1 \, dz}_{\mathcal{L}(\gamma_{k}^{(n)}) } &lt;/math&gt;
::&lt;math&gt;\leq 
4^n \cdot \epsilon \cdot \frac{1}{2^n} \cdot L(\gamma) \cdot \mathcal{L}(\gamma_{k}^{(n)} ) \leq 
4^n \cdot \epsilon \cdot \frac{\mathcal{L}(\gamma)}{4^n} = \epsilon \cdot \mathcal{L}(\gamma) &lt;/math&gt; for all &lt;math&gt;\epsilon &gt;0&lt;/math&gt;
:: Justification: (S19,LIW,IAL)
* (C1) &lt;math&gt;\Rightarrow&lt;/math&gt; &lt;math&gt;\int_{{{\left\langle{z}_{{1}},{z}_{{2}},{z}_{{3}}\right\rangle}}}{f{{\left({z}\right)}}}{d}{z}={0}&lt;/math&gt;


== Abbreviations for justifications ==
* (DU) &lt;math&gt;\forall_{a,b \in\mathbb{C}} : | a+b | \leq |a| + |b| &lt;/math&gt;
&lt;!--
* (DG) &lt;math&gt;\forall_{a,b,c \in\mathbb{M}} : a \cdot (b+c) = a \cdot b + a \cdot c &lt;/math&gt;
* (AG&lt;math&gt;+&lt;/math&gt;) &lt;math&gt;\forall_{a,b,c \in\mathbb{M}} : a+ (b + c) = (a + b) + c &lt;/math&gt;
* (AG&lt;math&gt;\cdot&lt;/math&gt;) &lt;math&gt;\forall_{a,b,c \in\mathbb{M}} : a\cdot (b \cdot c) = (a \cdot b) \cdot c &lt;/math&gt;
--&gt;
* (DI) Definition: Let &lt;math&gt; M\subset{C}&lt;/math&gt; be a set &lt;math&gt;\text{diam}(M) :=\text{sup} \lbrace |b-a| \, :\, a,b \in M \rbrace &lt;/math&gt;
* (WE) Definition (Path): Let &lt;math&gt;{U}\subseteq\mathbb{C}&lt;/math&gt; be a subset and &lt;math&gt; a,b \in\mathbb{R}&lt;/math&gt; with &lt;math&gt;a &lt; b &lt;/math&gt;. A path &lt;math&gt;\gamma&lt;/math&gt; in &lt;math&gt;U \subseteq \mathbb{C}&lt;/math&gt; is a continuous mapping &lt;math&gt;\gamma: [a,b] \to U &lt;/math&gt;.
* (SPU) Definition (Trace): Let &lt;math&gt;\gamma: [a,b] \to U&lt;/math&gt; be a path in &lt;math&gt;{U}\subseteq\mathbb{C}&lt;/math&gt;. The trace of &lt;math&gt;\gamma&lt;/math&gt; is defined as: &lt;math&gt;\text{Spur}(\gamma) := \lbrace \gamma(t)\in\mathbb{C} \, {\mid} \, t \in [a,b] \rbrace &lt;/math&gt;.
* (WZ) Definition (Path-connected): Let &lt;math&gt;{U}\subseteq\mathbb{C}&lt;/math&gt; be a subset. &lt;math&gt;{U}&lt;/math&gt; is called path-connected if there exists a path &lt;math&gt;\gamma:[a,b] \to U&lt;/math&gt; in &lt;math&gt;U \subseteq \mathbb{C}&lt;/math&gt; with &lt;math&gt; \gamma(a) = z_1 &lt;/math&gt;, &lt;math&gt;\gamma(b) = z_2 &lt;/math&gt; and &lt;math&gt;\text{Spur}(\gamma) \subseteq U&lt;/math&gt;.
* (GE) Definition (Domain): A subset &lt;math&gt;{G}\subseteq\mathbb{C}&lt;/math&gt; is called a domain if (1) &lt;math&gt;{G}&lt;/math&gt; is open, (2) &lt;math&gt;{G}\ne\emptyset&lt;/math&gt; and (3) &lt;math&gt;{G}&lt;/math&gt; is path-connected.
* (WG1) Definition (Smooth path): A path &lt;math&gt;\gamma: [a,b] \to\mathbb{C}&lt;/math&gt; is smooth if it is continuously differentiable.
* (UT) Definition (Subdivision): Let &lt;math&gt;[a,b]&lt;/math&gt; be an interval, &lt;math&gt;n \in\mathbb{N}&lt;/math&gt; and &lt;math&gt;{a}={u}_{{0}} &lt; {\ldots} &lt; {{u}}_{n}={b}&lt;/math&gt;. &lt;math&gt;{\left({u}_{{0}},\ldots,{u}_{{{n}}}\right)}\in\mathbb{R}^{n+1}&lt;/math&gt; is called a subdivision of &lt;math&gt;{\left[{a},{b}\right]}&lt;/math&gt;.
* (WG2) Definition (Path subdivision): Let &lt;math&gt;\gamma: [a,b] \to\mathbb{C}&lt;/math&gt; be a path in &lt;math&gt;{U}\subseteq\mathbb{C}&lt;/math&gt;, &lt;math&gt;{n}\in\mathbb{N}&lt;/math&gt;, &lt;math&gt;{\left({u}_{{0}},\ldots,{u}_{{{n}}}\right)}&lt;/math&gt; a subdivision of &lt;math&gt;[a,b]&lt;/math&gt;, &lt;math&gt;\gamma_{{k}}:{\left[{u}_{{{k}-{1}}},{u}_{{k}}\right]}\to\mathbb{C}&lt;/math&gt; for all &lt;math&gt;{k}\in{\left\lbrace{1},\ldots,{n}\right\rbrace}&lt;/math&gt; a path in &lt;math&gt;{U}&lt;/math&gt;. &lt;math&gt;{\left(\gamma_{{{1}}},\ldots,\gamma_{{{n}}}\right)}&lt;/math&gt; is called a path subdivision of &lt;math&gt;\gamma&lt;/math&gt; if &lt;math&gt;\gamma_{{n}}{\left({b}\right)}=\gamma{\left({b}\right)}&lt;/math&gt; and &lt;math&gt;\forall_{{{k}\in{\left\lbrace{1},\ldots,{n}\right\rbrace}}}\forall_{{{t}\in{\left[{u}_{{{k}-{1}}},{u}_{{k}}\right)}}}:\gamma_{{k}}{\left({t}\right)}=\gamma{\left({t}\right)}\wedge\gamma_{{k}}{\left({u}_{{{k}-{1}}}\right)}=\gamma_{{{k}-{1}}}{\left({u}_{{k}}\right)}&lt;/math&gt;.
* (WG3) Definition (Piecewise smooth path): A path &lt;math&gt;\gamma:{\left[{a},{b}\right]}\to\mathbb{C}&lt;/math&gt; is piecewise smooth if there exists a path subdivision &lt;math&gt;{\left(\gamma_{{1}},\ldots\gamma_{{n}}\right)}&lt;/math&gt; of &lt;math&gt;\gamma&lt;/math&gt; consisting of smooth paths &lt;math&gt;\gamma_{{k}}&lt;/math&gt; for all &lt;math&gt;{k}\in{\left\lbrace{1},\ldots,{n}\right\rbrace}&lt;/math&gt;.
* (WG4) Definition (Path integral): Let &lt;math&gt;f : U \to\mathbb{C}&lt;/math&gt; be a continuous function and &lt;math&gt;\gamma: [a,b] \to U &lt;/math&gt; a smooth path, then the path integral is defined as: &lt;math&gt;\int_{\gamma} f := \int_{\gamma} f(z) \, dz := \int_a^b f(\gamma(t)) \cdot\gamma'(t)\, dt &lt;/math&gt;. If &lt;math&gt;\gamma&lt;/math&gt; is only piecewise smooth with respect to a path subdivision &lt;math&gt;( \gamma_1 ,\ldots,\gamma_n ) &lt;/math&gt;, then we define &lt;math&gt;\int_{\gamma} f(z) \, dz :=\sum_{k=1}^{n} \int_{\gamma_k} f(z) \, dz&lt;/math&gt;.
* (SF) Theorem (Primitive with closed paths): If a continuous function &lt;math&gt;f : U \to\mathbb{C}&lt;/math&gt; has a primitive &lt;math&gt;F : U \to\mathbb{C}&lt;/math&gt;, then for a piecewise smooth path &lt;math&gt;\gamma: [a,b] \to U&lt;/math&gt; we have &lt;math&gt;\int_{\gamma} f(z) \, dz =F(b) - F(a)&lt;/math&gt;.
* (LIW) Length of the integration path: Let &lt;math&gt;\gamma:{\left[{a},{b}\right]}\to\mathbb{C}&lt;/math&gt; be a smooth path, then the &lt;math&gt;\mathcal{L}(\gamma)&lt;/math&gt; is defined as:
:: &lt;math&gt;\mathcal{L}(\gamma) := \int_{a}^{b} |\gamma'(t)| \, dt&lt;/math&gt;. 
: If &lt;math&gt;\gamma:{\left[{a},{b}\right]}\to\mathbb{C}&lt;/math&gt; is a general integration path with the path subdivision &lt;math&gt;{\left(\gamma_{{1}},\ldots\gamma_{{n}}\right)}&lt;/math&gt; of smooth paths &lt;math&gt;\gamma_{{k}}&lt;/math&gt;, then &lt;math&gt;\mathcal{L}(\gamma)&lt;/math&gt; is defined as the sum of the lengths of the smooth paths &lt;math&gt;\gamma_{{k}}&lt;/math&gt;, i.e.:
:: &lt;math&gt;\mathcal{L}(\gamma) := \sum_{k=1}^{n} \mathcal{L}(\gamma_{k})&lt;/math&gt;
* (IAL) Integral estimate over the length of the integration path: Let &lt;math&gt;\gamma:{\left[{a},{b}\right]}\to\mathbb{G}&lt;/math&gt; be an integration path on the domain &lt;math&gt;G \subseteq \mathbb{C}&lt;/math&gt;, then for a continuous function &lt;math&gt;f&lt;/math&gt; on &lt;math&gt;\text{Spur}(\gamma)&lt;/math&gt; we have the estimate:
::&lt;math&gt;\left| \int_{\gamma} f(z) \, dz \right| \leq \max_{z \in \text{Spur}(\gamma)} |f(z)| \cdot \mathcal{L}(\gamma)&lt;/math&gt;

== Literature ==
* Eberhard Freitag &amp; Rolf Busam: ''Funktionentheorie 1'', Springer-Verlag, Berlin

== See also ==
* [[Complex Analysis/Goursat's Lemma]]-Short form
* [[Complex Analysis/Path of Integration|Path of Integration]]

[[Category:Complex Analysis]]
[[Category:Theorem (Mathematics)|Goursat, Lemma of]]


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      <text bytes="18516" sha1="drhw9jxuixww0wluggb0536mycqcjv1" xml:space="preserve">Goursat's Lemma, also known as the Goursat's Theorem, is a theorem in [[Complex analysis]].
Goursat's lemma is a precursor to the [[w:en:Cauchy's integral theorem|Cauchy's integral theorem]] and is often used in its proof. It plays an important role in the development of complex analysis. Remarkably, the lemma only requires  [[Holomorphic function|Complex differentiability]] but not [[w:en:Continuity|continuous]] differentiability. The lemma was proved in its rectangular form by [[w:en:Édouard Goursat|Édouard Goursat]] ([[w:en:1858|1858]]–[[w:en:1936|1936]]) and published in [[w:en:1884|1884]]. The triangular form commonly used today was introduced by [[w:en:Alfred Pringsheim|Alfred Pringsheim]].
== Goursat's Lemma ==
Given the following assumptions:

* (P1) Let &lt;math&gt;{U}\subseteq\mathbb{C}&lt;/math&gt; be an open subset,
* (P2) Let &lt;math&gt;{z}_{{1}},{z}_{{2}},{z}_{{3}}\in\mathbb{C}&lt;/math&gt; be three non-collinear points that define the triangle
:&lt;math&gt;\Delta{\left({z}_{{{1}}},{z}_{{{2}}},{z}_{{{3}}}\right)} := \left\{ \sum_{k=1}^{3} \lambda_{k} \cdot{z}_{k} {\mid} {\left({\sum_{{{k} {1}}}^{{3}}}\lambda_{{k}}={1}\right)}\wedge\forall{k}\in{\left\lbrace{1},{2},{3} \right\rbrace} \lambda_{{k}}\in [{0},{1}]\right\} \subset{U} &lt;/math&gt;
* (P3) Let &lt;math&gt;{f}:{U}\to\mathbb{C}&lt;/math&gt; be a holomorphic function,
* (P4) Let &lt;math&gt;{\left\langle{z}_{{1}},{z}_{{2}},{z}_{{3}}\right\rangle}:{\left[{0},{3}\right]}\to\mathbb{C}&lt;/math&gt; be the closed path over the triangle edge of &lt;math&gt;\Delta{\left({z}_{{{1}}},{z}_{{{2}}},{z}_{{{3}}}\right)}&lt;/math&gt; with starting point &lt;math&gt;{z}_{{1}}&lt;/math&gt;,
then the following statements hold:

* (C1) &lt;math&gt;\int_{{{\left\langle{z}_{{1}},{z}_{{2}},{z}_{{3}}\right\rangle}}}{f{{\left({z}\right)}}}{d}{z}={0}&lt;/math&gt;
== Proof ==
[[File:Dreiecksweg.svg|thumb|Integration path along the triangle boundary]]
[[File:Lemma goursat2 seitenmitten m1m2m3.svg|thumb|Subdivision of the outer paths and insertion of additional paths between the midpoints of the sides, which cancel out in the line integral due to the reversed direction of the integration path, resulting in a sum of 0 and leaving the total integral unchanged.]]
[[File:Lemma goursat3 wege.svg|thumb|Inductive definition of the paths. The subtriangles are [[w:de:Ähnlichkeit_(Geometrie)|similar]] to the original triangle. By using the midpoints of the sides, the perimeter of a triangle &lt;math&gt;\Delta^{(n)}&lt;/math&gt; is halved with each step to &lt;math&gt;\Delta^{(n+1)}&lt;/math&gt;.]]
(S1) We define a sequence of triangular paths recursively as &lt;math&gt;\gamma^{(n)} := {\left\langle z_{1}^{(n)} , z_{2}^{(n)} , z_{3}^{(n)} \right\rangle}&lt;/math&gt;.
=== Proof part 1: Definition of the triangle paths ===
* (S2) (DEF) For &lt;math&gt;{n}={0}&lt;/math&gt; let the closed triangle path &lt;math&gt;\gamma^{(0)} : [0,3] \to \mathbb{C}&lt;/math&gt; be defined as:
:&lt;math&gt;
\gamma^{(0)}(t) := \left\langle z_1 ,z_2 ,z_3 \right\rangle (t) := 
\begin{cases}
 (1-t)\cdot z_1 + t\cdot z_2 &amp; \text{for } t \in [0,1] \\
 (2-t)\cdot z_2 + (t-1) \cdot z_3 &amp; \text{for } t \in (1,2] \\ 
 (3-t)\cdot z_3 + (t-2) \cdot z_1 &amp; \text{for } t \in (2,3] \\ 
\end{cases}
&lt;/math&gt; 
Furthermore, let &lt;math&gt;\gamma^{(n)}&lt;/math&gt; be already defined. We define &lt;math&gt;\gamma^{(n+1)}&lt;/math&gt; inductively.
:: Justification: (P4,UT)
* (S3) (DEF) Definition: Triangle path &lt;math&gt;{\gamma_{1}^{(n)}} := {\left\langle\frac{{ z_{1}^{(n)} + z_{2}^{(n)} }}{{2}}, z_{2}^{(n)} ,\frac{{ z_{2}^{(n)} + z_{3}^{(n)} }}{{2}}\right\rangle}&lt;/math&gt;,
:: Justification: (S3,S4,S5)
* (S4) (DEF) Definition: Triangle path &lt;math&gt;{\gamma_{2}^{(n)}} := {\left\langle\frac{{ z_{2}^{(n)} + z_{3}^{(n)} }}{{2}}, z_{3}^{(n)} ,\frac{{ z_{1}^{(n)} + z_{3}^{(n)} }}{{2}}\right\rangle}&lt;/math&gt;,
* (S5) (DEF) Definition: Triangle path &lt;math&gt;{\gamma_{3}^{(n)}} := {\left\langle\frac{{ z_{1}^{(n)} + z_{3}^{(n)} }}{{2}}, z_{1}^{(n)} , 
\frac{{ z_{1}^{(n)} + z_{2}^{(n)} }}{{2}}\right\rangle}&lt;/math&gt;,
* (S6) (DEF) Definition: Triangle path &lt;math&gt;{\gamma_{4}^{(n)}} := {\left\langle\frac{{ z_{1}^{(n)} + z_{2}^{(n)} }}{{2}},\frac{{ z_{2}^{(n)} + z_{3}^{(n)} }}{{2}},\frac{{ z_{1}^{(n)} + z_{3}^{(n)} }}{{2}}\right\rangle}&lt;/math&gt;
* (S7) (DEF) Let &lt;math&gt;{i}\in{\left\lbrace{1},{2},{3},{4}\right\rbrace}&lt;/math&gt; be the smallest index with &lt;math&gt;\forall_{{{k}\in{\left\lbrace{1},,{2},{3},{4}\right\rbrace}}}:{\left|\int_{{{\gamma_{{k}}^{(n)}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}\le{\left|\int_{{{\gamma_{{i}}^{(n)}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}&lt;/math&gt; and &lt;math&gt;\gamma^{{{\left({n}+{1}\right)}}} := {\gamma_{{i}}^{(n)}}&lt;/math&gt;

=== Proof part 2: Estimates ===

* (S8) &lt;math&gt;\Rightarrow&lt;/math&gt; &lt;math&gt; \int_{\gamma^{(n)}} f(z) \, dz = \sum_{k=1}^{4} \int_{\gamma_k}^{(n)} f(z) \, dz&lt;/math&gt;
* (S9) &lt;math&gt;\Rightarrow&lt;/math&gt; &lt;math&gt;{\left|\int_{{\gamma^{{{n}}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}={\left|{\sum_{{{k}={1}}}^{{4}}}\int_{{{\gamma_{{k}}^{(n)}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}\le{\sum_{{{k}={1}}}^{{4}}}{\left|\int_{{{\gamma_{{k}}^{(n)}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}\le{4}\cdot{\left|\int_{{{\gamma_{{k}}^{(n)}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}&lt;/math&gt; for all &lt;math&gt;{n}\in\mathbb{N}&lt;/math&gt;
:: Justification: (S7,WG4,DU)
* (S10) &lt;math&gt;\Rightarrow&lt;/math&gt; &lt;math&gt;{0}\le{\left|\int_{{{\left\langle{z}_{{1}},{z}_{{2}},{z}_{{3}}\right\rangle}}}{f{{\left({z}\right)}}}{d}{z}\right|}={\left|\int_{\gamma^{(0)}} f(z) {\left.{d}{z}\right.}\right|}\le{4}\cdot{\left|\int_{{\gamma^{{{\left({1}\right)}}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}\le\ldots\le{4}^{{n}}\cdot{\left|\int_{{{\gamma_{{i}}^{(n)}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}={4}^{{n}}\cdot{\left|\int_{{\gamma^{{{\left({n}+{1}\right)}}}}}{f{{\left({z}\right)}}}{\left.{d}{z}\right.}\right|}&lt;/math&gt;

=== Proof part 3: Diameter of the sub-triangles === 
* (S11) The nested definition of the sub-triangles yields for all &lt;math&gt;{n}\in\mathbb{N}&lt;/math&gt;: &lt;math&gt;\Delta{\left({{z}_{1}^{(n)}},{{z}_{2}^{(n)}},{{z}_{3}^{(n)}}\right)}\supset\Delta{\left({{z}_{1}^{{{\left({n}+{1}\right)}}}},{{z}_{2}^{{{\left({n}+{1}\right)}}}},{{z}_{3}^{{{\left({n}+{1}\right)}}}}\right)}&lt;/math&gt; and 
:: &lt;math&gt;\lim_{n\to\infty} \, \text{diam} \left(\Delta{\left({{z}_{1}^{(n)}},{{z}_{2}^{(n)}},{{z}_{3}^{(n)}}\right)}\right) = 0 &lt;/math&gt;
* (S12) &lt;math&gt;\Rightarrow&lt;/math&gt; &lt;math&gt;\exists_{{{z}_{{0}}\in{U}}}\forall_{{{n}\in\mathbb{N}}}:{z}_{{0}}\in\Delta^{(n)} := \Delta{\left({{z}_{1}^{(n)}},{{z}_{2}^{(n)}},{{z}_{3}^{(n)}}\right)}&lt;/math&gt; and &lt;math&gt;{\left\lbrace{z}_{{0}}\right\rbrace}=\bigcap_{{{n}\in\mathbb{N}}}\Delta^{(n)}&lt;/math&gt;

=== Proof part 4: Use of holomorphism (P3) ===
* (S13) We use the holomorphism of &lt;math&gt;f&lt;/math&gt; in &lt;math&gt;z_0 \in U &lt;/math&gt; for further steps with 
: &lt;math&gt;f(z) := f(z_0) +f'(z_0) \cdot ( z - z_0 ) + r(z)&lt;/math&gt; and &lt;math&gt;\lim_{z \to z_0} \frac{r(z)}{z - z_0} = 0 &lt;/math&gt;
:: Justification: (P3)
* (S14) &lt;math&gt;\Rightarrow&lt;/math&gt; The function &lt;math&gt;{h}:{U}\to\mathbb{C}&lt;/math&gt; with &lt;math&gt;h(z) := f(z_0) +f'(z_0) \cdot ( z - z_0 )&lt;/math&gt; has a primitive &lt;math&gt;H(z) := f(z_0) +f'(z_0) \cdot \frac{1}{2}\cdot (z - z_0)^2 &lt;/math&gt;
:: Justification: since &lt;math&gt;h(z)&lt;/math&gt; is a polynomial of degree 1.
* (S15) &lt;math&gt;\Rightarrow&lt;/math&gt; The path integral over the closed paths &lt;math&gt;\gamma^{(n)}&lt;/math&gt; of the function &lt;math&gt;{h}:{U}\to\mathbb{C}&lt;/math&gt; is thus &lt;math&gt;\int_{{{\gamma_{{k}}^{(n)}}}}{h}{\left({z}\right)}={0}&lt;/math&gt;
:: Justification: (SF)
* (S16) &lt;math&gt;\Rightarrow&lt;/math&gt; For the path integral over the closed paths &lt;math&gt;\gamma^{(n)}&lt;/math&gt; of the function &lt;math&gt;{f}:{U}\to\mathbb{C}&lt;/math&gt; we have &lt;math&gt;\int_{{{\gamma_{{k}}^{(n)}}}}{f{{\left({z}\right)}{\left.{d}{z}\right.}}}=\int_{{{\gamma_{{k}}^{(n)}}}}{h}{\left({z}\right)}+{r}{\left({z}\right)}{\left.{d}{z}\right.}=\int_{{{\gamma_{{k}}^{(n)}}}}{r}{\left({z}\right)}{\left.{d}{z}\right.}&lt;/math&gt;

=== Proof part 4: Estimate of the remainder term &lt;math&gt;r(z)&lt;/math&gt; ===
* (S17) &lt;math&gt;\Rightarrow&lt;/math&gt; With &lt;math&gt;\lim_{z \to z_0} \frac{r(z)}{ z - z_0 }= 0 &lt;/math&gt; we have: For all &lt;math&gt;\epsilon&gt;{0}&lt;/math&gt; there exists a &lt;math&gt;\delta&gt;{0}&lt;/math&gt;
:: &lt;math&gt; | z - z_0 | &lt; \delta \Rightarrow \left| \frac{ r(z) }{ z - z_0 } \right| &lt; \epsilon&lt;/math&gt;
: Justification: &lt;math&gt;\epsilon&lt;/math&gt;-&lt;math&gt;\delta&lt;/math&gt;-criterion applied to &lt;math&gt;g(z):=\frac{r(z)}{ z - z_0 }&lt;/math&gt; and continuity of &lt;math&gt;g&lt;/math&gt; in &lt;math&gt;z_0&lt;/math&gt; 
* (S18) &lt;math&gt;\Rightarrow&lt;/math&gt; For all &lt;math&gt;\epsilon&gt;{0}&lt;/math&gt; there exists a &lt;math&gt;\delta &gt; 0&lt;/math&gt;: &lt;math&gt;| z - z_0 | &lt; \delta\Rightarrow | r(z) | &lt; \epsilon \cdot | z - z_0 |&lt;/math&gt; 
:: &lt;math&gt;0 \leq \left|\int_{\left\langle z_1 , z_2 , z_3 \right\rangle} f(z) \, dz \right| \leq 4^n \cdot \left| \int_{ \gamma_{k}^{(n)} } f(z)\, dz \right| = 4^n \cdot \left| \int_{ \gamma_{k}^{(n)} } r(z) \, dz \right| \leq 4^n \cdot \int_{ \gamma_{k}^{(n)} } |r(z)| \, dz \leq 4^n \cdot \int_{ \gamma_{k}^{(n)} } \epsilon \cdot | z - z_0 | \, dz&lt;/math&gt;
:: Justification: (S2)
* (S20) From the condition &lt;math&gt;\lim_{n\to\infty}\text{diam} \left( \Delta^{(n)} \right) = 0&lt;/math&gt; there exists for all &lt;math&gt;\epsilon &gt; 0&lt;/math&gt; an &lt;math&gt;n_{\delta} \in\mathbb{N}&lt;/math&gt; with &lt;math&gt;\Delta^{(n)}\subseteq {D}_{\delta}(z_0)&lt;/math&gt; for all &lt;math&gt;n &gt; n_{\delta}&lt;/math&gt;.
* (S21) &lt;math&gt;\Rightarrow&lt;/math&gt; &lt;math&gt;| z - z_0 | &lt; L \left(\gamma^{(n)}\right) = \frac{1}{2^n} \cdot L \left(\gamma\right) &lt;/math&gt; for all &lt;math&gt; n \in\mathbb{N}&lt;/math&gt; and all &lt;math&gt; z \in \Delta^{(n)}&lt;/math&gt;
:: Justification: The factor &lt;math&gt;\frac{1}{2^n}&lt;/math&gt; arises from the continued halving of the sides of the triangles &lt;math&gt;\Delta^{(n)}&lt;/math&gt;
* (S22) This implies:
:&lt;math&gt; 0 \leq \left|\int_{ \langle z_1 , z_2 , z_3 \rangle } f(z) \, dz \right| \leq 4^n \cdot \int_{ \gamma_{k}^{(n)} } \epsilon \cdot | z - z_0 | \, dz \leq 4^n \cdot \epsilon \cdot \int_{ \gamma_{k}^{(n)} } \frac{1}{2^n} \cdot\mathcal{L}(\gamma) \, dz = 4^n \cdot \epsilon \cdot \frac{1}{2^n} \cdot \mathcal{L}(\gamma) \underbrace{\leq \int_{ \gamma_{k}^{(n)} } 1 \, dz}_{\mathcal{L}(\gamma_{k}^{(n)}) } &lt;/math&gt;
::&lt;math&gt;\leq 
4^n \cdot \epsilon \cdot \frac{1}{2^n} \cdot L(\gamma) \cdot \mathcal{L}(\gamma_{k}^{(n)} ) \leq 
4^n \cdot \epsilon \cdot \frac{\mathcal{L}(\gamma)}{4^n} = \epsilon \cdot \mathcal{L}(\gamma) &lt;/math&gt; for all &lt;math&gt;\epsilon &gt;0&lt;/math&gt;
:: Justification: (S19,LIW,IAL)
* (C1) &lt;math&gt;\Rightarrow&lt;/math&gt; &lt;math&gt;\int_{{{\left\langle{z}_{{1}},{z}_{{2}},{z}_{{3}}\right\rangle}}}{f{{\left({z}\right)}}}{d}{z}={0}&lt;/math&gt;


== Abbreviations for justifications ==
* (DU) &lt;math&gt;\forall_{a,b \in\mathbb{C}} : | a+b | \leq |a| + |b| &lt;/math&gt;
&lt;!--
* (DG) &lt;math&gt;\forall_{a,b,c \in\mathbb{M}} : a \cdot (b+c) = a \cdot b + a \cdot c &lt;/math&gt;
* (AG&lt;math&gt;+&lt;/math&gt;) &lt;math&gt;\forall_{a,b,c \in\mathbb{M}} : a+ (b + c) = (a + b) + c &lt;/math&gt;
* (AG&lt;math&gt;\cdot&lt;/math&gt;) &lt;math&gt;\forall_{a,b,c \in\mathbb{M}} : a\cdot (b \cdot c) = (a \cdot b) \cdot c &lt;/math&gt;
--&gt;
* (DI) Definition: Let &lt;math&gt; M\subset{C}&lt;/math&gt; be a set &lt;math&gt;\text{diam}(M) :=\text{sup} \lbrace |b-a| \, :\, a,b \in M \rbrace &lt;/math&gt;
* (WE) Definition (Path): Let &lt;math&gt;{U}\subseteq\mathbb{C}&lt;/math&gt; be a subset and &lt;math&gt; a,b \in\mathbb{R}&lt;/math&gt; with &lt;math&gt;a &lt; b &lt;/math&gt;. A path &lt;math&gt;\gamma&lt;/math&gt; in &lt;math&gt;U \subseteq \mathbb{C}&lt;/math&gt; is a continuous mapping &lt;math&gt;\gamma: [a,b] \to U &lt;/math&gt;.
* (SPU) Definition (Trace): Let &lt;math&gt;\gamma: [a,b] \to U&lt;/math&gt; be a path in &lt;math&gt;{U}\subseteq\mathbb{C}&lt;/math&gt;. The trace of &lt;math&gt;\gamma&lt;/math&gt; is defined as: &lt;math&gt;\text{Spur}(\gamma) := \lbrace \gamma(t)\in\mathbb{C} \, {\mid} \, t \in [a,b] \rbrace &lt;/math&gt;.
* (WZ) Definition (Path-connected): Let &lt;math&gt;{U}\subseteq\mathbb{C}&lt;/math&gt; be a subset. &lt;math&gt;{U}&lt;/math&gt; is called path-connected if there exists a path &lt;math&gt;\gamma:[a,b] \to U&lt;/math&gt; in &lt;math&gt;U \subseteq \mathbb{C}&lt;/math&gt; with &lt;math&gt; \gamma(a) = z_1 &lt;/math&gt;, &lt;math&gt;\gamma(b) = z_2 &lt;/math&gt; and &lt;math&gt;\text{Spur}(\gamma) \subseteq U&lt;/math&gt;.
* (GE) Definition (Domain): A subset &lt;math&gt;{G}\subseteq\mathbb{C}&lt;/math&gt; is called a domain if (1) &lt;math&gt;{G}&lt;/math&gt; is open, (2) &lt;math&gt;{G}\ne\emptyset&lt;/math&gt; and (3) &lt;math&gt;{G}&lt;/math&gt; is path-connected.
* (WG1) Definition (Smooth path): A path &lt;math&gt;\gamma: [a,b] \to\mathbb{C}&lt;/math&gt; is smooth if it is continuously differentiable.
* (UT) Definition (Subdivision): Let &lt;math&gt;[a,b]&lt;/math&gt; be an interval, &lt;math&gt;n \in\mathbb{N}&lt;/math&gt; and &lt;math&gt;{a}={u}_{{0}} &lt; {\ldots} &lt; {{u}}_{n}={b}&lt;/math&gt;. &lt;math&gt;{\left({u}_{{0}},\ldots,{u}_{{{n}}}\right)}\in\mathbb{R}^{n+1}&lt;/math&gt; is called a subdivision of &lt;math&gt;{\left[{a},{b}\right]}&lt;/math&gt;.
* (WG2) Definition (Path subdivision): Let &lt;math&gt;\gamma: [a,b] \to\mathbb{C}&lt;/math&gt; be a path in &lt;math&gt;{U}\subseteq\mathbb{C}&lt;/math&gt;, &lt;math&gt;{n}\in\mathbb{N}&lt;/math&gt;, &lt;math&gt;{\left({u}_{{0}},\ldots,{u}_{{{n}}}\right)}&lt;/math&gt; a subdivision of &lt;math&gt;[a,b]&lt;/math&gt;, &lt;math&gt;\gamma_{{k}}:{\left[{u}_{{{k}-{1}}},{u}_{{k}}\right]}\to\mathbb{C}&lt;/math&gt; for all &lt;math&gt;{k}\in{\left\lbrace{1},\ldots,{n}\right\rbrace}&lt;/math&gt; a path in &lt;math&gt;{U}&lt;/math&gt;. &lt;math&gt;{\left(\gamma_{{{1}}},\ldots,\gamma_{{{n}}}\right)}&lt;/math&gt; is called a path subdivision of &lt;math&gt;\gamma&lt;/math&gt; if &lt;math&gt;\gamma_{{n}}{\left({b}\right)}=\gamma{\left({b}\right)}&lt;/math&gt; and &lt;math&gt;\forall_{{{k}\in{\left\lbrace{1},\ldots,{n}\right\rbrace}}}\forall_{{{t}\in{\left[{u}_{{{k}-{1}}},{u}_{{k}}\right)}}}:\gamma_{{k}}{\left({t}\right)}=\gamma{\left({t}\right)}\wedge\gamma_{{k}}{\left({u}_{{{k}-{1}}}\right)}=\gamma_{{{k}-{1}}}{\left({u}_{{k}}\right)}&lt;/math&gt;.
* (WG3) Definition (Piecewise smooth path): A path &lt;math&gt;\gamma:{\left[{a},{b}\right]}\to\mathbb{C}&lt;/math&gt; is piecewise smooth if there exists a path subdivision &lt;math&gt;{\left(\gamma_{{1}},\ldots\gamma_{{n}}\right)}&lt;/math&gt; of &lt;math&gt;\gamma&lt;/math&gt; consisting of smooth paths &lt;math&gt;\gamma_{{k}}&lt;/math&gt; for all &lt;math&gt;{k}\in{\left\lbrace{1},\ldots,{n}\right\rbrace}&lt;/math&gt;.
* (WG4) Definition (Path integral): Let &lt;math&gt;f : U \to\mathbb{C}&lt;/math&gt; be a continuous function and &lt;math&gt;\gamma: [a,b] \to U &lt;/math&gt; a smooth path, then the path integral is defined as: &lt;math&gt;\int_{\gamma} f := \int_{\gamma} f(z) \, dz := \int_a^b f(\gamma(t)) \cdot\gamma'(t)\, dt &lt;/math&gt;. If &lt;math&gt;\gamma&lt;/math&gt; is only piecewise smooth with respect to a path subdivision &lt;math&gt;( \gamma_1 ,\ldots,\gamma_n ) &lt;/math&gt;, then we define &lt;math&gt;\int_{\gamma} f(z) \, dz :=\sum_{k=1}^{n} \int_{\gamma_k} f(z) \, dz&lt;/math&gt;.
* (SF) Theorem (Primitive with closed paths): If a continuous function &lt;math&gt;f : U \to\mathbb{C}&lt;/math&gt; has a primitive &lt;math&gt;F : U \to\mathbb{C}&lt;/math&gt;, then for a piecewise smooth path &lt;math&gt;\gamma: [a,b] \to U&lt;/math&gt; we have &lt;math&gt;\int_{\gamma} f(z) \, dz =F(b) - F(a)&lt;/math&gt;.
* (LIW) Length of the integration path: Let &lt;math&gt;\gamma:{\left[{a},{b}\right]}\to\mathbb{C}&lt;/math&gt; be a smooth path, then the &lt;math&gt;\mathcal{L}(\gamma)&lt;/math&gt; is defined as:
:: &lt;math&gt;\mathcal{L}(\gamma) := \int_{a}^{b} |\gamma'(t)| \, dt&lt;/math&gt;. 
: If &lt;math&gt;\gamma:{\left[{a},{b}\right]}\to\mathbb{C}&lt;/math&gt; is a general integration path with the path subdivision &lt;math&gt;{\left(\gamma_{{1}},\ldots\gamma_{{n}}\right)}&lt;/math&gt; of smooth paths &lt;math&gt;\gamma_{{k}}&lt;/math&gt;, then &lt;math&gt;\mathcal{L}(\gamma)&lt;/math&gt; is defined as the sum of the lengths of the smooth paths &lt;math&gt;\gamma_{{k}}&lt;/math&gt;, i.e.:
:: &lt;math&gt;\mathcal{L}(\gamma) := \sum_{k=1}^{n} \mathcal{L}(\gamma_{k})&lt;/math&gt;
* (IAL) Integral estimate over the length of the integration path: Let &lt;math&gt;\gamma:{\left[{a},{b}\right]}\to\mathbb{G}&lt;/math&gt; be an integration path on the domain &lt;math&gt;G \subseteq \mathbb{C}&lt;/math&gt;, then for a continuous function &lt;math&gt;f&lt;/math&gt; on &lt;math&gt;\text{Spur}(\gamma)&lt;/math&gt; we have the estimate:
::&lt;math&gt;\left| \int_{\gamma} f(z) \, dz \right| \leq \max_{z \in \text{Spur}(\gamma)} |f(z)| \cdot \mathcal{L}(\gamma)&lt;/math&gt;

== Literature ==
* Eberhard Freitag &amp; Rolf Busam: ''Funktionentheorie 1'', Springer-Verlag, Berlin

== See also ==
* [[Complex Analysis/Goursat's Lemma]]-Short form
* [[Complex Analysis/Path of Integration|Path of Integration]]

[[Category:Complex Analysis]]
[[Category:Theorem (Mathematics)|Goursat, Lemma of]]


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&lt;noinclude&gt;[[de:Kurs:Funktionentheorie/Lemma von Goursat (Details)]]&lt;/noinclude&gt;</text>
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    <title>Complex Analysis/Goursat's Lemma</title>
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      <text bytes="6233" sha1="51iabq68czw4wac5ee0t81ydk8x0eel" xml:space="preserve">Goursat's Lemma is a crucial result in the proof of the  [[w:en:Cauchy's integral theorem|Cauchy's integral theorem]].It restricts the integration paths to triangles, making it provable via a [[w:en:Complex Analysis/Lemma_von_Goursat_(Details)|geometric subdivision argument]].

== Statement ==
Let &lt;math&gt;D \subseteq \mathbb{C}&lt;/math&gt; be a closed triangle, &lt;math&gt;G \supseteq D&lt;/math&gt; an open set, and &lt;math&gt;f \colon U \to \mathbb{C}&lt;/math&gt; a [[w:en:Holomorphic_function|holomorphic]] function. Then:
&lt;math&gt;\int_{\partial D} f(z), dz = 0.&lt;/math&gt;

== Proof ==
Set &lt;math&gt;\Delta_0 := D&lt;/math&gt;. We inductively construct a sequence &lt;math&gt;(\Delta_n)_{n \geq 0}&lt;/math&gt; with the properties:

1. &lt;math&gt;\Delta_n \subseteq \Delta_{n-1}&lt;/math&gt;

2. &lt;math&gt;\mathcal{L}(\partial \Delta_n) = 2^{-n}\mathcal{L}(\partial D)&lt;/math&gt;, where &lt;math&gt;\mathcal{L}&lt;/math&gt; represents the [[Course:Complex Analysis/Curve|length of a curve]]

3. &lt;math&gt;\left|\int_{\partial D} f(z), dz\right| \leq 4^n \left|\int_{\partial \Delta_n} f(z), dz\right|&lt;/math&gt;

For &lt;math&gt;n \geq 0&lt;/math&gt; and &lt;math&gt;\Delta_n&lt;/math&gt; already constructed, we subdivide &lt;math&gt;\Delta_n&lt;/math&gt; by connecting the midpoints of its sides, forming four subtriangles &lt;math&gt;\Delta_{n+1}^i&lt;/math&gt;, &lt;math&gt;1 \leq i \leq 4&lt;/math&gt;. Since the contributions of the midpoints cancel out in the integration, we have:
&lt;center&gt;&lt;math&gt;\begin{array}{rl}
  \displaystyle\left|\int_{\partial \Delta_n} f(z)\, dz\right| &amp;= \displaystyle\left|\sum_{i=1}^4 \int_{\partial \Delta_{n+1}^i} f(z)\, dz\right|\\
                &amp;\le \displaystyle\sum_{i=1}^4 \left|\int_{\partial \Delta_{n+1}^i} f(z)\, dz\right|\\
                &amp;\le \displaystyle\max_i \left|\int_{\partial \Delta_{n+1}^i} f(z)\, dz\right|
\end{array}&lt;/math&gt;&lt;/center&gt;
Choose &lt;math&gt;1 \leq i \leq 4&lt;/math&gt; such that &lt;math&gt;\left|\int_{\partial \Delta_{n+1}^i} f(z), dz\right| = \max_i\left|\int_{\partial \Delta_{n+1}^i} f(z), dz\right|&lt;/math&gt; and set &lt;math&gt;\Delta_{n+1} := \Delta_{n+1}^i&lt;/math&gt;. Then, by construction:
&lt;math&gt;\Delta_{n+1} \subseteq \Delta_n&lt;/math&gt;,
&lt;math&gt;\mathcal{L}(\partial \Delta_{n+1}) = \frac{1}{2}\mathcal{L}(\partial \Delta_n) = 2^{-(n+1)}\mathcal{L}(\partial D)&lt;/math&gt;, and
&lt;math&gt;\left|\int_{\partial D} f(z), dz\right| \leq 4^n \left|\int_{\partial \Delta_n} f(z), dz\right| \leq 4^{n+1} \left|\int_{\partial \Delta_{n+1}} f(z), dz\right|.&lt;/math&gt;

This ensures &lt;math&gt;\Delta_{n+1}&lt;/math&gt; has the required properties.

Since all &lt;math&gt;\Delta_n&lt;/math&gt; are compact, &lt;math&gt;\bigcap_{n\geq 0} \Delta_n \neq \emptyset&lt;/math&gt;. Let &lt;math&gt;z_0 \in \bigcap_{n\geq 0} \Delta_n&lt;/math&gt;. As &lt;math&gt;f&lt;/math&gt; is holomorphic at &lt;math&gt;z_0&lt;/math&gt;, there exists a neighborhood &lt;math&gt;V&lt;/math&gt; of &lt;math&gt;z_0&lt;/math&gt; and a continuous function &lt;math&gt;A \colon V \to \mathbb{C}&lt;/math&gt; with &lt;math&gt;A(z_0) = 0&lt;/math&gt; such that:
&lt;math&gt;f(z) = f(z_0) + (z-z_0)f'(z_0) + A(z)(z-z_0), \qquad z \in V.&lt;/math&gt;

Since the function &lt;math&gt;z \mapsto f(z_0) + (z-z_0)f'(z_0)&lt;/math&gt; has a primitive, it follows for &lt;math&gt;n \geq 0&lt;/math&gt; with &lt;math&gt;\Delta_n \subseteq V&lt;/math&gt; that:
&lt;math&gt;\int_{\partial \Delta_n} f(z), dz = \int_{\partial \Delta_n} f(z_0) + (z-z_0)f'(z_0) + A(z)(z-z_0) , dz = \int_{\partial \Delta_n} A(z)(z-z_0) , dz.&lt;/math&gt;

Thus, due to the continuity of &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;A(z_0) = 0&lt;/math&gt;, we obtain:
&lt;center&gt;&lt;math&gt; \begin{array}{rl}
  \displaystyle\left|\int_{\partial D} f(z)\, dz\right| &amp;\le 
 \displaystyle 4^n\left|\int_{\partial \Delta_{n}} f(z)\, dz\right|\\ &amp;= \displaystyle 4^n\left|\int_{\partial \Delta_{n}} A(z)(z-z_0)\, dz\right|\\
            &amp;\le\displaystyle 4^n \cdot \mathcal{L}(\partial \Delta_n) \max_{z\in \partial \Delta_n} |z-z_0||A(z)|\\
            &amp;\le\displaystyle 4^n \cdot \mathcal{L}(\partial \Delta_n)^2 \max_{z\in \partial \Delta_n} |A(z)|\\
            &amp;=\displaystyle \mathcal{L}(\partial D) \max_{z\in \partial \Delta_n} |A(z)|\\
            &amp;\to\displaystyle \mathcal{L}(\partial D) |A(z_0)| = 0, \qquad n \to \infty.
\end{array}&lt;/math&gt;&lt;/center&gt;
== Notation in the Proof ==

&lt;math&gt;\Delta_n&lt;/math&gt; is the &lt;math&gt;n&lt;/math&gt;-th subtriangle of the original triangle, with side lengths scaled by a factor of &lt;math&gt;\frac{1}{2^n}&lt;/math&gt;.

&lt;math&gt;\partial \Delta_n&lt;/math&gt; is the integration path along the boundary of the &lt;math&gt;n&lt;/math&gt;-th subtriangle, with perimeter &lt;math&gt;\mathcal{L}(\partial \Delta_n) = \frac{1}{2^n} \cdot \mathcal{L}(\partial \Delta_0)&lt;/math&gt;.


== See Also ==
* [[Complex Analysis/Goursat's Lemma (Details)|Goursat's Lemma (Details)]]
* [[Complex Analysis/rectifiable curve|rectifiable curve]]


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      <text bytes="6252" sha1="sq8m72g2gm320rkhf3i0dv05p20diez" xml:space="preserve">== Introduction ==
Goursat's Lemma is a crucial result in the proof of the  [[w:en:Cauchy's integral theorem|Cauchy's integral theorem]].It restricts the integration paths to triangles, making it provable via a [[w:en:Complex Analysis/Lemma_von_Goursat_(Details)|geometric subdivision argument]].

== Statement ==
Let &lt;math&gt;D \subseteq \mathbb{C}&lt;/math&gt; be a closed triangle, &lt;math&gt;G \supseteq D&lt;/math&gt; an open set, and &lt;math&gt;f \colon U \to \mathbb{C}&lt;/math&gt; a [[w:en:Holomorphic_function|holomorphic]] function. Then:
&lt;math&gt;\int_{\partial D} f(z), dz = 0.&lt;/math&gt;

== Proof ==
Set &lt;math&gt;\Delta_0 := D&lt;/math&gt;. We inductively construct a sequence &lt;math&gt;(\Delta_n)_{n \geq 0}&lt;/math&gt; with the properties:

1. &lt;math&gt;\Delta_n \subseteq \Delta_{n-1}&lt;/math&gt;

2. &lt;math&gt;\mathcal{L}(\partial \Delta_n) = 2^{-n}\mathcal{L}(\partial D)&lt;/math&gt;, where &lt;math&gt;\mathcal{L}&lt;/math&gt; represents the [[Course:Complex Analysis/Curve|length of a curve]]

3. &lt;math&gt;\left|\int_{\partial D} f(z), dz\right| \leq 4^n \left|\int_{\partial \Delta_n} f(z), dz\right|&lt;/math&gt;

For &lt;math&gt;n \geq 0&lt;/math&gt; and &lt;math&gt;\Delta_n&lt;/math&gt; already constructed, we subdivide &lt;math&gt;\Delta_n&lt;/math&gt; by connecting the midpoints of its sides, forming four subtriangles &lt;math&gt;\Delta_{n+1}^i&lt;/math&gt;, &lt;math&gt;1 \leq i \leq 4&lt;/math&gt;. Since the contributions of the midpoints cancel out in the integration, we have:
&lt;center&gt;&lt;math&gt;\begin{array}{rl}
  \displaystyle\left|\int_{\partial \Delta_n} f(z)\, dz\right| &amp;= \displaystyle\left|\sum_{i=1}^4 \int_{\partial \Delta_{n+1}^i} f(z)\, dz\right|\\
                &amp;\le \displaystyle\sum_{i=1}^4 \left|\int_{\partial \Delta_{n+1}^i} f(z)\, dz\right|\\
                &amp;\le \displaystyle\max_i \left|\int_{\partial \Delta_{n+1}^i} f(z)\, dz\right|
\end{array}&lt;/math&gt;&lt;/center&gt;
Choose &lt;math&gt;1 \leq i \leq 4&lt;/math&gt; such that &lt;math&gt;\left|\int_{\partial \Delta_{n+1}^i} f(z), dz\right| = \max_i\left|\int_{\partial \Delta_{n+1}^i} f(z), dz\right|&lt;/math&gt; and set &lt;math&gt;\Delta_{n+1} := \Delta_{n+1}^i&lt;/math&gt;. Then, by construction:
&lt;math&gt;\Delta_{n+1} \subseteq \Delta_n&lt;/math&gt;,
&lt;math&gt;\mathcal{L}(\partial \Delta_{n+1}) = \frac{1}{2}\mathcal{L}(\partial \Delta_n) = 2^{-(n+1)}\mathcal{L}(\partial D)&lt;/math&gt;, and
&lt;math&gt;\left|\int_{\partial D} f(z), dz\right| \leq 4^n \left|\int_{\partial \Delta_n} f(z), dz\right| \leq 4^{n+1} \left|\int_{\partial \Delta_{n+1}} f(z), dz\right|.&lt;/math&gt;

This ensures &lt;math&gt;\Delta_{n+1}&lt;/math&gt; has the required properties.

Since all &lt;math&gt;\Delta_n&lt;/math&gt; are compact, &lt;math&gt;\bigcap_{n\geq 0} \Delta_n \neq \emptyset&lt;/math&gt;. Let &lt;math&gt;z_0 \in \bigcap_{n\geq 0} \Delta_n&lt;/math&gt;. As &lt;math&gt;f&lt;/math&gt; is holomorphic at &lt;math&gt;z_0&lt;/math&gt;, there exists a neighborhood &lt;math&gt;V&lt;/math&gt; of &lt;math&gt;z_0&lt;/math&gt; and a continuous function &lt;math&gt;A \colon V \to \mathbb{C}&lt;/math&gt; with &lt;math&gt;A(z_0) = 0&lt;/math&gt; such that:
&lt;math&gt;f(z) = f(z_0) + (z-z_0)f'(z_0) + A(z)(z-z_0), \qquad z \in V.&lt;/math&gt;

Since the function &lt;math&gt;z \mapsto f(z_0) + (z-z_0)f'(z_0)&lt;/math&gt; has a primitive, it follows for &lt;math&gt;n \geq 0&lt;/math&gt; with &lt;math&gt;\Delta_n \subseteq V&lt;/math&gt; that:
&lt;math&gt;\int_{\partial \Delta_n} f(z), dz = \int_{\partial \Delta_n} f(z_0) + (z-z_0)f'(z_0) + A(z)(z-z_0) , dz = \int_{\partial \Delta_n} A(z)(z-z_0) , dz.&lt;/math&gt;

Thus, due to the continuity of &lt;math&gt;A&lt;/math&gt; and &lt;math&gt;A(z_0) = 0&lt;/math&gt;, we obtain:
&lt;center&gt;&lt;math&gt; \begin{array}{rl}
  \displaystyle\left|\int_{\partial D} f(z)\, dz\right| &amp;\le 
 \displaystyle 4^n\left|\int_{\partial \Delta_{n}} f(z)\, dz\right|\\ &amp;= \displaystyle 4^n\left|\int_{\partial \Delta_{n}} A(z)(z-z_0)\, dz\right|\\
            &amp;\le\displaystyle 4^n \cdot \mathcal{L}(\partial \Delta_n) \max_{z\in \partial \Delta_n} |z-z_0||A(z)|\\
            &amp;\le\displaystyle 4^n \cdot \mathcal{L}(\partial \Delta_n)^2 \max_{z\in \partial \Delta_n} |A(z)|\\
            &amp;=\displaystyle \mathcal{L}(\partial D) \max_{z\in \partial \Delta_n} |A(z)|\\
            &amp;\to\displaystyle \mathcal{L}(\partial D) |A(z_0)| = 0, \qquad n \to \infty.
\end{array}&lt;/math&gt;&lt;/center&gt;
== Notation in the Proof ==

&lt;math&gt;\Delta_n&lt;/math&gt; is the &lt;math&gt;n&lt;/math&gt;-th subtriangle of the original triangle, with side lengths scaled by a factor of &lt;math&gt;\frac{1}{2^n}&lt;/math&gt;.

&lt;math&gt;\partial \Delta_n&lt;/math&gt; is the integration path along the boundary of the &lt;math&gt;n&lt;/math&gt;-th subtriangle, with perimeter &lt;math&gt;\mathcal{L}(\partial \Delta_n) = \frac{1}{2^n} \cdot \mathcal{L}(\partial \Delta_0)&lt;/math&gt;.


== See Also ==
* [[Complex Analysis/Goursat's Lemma (Details)|Goursat's Lemma (Details)]]
* [[Complex Analysis/rectifiable curve|rectifiable curve]]


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      <text bytes="13630" sha1="akdhuvo8hj8ibpw5nv7yu0mkewx5p0b" xml:space="preserve">The '''Cauchy Integral Formula''' (named after [[w:en:Augustin-Louis Cauchy|Augustin-Louis Cauchy]]) is one of the fundamental results of [[w:en:Complex analysis|complex analysis]], a branch of [[w:en:Mathematics|mathematics]]. In its weakest form, it states that the values of a [[w:en:Holomorphic function|holomorphic function]] &lt;math&gt;f&lt;/math&gt; inside a disk are completely determined by its values on the boundary of that disk. A powerful generalization of this is the [[w:en:Residue theorem|Residue theorem]].

== Cauchy Integral Formula for Disks ==
=== Statement ===
Let &lt;math&gt;G \subseteq \mathbb{C}&lt;/math&gt; be open, &lt;math&gt;f\colon G \to \mathbb{C}&lt;/math&gt; holomorphic, &lt;math&gt;z_0 \in G&lt;/math&gt; a point in &lt;math&gt;G&lt;/math&gt;, and &lt;math&gt;U := D_r(z_0) \subset G&lt;/math&gt; a bounded disk in &lt;math&gt;G&lt;/math&gt;. Then for all &lt;math&gt;z \in D_r(z_0)&lt;/math&gt; (i.e., for all &lt;math&gt;z&lt;/math&gt; with &lt;math&gt;|z - z_0| &lt; r&lt;/math&gt;), the following holds: :&lt;math&gt;f(z) = \frac{1}{2\pi\mathrm{i}} \oint_{\partial U} \frac{f(\zeta)}{\zeta - z} \mathrm{d}\zeta&lt;/math&gt; Here, &lt;math&gt;\partial U&lt;/math&gt; denotes the positively oriented curve &lt;math&gt;t \mapsto z_0 + r e^{\mathrm{i}t}&lt;/math&gt; for &lt;math&gt;t \in [0, 2\pi]&lt;/math&gt; along the boundary of the disk &lt;math&gt;U&lt;/math&gt;.
=== Proof 1 ===
For a fixed &lt;math&gt;z \in U&lt;/math&gt;, the function &lt;math&gt;g\colon U\to\mathbb{C}&lt;/math&gt; defined by &lt;math&gt;w\mapsto\tfrac{f(w)-f(z)}{w-z}&lt;/math&gt; for &lt;math&gt;w\neq z&lt;/math&gt; und &lt;math&gt;w\mapsto f'(z)&lt;/math&gt; for &lt;math&gt;w=z&lt;/math&gt;. &lt;math&gt;g&lt;/math&gt; is steadily on  &lt;math&gt;U&lt;/math&gt; and holomorphic on &lt;math&gt;U\setminus\{z\}&lt;/math&gt;.  By the [[w:en:Cauchy Integral Theorem|Cauchy Integral Theorem]], we now have: :&lt;math&gt;0 = \oint_{\partial U} g = \oint_{\partial U}\frac{f(\zeta)}{\zeta-z} \mathrm{d}\zeta - f(z)\oint_{\partial U}\frac{\mathrm{d}\zeta}{\zeta-z}&lt;/math&gt;.

=== Proof 2 ===
The function &lt;math&gt;h\colon U \to \mathbb{C}&lt;/math&gt;, &lt;math&gt;\textstyle w \mapsto \oint_{\partial U} \frac{\mathrm{d}\zeta}{\zeta-w}&lt;/math&gt; is holomorphic with the derivative &lt;math&gt;\textstyle h'(w) = \oint_{\partial U} \frac{\mathrm{d}\zeta}{\left(\zeta-w\right)^2}&lt;/math&gt;, which vanishes since the integrand has an antiderivative (namely &lt;math&gt;\zeta \mapsto -\frac{1}{\zeta-w}&lt;/math&gt;). Therefore, &lt;math&gt;h&lt;/math&gt; is constant, and since &lt;math&gt;h(a) = 2\pi i&lt;/math&gt;, we have &lt;math&gt;h(z) = 2\pi i&lt;/math&gt;.

== Consequences of the Cauchy Integral Theorem ==
The Cauchy Integral Theorem (CIS) leads to the following corollaries:
=== Representation of the Function at the Center of the Disk ===
For every holomorphic function, the function value at the center of a circle is the average of the function values on the circle's boundary. Use &lt;math&gt;\zeta(t) = z_o + r e^{\mathrm{i}t},\ \mathrm{d}\zeta = \mathrm{i} r e^{\mathrm{i}t} \mathrm{d}t&lt;/math&gt;.
Test: :&lt;math&gt; \begin{align} f|{U}(z_o) &amp;= \frac{1}{2\pi \mathrm{i}} \oint{\partial U} \frac{f(\zeta)}{\zeta - z_o} \mathrm{d}\zeta = \frac{1}{2\pi \mathrm{i}} \int_{0}^{2\pi} \frac{f(a + r e^{\mathrm{i}t})}{r e^{\mathrm{i}t}} \mathrm{i} r e^{\mathrm{i}t} , \mathrm{d}t \ &amp;= \frac{1}{2\pi} \int_{0}^{2\pi} f(z_o + r e^{\mathrm{i}t}) , \mathrm{d}t \end{align}&lt;/math&gt;

=== Derivatives - Cauchy Integral Formula - CIF ===
Every holomorphic function is infinitely complex differentiable, and each of these derivatives is also holomorphic. Expressed using the integral formula, this means for &lt;math&gt;|z - z_o| &lt; r&lt;/math&gt; and &lt;math&gt;n \in \mathbb{N}{0}&lt;/math&gt;: :&lt;math&gt;f^{(n)}(z) = \frac{n!}{2\pi \mathrm{i}} \oint{\partial U} \frac{f(\zeta)}{(\zeta - z)^{n+1}} \mathrm{d}\zeta.&lt;/math&gt;
=== Local Developability in Power Series ===
Every holomorphic function can be locally expanded into a [[w:en:Power Series|power series]] for &lt;math&gt;|z - a| &lt; r&lt;/math&gt;.
:&lt;math&gt;f(z) = \sum\limits_{n=0}^\infty \left( \frac{1}{2\pi \mathrm{i}} \oint_{\partial U} \frac{f(\zeta)}{(\zeta - a)^{n+1}} \mathrm{d}\zeta \right) (z - a)^n = \sum\limits_{n=0}^\infty a_n (z - a)^n.&lt;/math&gt;
Using the integral formula for &lt;math&gt;f^{(n)}&lt;/math&gt;, it immediately follows that the coefficients &lt;math&gt;a_n&lt;/math&gt; are exactly the [[w:en:Taylor series|Taylor coefficients]].

=== Estimation of the Taylor Series Coefficients ===
For the coefficients, the following estimate holds when &lt;math&gt;|f(z)| \leq M&lt;/math&gt; for &lt;math&gt;|z - a| &lt; r \ \Leftrightarrow z \in U_r(a)&lt;/math&gt;: :&lt;math&gt;|a_n| \leq \frac{M}{r^n}&lt;/math&gt;
The [[w:en:Liouville's Theorem|Liouville Theorem]] (every [[w:en:Entire Function|holomorphic function bounded on the entire complex plane]] is constant) can be easily proven using the integral formula. This can then be used to easily prove the [[w:en:Fundamental Theorem of Algebra|Fundamental Theorem of Algebra]] (every polynomial in &lt;math&gt;\mathbb{C}&lt;/math&gt; factors into linear factors).
Here's the translation with the specified conditions:

=== Proof 1 ===
The Cauchy integral formula is differentiated partially, allowing differentiation and integration to be swapped:
:&lt;math&gt;\begin{align} f^{(n)}|_{U}(z) &amp; =\frac{\partial^{n}f}{\partial z^{n}}|_{U}(z)=\frac{1}{2\pi\mathrm{i}}\frac{\partial^{n}}{\partial z^{n}}\oint_{\partial U}\frac{f(\zeta)}{\zeta-z}\mathrm{d}\zeta\\  &amp; =\frac{1}{2\pi\mathrm{i}}\oint_{\partial U}f(\zeta)\underbrace{\frac{\partial^{n}}{\partial z^{n}}\frac{1}{\zeta-z}}_{n!/(\zeta-z)^{1+n}}\mathrm{d}\zeta=\frac{n!}{2\pi\mathrm{i}}\oint_{\partial U}\frac{f(\zeta)}{(\zeta-z)^{1+n}}\mathrm{d}\zeta\end{align} &lt;/math&gt;
=== Proof 2a: Cauchy Kernel ===
Developing &lt;math&gt;\frac{1}{\zeta - z}&lt;/math&gt; in the Cauchy integral formula using the geometric series gives (Cauchy kernel):
:&lt;math&gt; \frac{1}{1 - \frac{z - z_o}{\zeta - z_o}} = \sum_{n=0}^{\infty} \left( \frac{z - z_o}{\zeta - z_o} \right)^{n} &lt;/math&gt;

=== Proof 2: Cauchy Kernel - Taylor Series ===
:&lt;math&gt;\begin{align} f|_{U}(z) &amp; =\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{r}(z_o)}\frac{f(\zeta)}{\zeta-z}\mathrm{d}\zeta=\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{r}(z_o)}\frac{f(\zeta)}{\zeta-z_o-(z-z_o)}\mathrm{d}\zeta \\ &amp; {=} \frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{r}(z_o)}\frac{f(\zeta)}{\zeta-z_o}\cdot \frac{1}{1-\frac{z-z_o}{\zeta-z_o}}\mathrm{d}\zeta\, \\ &amp;\overset{|\frac{z-z_o}{\zeta-z_o}|&lt;1}{=} \frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{r}(z_o)}\frac{f(\zeta)}{\zeta-z_o}\sum_{n=0}^{\infty}\left(\frac{z-z_o}{\zeta-z_o}\right)^{n}\mathrm{d}\zeta\\ &amp; =\sum_{n=0}^{\infty}\underbrace{\left(\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{r}(z_o)}\frac{f(\zeta)}{(\zeta-z_o)^{n+1}}\mathrm{d}\zeta\right)}_{a_{n}}(z-z_o)^{n}\end{align}&lt;/math&gt;

=== Proof 2b: Cauchy Kernel ===
Since the geometric series converges uniformly for &lt;math&gt;|z - z_o| &lt; |\zeta - z_o| = r&lt;/math&gt;, one can integrate term by term, i.e., swap the sum and the integral. The development coefficients are:
:&lt;math&gt;\begin{align} a_{n} &amp; =\frac{1}{n!}f^{(n)}|_{U}(z_o)=\frac{1}{2\pi\mathrm{i}}\oint_{\partial U_{r}(a)}\frac{f(\zeta)}{(\zeta-z_o)^{n+1}}\mathrm{d}\zeta\\ &amp; =\frac{1}{2\pi\mathrm{i}}\int_{0}^{2\pi}\frac{f(z_o+re^{\mathrm{i}t})}{(re^{\mathrm{i}t})^{n+1}}\mathrm{i}re^{\mathrm{i}t}\,\mathrm{d}t=\frac{1}{2\pi r^{n}}\int_{0}^{2\pi}f(z_o+re^{\mathrm{i}t})e^{-\mathrm{i}nt}\,\mathrm{d}t\end{align}&lt;/math&gt;

=== Proof 3: Estimation of the Coefficients ===
For the coefficients &lt;math&gt;a_n \in \mathbb{C}&lt;/math&gt;, the following estimate holds. There exists a &lt;math&gt;M &gt; 0&lt;/math&gt; such that &lt;math&gt;|f(z)| \leq M&lt;/math&gt; for &lt;math&gt;|z - z_o| = r&lt;/math&gt;. Then, for &lt;math&gt;n \in \mathbb{N}0&lt;/math&gt;, we have:
:&lt;math&gt;\begin{align} |a_{n}|&amp;=\left|\frac{1}{2\pi r^{n}}\int_{0}^{2\pi}f(z_o+re^{\mathrm{i}t})e^{-\mathrm{i}nt}\,\mathrm{d}t\right|\\ &amp;\leq\frac{1}{2\pi r^n}\int_0^{2\pi}\underbrace{|f(z_o+re^{\mathrm{i} t})|}_{\leq M}\,\mathrm{d}t\leq \frac{M}{r^{n}}\end{align}&lt;/math&gt;

=== Proof 4: Liouville's Theorem ===
If &lt;math&gt;f&lt;/math&gt; is holomorphic on all of &lt;math&gt;\mathbb{C}&lt;/math&gt; and bounded, i.e., &lt;math&gt;|f(z)| = |\sum_{n=0}^{\infty} a_n z^n| \leq M&lt;/math&gt; for all &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;, then, as before, for all &lt;math&gt;r &gt; 0&lt;/math&gt;, we have:
:&lt;math&gt;|a_n| \leq \frac{M}{r^n}&lt;/math&gt;
Since &lt;math&gt;r&lt;/math&gt; was arbitrary, it follows that &lt;math&gt;a_n = 0&lt;/math&gt; for all &lt;math&gt;n \in \mathbb{N}&lt;/math&gt;. Therefore, from the boundedness of &lt;math&gt;f&lt;/math&gt;, we conclude:
: &lt;math&gt;f(z) = a_0&lt;/math&gt;
Thus, every bounded holomorphic function on all of &lt;math&gt;\mathbb{C}&lt;/math&gt; is constant (Liouville's theorem).

=== Example ===
Using the integral formula, integrals can also be computed:
:&lt;math&gt; \oint_{\partial U_2(0)} \frac{e^{2\zeta}}{(\zeta + 1)^4} \mathrm{d}\zeta = \frac{2\pi \mathrm{i}}{3!} \frac{\mathrm{d}^3}{\mathrm{d}z^3} e^{2z} |_{z = -1} = \frac{8 \pi \mathrm{i}}{3 e^2} &lt;/math&gt;

== Cauchy Integral Formula for Cycles ==
A generalization of the integral formula for circular contours is the version for cycles:

Let &lt;math&gt;G \subseteq \mathbb{C}&lt;/math&gt; be a domain, &lt;math&gt;f \colon G \to \mathbb{C}&lt;/math&gt; holomorphic, and &lt;math&gt;\Gamma&lt;/math&gt; a [[w:en:zero homologous|zero homologous]] [[w:en:cycle|cycle]] in &lt;math&gt;D&lt;/math&gt;. Then, for all &lt;math&gt;z \in D&lt;/math&gt; not on &lt;math&gt;\Gamma&lt;/math&gt;, the following integral formula holds:
:&lt;math&gt; n(\Gamma, z) \cdot f(z) = \frac{1}{2\pi \mathrm{i}} \int_\Gamma \frac{f(\zeta)}{\zeta - z} \mathrm{d}\zeta &lt;/math&gt;

Here, &lt;math&gt;n(\Gamma, z)&lt;/math&gt; denotes the [[w:en:winding number|winding number]] or [[w:en:revolution|revolution]] of &lt;math&gt;\Gamma&lt;/math&gt; around &lt;math&gt;z&lt;/math&gt;.

== Cauchy Integral Formula for Polycycles ==
The Cauchy integral formula has been generalized to the multidimensional complex space &lt;math&gt;\mathbb{C}^n&lt;/math&gt;. Let &lt;math&gt;U_1, \ldots, U_n&lt;/math&gt; be disk domains in &lt;math&gt;\mathbb{C}&lt;/math&gt;, then &lt;math&gt; U := \prod_{i=1}^n U_i &lt;/math&gt; is a [[w:en:Polycylinder|Polycylinder]] in &lt;math&gt;\mathbb{C}^n&lt;/math&gt;. Let &lt;math&gt;f \colon U \to \mathbb{C}&lt;/math&gt; be a holomorphic function and &lt;math&gt;\xi \in U&lt;/math&gt;. The Cauchy integral formula is given by
:&lt;math&gt; f(z_1, \ldots, z_n) = \frac{1}{(2\pi \mathrm{i})^n} \oint_{\partial U_n} \cdots \oint_{\partial U_1} \frac{f(\xi_1, \ldots, \xi_n)}{(\xi_1 - z_1) \cdots (\xi_n - z_n)} \mathrm{d} \xi_1 \cdots \mathrm{d} \xi_n &lt;/math&gt;

=== Restrictions in Multidimensional Space ===
Since the Cauchy integral theorem does not hold in higher-dimensional space, this formula cannot be derived analogously to the one-dimensional case. Therefore, this integral formula is derived using [[w:en:Induction (Mathematik)|induction]] from the Cauchy integral formula for disk domains. Using the [[w:en:Multiindex|multi-index]] notation, the formula can be simplified to:
:&lt;math&gt; f(z) = \frac{1}{(2\pi \mathrm{i})^n} \oint_{\partial U} \frac{f(\xi)}{(\xi - z)} , \mathrm{d} \xi &lt;/math&gt;
with &lt;math&gt;\partial U = \partial U_1 \times \cdots \times \partial U_n&lt;/math&gt;.

=== Polycycles ===
Polycycles are defined using a vector of radii, where &lt;math&gt; M := \max_{\xi \in U} |f(\xi)| &lt;/math&gt; and &lt;math&gt; r = (r_1, \ldots, r_n) &lt;/math&gt; is the radius of the polycycle &lt;math&gt; U := \prod_{i=1}^n U_i &lt;/math&gt;.&lt;ref&gt;
for the derivatives of the holomorphic Function &lt;math&gt;f&lt;/math&gt; as well as Cauchy's inequality 
:&lt;math&gt;\left|D^k f(z)\right |\le \frac{M \cdot k!}{r^k},&lt;/math&gt;

== See also ==

*[[cycle]]

*[[Cauchy's Integral Theorem for Cycles]]

*[[null-homologous|zero homologous]]

== References == 
 &lt;references /&gt;

== Literature ==
*Kurt Endl, [[w:de:Wolfgang Luh|Wolfgang Luh]]: ''Analysis.'' Volume 3: ''Function Theory, Differential Equations.'' 6th revised edition. Aula-Verlag, Wiesbaden 1987, ISBN 3-89104-456-9, p. 153, Theorem 4.9.1.
*Wolfgang Fischer, [[w:de:Ingo Lieb|Ingo Lieb]]: ''Function Theory.'' 7th improved edition. Vieweg, Braunschweig, 1994, ISBN 3-528-67247-1, p. 60, Chapter 3, Theorem 2.2 (''Vieweg-Studium. Advanced Mathematics Course'' 47).
[[Category: Function Theory]] [[Category: Theorem (Mathematics)|Cauchy's Integral Formula]]

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      <text bytes="14296" sha1="lbr9f79tr8093fa6tqtxs5t5dxj747o" xml:space="preserve">The '''Cauchy Integral Formula''' (named after [[w:en:Augustin-Louis Cauchy|Augustin-Louis Cauchy]]) is one of the fundamental results of [[w:en:Complex analysis|complex analysis]], a branch of [[w:en:Mathematics|mathematics]]. In its weakest form, it states that the values of a [[w:en:Holomorphic function|holomorphic function]] &lt;math&gt;f&lt;/math&gt; inside a disk are completely determined by its values on the boundary of that disk. A powerful generalization of this is the [[w:en:Residue theorem|Residue theorem]].

== Cauchy Integral Formula for Disks ==
=== Statement ===
Let &lt;math&gt;G \subseteq \mathbb{C}&lt;/math&gt; be open, &lt;math&gt;f\colon G \to \mathbb{C}&lt;/math&gt; holomorphic, &lt;math&gt;z_0 \in G&lt;/math&gt; a point in &lt;math&gt;G&lt;/math&gt;, and &lt;math&gt;U := D_r(z_0) \subset G&lt;/math&gt; a bounded disk in &lt;math&gt;G&lt;/math&gt;. Then for all &lt;math&gt;z \in D_r(z_0)&lt;/math&gt; (i.e., for all &lt;math&gt;z&lt;/math&gt; with &lt;math&gt;|z - z_0| &lt; r&lt;/math&gt;), the following holds: :&lt;math&gt;f(z) = \frac{1}{2\pi\mathrm{i}} \oint_{\partial U} \frac{f(\zeta)}{\zeta - z} \mathrm{d}\zeta&lt;/math&gt; Here, &lt;math&gt;\partial U&lt;/math&gt; denotes the positively oriented curve &lt;math&gt;t \mapsto z_0 + r e^{\mathrm{i}t}&lt;/math&gt; for &lt;math&gt;t \in [0, 2\pi]&lt;/math&gt; along the boundary of the disk &lt;math&gt;U&lt;/math&gt;.
=== Proof 1 ===
For a fixed &lt;math&gt;z \in U&lt;/math&gt;, the function &lt;math&gt;g\colon U\to\mathbb{C}&lt;/math&gt; defined by &lt;math&gt;w\mapsto\tfrac{f(w)-f(z)}{w-z}&lt;/math&gt; for &lt;math&gt;w\neq z&lt;/math&gt; und &lt;math&gt;w\mapsto f'(z)&lt;/math&gt; for &lt;math&gt;w=z&lt;/math&gt;. &lt;math&gt;g&lt;/math&gt; is steadily on  &lt;math&gt;U&lt;/math&gt; and holomorphic on &lt;math&gt;U\setminus\{z\}&lt;/math&gt;.  By the [[w:en:Cauchy Integral Theorem|Cauchy Integral Theorem]], we now have: :&lt;math&gt;0 = \oint_{\partial U} g = \oint_{\partial U}\frac{f(\zeta)}{\zeta-z} \mathrm{d}\zeta - f(z)\oint_{\partial U}\frac{\mathrm{d}\zeta}{\zeta-z}&lt;/math&gt;.

=== Proof 2 ===
The function &lt;math&gt;h\colon U \to \mathbb{C}&lt;/math&gt;, &lt;math&gt;\textstyle w \mapsto \oint_{\partial U} \frac{\mathrm{d}\zeta}{\zeta-w}&lt;/math&gt; is holomorphic with the derivative &lt;math&gt;\textstyle h'(w) = \oint_{\partial U} \frac{\mathrm{d}\zeta}{\left(\zeta-w\right)^2}&lt;/math&gt;, which vanishes since the integrand has an antiderivative (namely &lt;math&gt;\zeta \mapsto -\frac{1}{\zeta-w}&lt;/math&gt;). Therefore, &lt;math&gt;h&lt;/math&gt; is constant, and since &lt;math&gt;h(a) = 2\pi i&lt;/math&gt;, we have &lt;math&gt;h(z) = 2\pi i&lt;/math&gt;.

== Consequences of the Cauchy Integral Theorem ==
The Cauchy Integral Theorem (CIS) leads to the following corollaries:
=== Representation of the Function at the Center of the Disk ===
For every holomorphic function, the function value at the center of a circle is the average of the function values on the circle's boundary. Use &lt;math&gt;\zeta(t) = z_o + r e^{\mathrm{i}t},\ \mathrm{d}\zeta = \mathrm{i} r e^{\mathrm{i}t} \mathrm{d}t&lt;/math&gt;.
Test: :&lt;math&gt; \begin{align} f|{U}(z_o) &amp;= \frac{1}{2\pi \mathrm{i}} \oint{\partial U} \frac{f(\zeta)}{\zeta - z_o} \mathrm{d}\zeta = \frac{1}{2\pi \mathrm{i}} \int_{0}^{2\pi} \frac{f(a + r e^{\mathrm{i}t})}{r e^{\mathrm{i}t}} \mathrm{i} r e^{\mathrm{i}t} , \mathrm{d}t \ &amp;= \frac{1}{2\pi} \int_{0}^{2\pi} f(z_o + r e^{\mathrm{i}t}) , \mathrm{d}t \end{align}&lt;/math&gt;

=== Derivatives - Cauchy Integral Formula - CIF ===
Every holomorphic function is infinitely complex differentiable, and each of these derivatives is also holomorphic. Expressed using the integral formula, this means for &lt;math&gt;|z - z_o| &lt; r&lt;/math&gt; and &lt;math&gt;n \in \mathbb{N}{0}&lt;/math&gt;: :&lt;math&gt;f^{(n)}(z) = \frac{n!}{2\pi \mathrm{i}} \oint{\partial U} \frac{f(\zeta)}{(\zeta - z)^{n+1}} \mathrm{d}\zeta.&lt;/math&gt;
=== Local Developability in Power Series ===
Every holomorphic function can be locally expanded into a [[w:en:Power Series|power series]] for &lt;math&gt;|z - a| &lt; r&lt;/math&gt;.
:&lt;math&gt;f(z) = \sum\limits_{n=0}^\infty \left( \frac{1}{2\pi \mathrm{i}} \oint_{\partial U} \frac{f(\zeta)}{(\zeta - a)^{n+1}} \mathrm{d}\zeta \right) (z - a)^n = \sum\limits_{n=0}^\infty a_n (z - a)^n.&lt;/math&gt;
Using the integral formula for &lt;math&gt;f^{(n)}&lt;/math&gt;, it immediately follows that the coefficients &lt;math&gt;a_n&lt;/math&gt; are exactly the [[w:en:Taylor series|Taylor coefficients]].

=== Estimation of the Taylor Series Coefficients ===
For the coefficients, the following estimate holds when &lt;math&gt;|f(z)| \leq M&lt;/math&gt; for &lt;math&gt;|z - a| &lt; r \ \Leftrightarrow z \in U_r(a)&lt;/math&gt;: :&lt;math&gt;|a_n| \leq \frac{M}{r^n}&lt;/math&gt;
The [[w:en:Liouville's Theorem|Liouville Theorem]] (every [[w:en:Entire Function|holomorphic function bounded on the entire complex plane]] is constant) can be easily proven using the integral formula. This can then be used to easily prove the [[w:en:Fundamental Theorem of Algebra|Fundamental Theorem of Algebra]] (every polynomial in &lt;math&gt;\mathbb{C}&lt;/math&gt; factors into linear factors).
Here's the translation with the specified conditions:

=== Proof 1 ===
The Cauchy integral formula is differentiated partially, allowing differentiation and integration to be swapped:
:&lt;math&gt;\begin{align} f^{(n)}|_{U}(z) &amp; =\frac{\partial^{n}f}{\partial z^{n}}|_{U}(z)=\frac{1}{2\pi\mathrm{i}}\frac{\partial^{n}}{\partial z^{n}}\oint_{\partial U}\frac{f(\zeta)}{\zeta-z}\mathrm{d}\zeta\\  &amp; =\frac{1}{2\pi\mathrm{i}}\oint_{\partial U}f(\zeta)\underbrace{\frac{\partial^{n}}{\partial z^{n}}\frac{1}{\zeta-z}}_{n!/(\zeta-z)^{1+n}}\mathrm{d}\zeta=\frac{n!}{2\pi\mathrm{i}}\oint_{\partial U}\frac{f(\zeta)}{(\zeta-z)^{1+n}}\mathrm{d}\zeta\end{align} &lt;/math&gt;
=== Proof 2a: Cauchy Kernel ===
Developing &lt;math&gt;\frac{1}{\zeta - z}&lt;/math&gt; in the Cauchy integral formula using the geometric series gives (Cauchy kernel):
:&lt;math&gt; \frac{1}{1 - \frac{z - z_o}{\zeta - z_o}} = \sum_{n=0}^{\infty} \left( \frac{z - z_o}{\zeta - z_o} \right)^{n} &lt;/math&gt;

=== Proof 2: Cauchy Kernel - Taylor Series ===
:&lt;math&gt;\begin{align} f|_{U}(z) &amp; =\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{r}(z_o)}\frac{f(\zeta)}{\zeta-z}\mathrm{d}\zeta=\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{r}(z_o)}\frac{f(\zeta)}{\zeta-z_o-(z-z_o)}\mathrm{d}\zeta \\ &amp; {=} \frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{r}(z_o)}\frac{f(\zeta)}{\zeta-z_o}\cdot \frac{1}{1-\frac{z-z_o}{\zeta-z_o}}\mathrm{d}\zeta\, \\ &amp;\overset{|\frac{z-z_o}{\zeta-z_o}|&lt;1}{=} \frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{r}(z_o)}\frac{f(\zeta)}{\zeta-z_o}\sum_{n=0}^{\infty}\left(\frac{z-z_o}{\zeta-z_o}\right)^{n}\mathrm{d}\zeta\\ &amp; =\sum_{n=0}^{\infty}\underbrace{\left(\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{r}(z_o)}\frac{f(\zeta)}{(\zeta-z_o)^{n+1}}\mathrm{d}\zeta\right)}_{a_{n}}(z-z_o)^{n}\end{align}&lt;/math&gt;

=== Proof 2b: Cauchy Kernel ===
Since the geometric series converges uniformly for &lt;math&gt;|z - z_o| &lt; |\zeta - z_o| = r&lt;/math&gt;, one can integrate term by term, i.e., swap the sum and the integral. The development coefficients are:
:&lt;math&gt;\begin{align} a_{n} &amp; =\frac{1}{n!}f^{(n)}|_{U}(z_o)=\frac{1}{2\pi\mathrm{i}}\oint_{\partial U_{r}(a)}\frac{f(\zeta)}{(\zeta-z_o)^{n+1}}\mathrm{d}\zeta\\ &amp; =\frac{1}{2\pi\mathrm{i}}\int_{0}^{2\pi}\frac{f(z_o+re^{\mathrm{i}t})}{(re^{\mathrm{i}t})^{n+1}}\mathrm{i}re^{\mathrm{i}t}\,\mathrm{d}t=\frac{1}{2\pi r^{n}}\int_{0}^{2\pi}f(z_o+re^{\mathrm{i}t})e^{-\mathrm{i}nt}\,\mathrm{d}t\end{align}&lt;/math&gt;

=== Proof 3: Estimation of the Coefficients ===
For the coefficients &lt;math&gt;a_n \in \mathbb{C}&lt;/math&gt;, the following estimate holds. There exists a &lt;math&gt;M &gt; 0&lt;/math&gt; such that &lt;math&gt;|f(z)| \leq M&lt;/math&gt; for &lt;math&gt;|z - z_o| = r&lt;/math&gt;. Then, for &lt;math&gt;n \in \mathbb{N}0&lt;/math&gt;, we have:
:&lt;math&gt;\begin{align} |a_{n}|&amp;=\left|\frac{1}{2\pi r^{n}}\int_{0}^{2\pi}f(z_o+re^{\mathrm{i}t})e^{-\mathrm{i}nt}\,\mathrm{d}t\right|\\ &amp;\leq\frac{1}{2\pi r^n}\int_0^{2\pi}\underbrace{|f(z_o+re^{\mathrm{i} t})|}_{\leq M}\,\mathrm{d}t\leq \frac{M}{r^{n}}\end{align}&lt;/math&gt;

=== Proof 4: Liouville's Theorem ===
If &lt;math&gt;f&lt;/math&gt; is holomorphic on all of &lt;math&gt;\mathbb{C}&lt;/math&gt; and bounded, i.e., &lt;math&gt;|f(z)| = |\sum_{n=0}^{\infty} a_n z^n| \leq M&lt;/math&gt; for all &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;, then, as before, for all &lt;math&gt;r &gt; 0&lt;/math&gt;, we have:
:&lt;math&gt;|a_n| \leq \frac{M}{r^n}&lt;/math&gt;
Since &lt;math&gt;r&lt;/math&gt; was arbitrary, it follows that &lt;math&gt;a_n = 0&lt;/math&gt; for all &lt;math&gt;n \in \mathbb{N}&lt;/math&gt;. Therefore, from the boundedness of &lt;math&gt;f&lt;/math&gt;, we conclude:
: &lt;math&gt;f(z) = a_0&lt;/math&gt;
Thus, every bounded holomorphic function on all of &lt;math&gt;\mathbb{C}&lt;/math&gt; is constant (Liouville's theorem).

=== Example ===
Using the integral formula, integrals can also be computed:
:&lt;math&gt; \oint_{\partial U_2(0)} \frac{e^{2\zeta}}{(\zeta + 1)^4} \mathrm{d}\zeta = \frac{2\pi \mathrm{i}}{3!} \frac{\mathrm{d}^3}{\mathrm{d}z^3} e^{2z} |_{z = -1} = \frac{8 \pi \mathrm{i}}{3 e^2} &lt;/math&gt;

== Cauchy Integral Formula for Cycles ==
A generalization of the integral formula for circular contours is the version for cycles:

Let &lt;math&gt;G \subseteq \mathbb{C}&lt;/math&gt; be a domain, &lt;math&gt;f \colon G \to \mathbb{C}&lt;/math&gt; holomorphic, and &lt;math&gt;\Gamma&lt;/math&gt; a [[w:en:zero homologous|zero homologous]] [[w:en:cycle|cycle]] in &lt;math&gt;D&lt;/math&gt;. Then, for all &lt;math&gt;z \in D&lt;/math&gt; not on &lt;math&gt;\Gamma&lt;/math&gt;, the following integral formula holds:
:&lt;math&gt; n(\Gamma, z) \cdot f(z) = \frac{1}{2\pi \mathrm{i}} \int_\Gamma \frac{f(\zeta)}{\zeta - z} \mathrm{d}\zeta &lt;/math&gt;

Here, &lt;math&gt;n(\Gamma, z)&lt;/math&gt; denotes the [[w:en:winding number|winding number]] or [[w:en:revolution|revolution]] of &lt;math&gt;\Gamma&lt;/math&gt; around &lt;math&gt;z&lt;/math&gt;.

== Cauchy Integral Formula for Polycycles ==
The Cauchy integral formula has been generalized to the multidimensional complex space &lt;math&gt;\mathbb{C}^n&lt;/math&gt;. Let &lt;math&gt;U_1, \ldots, U_n&lt;/math&gt; be disk domains in &lt;math&gt;\mathbb{C}&lt;/math&gt;, then &lt;math&gt; U := \prod_{i=1}^n U_i &lt;/math&gt; is a [[w:en:Polycylinder|Polycylinder]] in &lt;math&gt;\mathbb{C}^n&lt;/math&gt;. Let &lt;math&gt;f \colon U \to \mathbb{C}&lt;/math&gt; be a holomorphic function and &lt;math&gt;\xi \in U&lt;/math&gt;. The Cauchy integral formula is given by
:&lt;math&gt; f(z_1, \ldots, z_n) = \frac{1}{(2\pi \mathrm{i})^n} \oint_{\partial U_n} \cdots \oint_{\partial U_1} \frac{f(\xi_1, \ldots, \xi_n)}{(\xi_1 - z_1) \cdots (\xi_n - z_n)} \mathrm{d} \xi_1 \cdots \mathrm{d} \xi_n &lt;/math&gt;

=== Restrictions in Multidimensional Space ===
Since the Cauchy integral theorem does not hold in higher-dimensional space, this formula cannot be derived analogously to the one-dimensional case. Therefore, this integral formula is derived using [[w:en:Induction (Mathematik)|induction]] from the Cauchy integral formula for disk domains. Using the [[w:en:Multiindex|multi-index]] notation, the formula can be simplified to:
:&lt;math&gt; f(z) = \frac{1}{(2\pi \mathrm{i})^n} \oint_{\partial U} \frac{f(\xi)}{(\xi - z)} , \mathrm{d} \xi &lt;/math&gt;
with &lt;math&gt;\partial U = \partial U_1 \times \cdots \times \partial U_n&lt;/math&gt;.

=== Polycycles ===
Polycycles are defined using a vector of radii, where &lt;math&gt; M := \max_{\xi \in U} |f(\xi)| &lt;/math&gt; and &lt;math&gt; r = (r_1, \ldots, r_n) &lt;/math&gt; is the radius of the polycycle &lt;math&gt; U := \prod_{i=1}^n U_i &lt;/math&gt;.&lt;ref&gt;
for the derivatives of the holomorphic Function &lt;math&gt;f&lt;/math&gt; as well as Cauchy's inequality 
:&lt;math&gt;\left|D^k f(z)\right |\le \frac{M \cdot k!}{r^k},&lt;/math&gt;

== See also ==

*[[cycle]]

*[[Cauchy's Integral Theorem for Cycles]]

*[[null-homologous|zero homologous]]

== References == 
 &lt;references /&gt;

== Literature ==
*Kurt Endl, [[w:de:Wolfgang Luh|Wolfgang Luh]]: ''Analysis.'' Volume 3: ''Function Theory, Differential Equations.'' 6th revised edition. Aula-Verlag, Wiesbaden 1987, ISBN 3-89104-456-9, p. 153, Theorem 4.9.1.
*Wolfgang Fischer, [[w:de:Ingo Lieb|Ingo Lieb]]: ''Function Theory.'' 7th improved edition. Vieweg, Braunschweig, 1994, ISBN 3-528-67247-1, p. 60, Chapter 3, Theorem 2.2 (''Vieweg-Studium. Advanced Mathematics Course'' 47).
[[Category: Function Theory]] [[Category: Theorem (Mathematics)|Cauchy's Integral Formula]]

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&lt;span type=&quot;translate&quot; src=&quot;Kurvenintegral&quot; srclang=&quot;de&quot; date=&quot;12/10/2024&quot; time=&quot;17:04&quot; status=&quot;inprogress&quot;&gt;&lt;/span&gt;


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      <text bytes="14421" sha1="1uo03706z4tn3n6wl0gbk6d7znpez1p" xml:space="preserve">The '''Cauchy Integral Formula''' (named after [[w:en:Augustin-Louis Cauchy|Augustin-Louis Cauchy]]) is one of the fundamental results of [[w:en:Complex analysis|complex analysis]], a branch of [[w:en:Mathematics|mathematics]]. In its weakest form, it states that the values of a [[w:en:Holomorphic function|holomorphic function]] &lt;math&gt;f&lt;/math&gt; inside a disk are completely determined by its values on the boundary of that disk. A powerful generalization of this is the [[w:en:Residue theorem|Residue theorem]].

== Cauchy Integral Formula for Disks ==
=== Statement ===
Let &lt;math&gt;G \subseteq \mathbb{C}&lt;/math&gt; be open, &lt;math&gt;f\colon G \to \mathbb{C}&lt;/math&gt; holomorphic, &lt;math&gt;z_0 \in G&lt;/math&gt; a point in &lt;math&gt;G&lt;/math&gt;, and &lt;math&gt;U := D_r(z_0) \subset G&lt;/math&gt; a bounded disk in &lt;math&gt;G&lt;/math&gt;. Then for all &lt;math&gt;z \in D_r(z_0)&lt;/math&gt; (i.e., for all &lt;math&gt;z&lt;/math&gt; with &lt;math&gt;|z - z_0| &lt; r&lt;/math&gt;), the following holds: :&lt;math&gt;f(z) = \frac{1}{2\pi\mathrm{i}} \oint_{\partial U} \frac{f(\zeta)}{\zeta - z} \mathrm{d}\zeta&lt;/math&gt; Here, &lt;math&gt;\partial U&lt;/math&gt; denotes the positively oriented curve &lt;math&gt;t \mapsto z_0 + r e^{\mathrm{i}t}&lt;/math&gt; for &lt;math&gt;t \in [0, 2\pi]&lt;/math&gt; along the boundary of the disk &lt;math&gt;U&lt;/math&gt;.
=== Proof 1 ===
For a fixed &lt;math&gt;z \in U&lt;/math&gt;, the function &lt;math&gt;g\colon U\to\mathbb{C}&lt;/math&gt; defined by &lt;math&gt;w\mapsto\tfrac{f(w)-f(z)}{w-z}&lt;/math&gt; for &lt;math&gt;w\neq z&lt;/math&gt; und &lt;math&gt;w\mapsto f'(z)&lt;/math&gt; for &lt;math&gt;w=z&lt;/math&gt;. &lt;math&gt;g&lt;/math&gt; is steadily on  &lt;math&gt;U&lt;/math&gt; and holomorphic on &lt;math&gt;U\setminus\{z\}&lt;/math&gt;.  By the [[w:en:Cauchy Integral Theorem|Cauchy Integral Theorem]], we now have: :&lt;math&gt;0 = \oint_{\partial U} g = \oint_{\partial U}\frac{f(\zeta)}{\zeta-z} \mathrm{d}\zeta - f(z)\oint_{\partial U}\frac{\mathrm{d}\zeta}{\zeta-z}&lt;/math&gt;.

=== Proof 2 ===
The function &lt;math&gt;h\colon U \to \mathbb{C}&lt;/math&gt;, &lt;math&gt;\textstyle w \mapsto \oint_{\partial U} \frac{\mathrm{d}\zeta}{\zeta-w}&lt;/math&gt; is holomorphic with the derivative &lt;math&gt;\textstyle h'(w) = \oint_{\partial U} \frac{\mathrm{d}\zeta}{\left(\zeta-w\right)^2}&lt;/math&gt;, which vanishes since the integrand has an antiderivative (namely &lt;math&gt;\zeta \mapsto -\frac{1}{\zeta-w}&lt;/math&gt;). Therefore, &lt;math&gt;h&lt;/math&gt; is constant, and since &lt;math&gt;h(a) = 2\pi i&lt;/math&gt;, we have &lt;math&gt;h(z) = 2\pi i&lt;/math&gt;.

== Consequences of the Cauchy Integral Theorem ==
The Cauchy Integral Theorem (CIS) leads to the following corollaries:
=== Representation of the Function at the Center of the Disk ===
For every holomorphic function, the function value at the center of a circle is the average of the function values on the circle's boundary. Use &lt;math&gt;\zeta(t) = z_o + r e^{\mathrm{i}t},\ \mathrm{d}\zeta = \mathrm{i} r e^{\mathrm{i}t} \mathrm{d}t&lt;/math&gt;.
Test: :&lt;math&gt; \begin{align} f|{U}(z_o) &amp;= \frac{1}{2\pi \mathrm{i}} \oint{\partial U} \frac{f(\zeta)}{\zeta - z_o} \mathrm{d}\zeta = \frac{1}{2\pi \mathrm{i}} \int_{0}^{2\pi} \frac{f(a + r e^{\mathrm{i}t})}{r e^{\mathrm{i}t}} \mathrm{i} r e^{\mathrm{i}t} , \mathrm{d}t \ &amp;= \frac{1}{2\pi} \int_{0}^{2\pi} f(z_o + r e^{\mathrm{i}t}) , \mathrm{d}t \end{align}&lt;/math&gt;

=== Derivatives - Cauchy Integral Formula - CIF ===
Every holomorphic function is infinitely complex differentiable, and each of these derivatives is also holomorphic. Expressed using the integral formula, this means for &lt;math&gt;|z - z_o| &lt; r&lt;/math&gt; and &lt;math&gt;n \in \mathbb{N}{0}&lt;/math&gt;: :&lt;math&gt;f^{(n)}(z) = \frac{n!}{2\pi \mathrm{i}} \oint{\partial U} \frac{f(\zeta)}{(\zeta - z)^{n+1}} \mathrm{d}\zeta.&lt;/math&gt;
=== Local Developability in Power Series ===
Every holomorphic function can be locally expanded into a [[w:en:Power Series|power series]] for &lt;math&gt;|z - a| &lt; r&lt;/math&gt;.
:&lt;math&gt;f(z) = \sum\limits_{n=0}^\infty \left( \frac{1}{2\pi \mathrm{i}} \oint_{\partial U} \frac{f(\zeta)}{(\zeta - a)^{n+1}} \mathrm{d}\zeta \right) (z - a)^n = \sum\limits_{n=0}^\infty a_n (z - a)^n.&lt;/math&gt;
Using the integral formula for &lt;math&gt;f^{(n)}&lt;/math&gt;, it immediately follows that the coefficients &lt;math&gt;a_n&lt;/math&gt; are exactly the [[w:en:Taylor series|Taylor coefficients]].

=== Estimation of the Taylor Series Coefficients ===
For the coefficients, the following estimate holds when &lt;math&gt;|f(z)| \leq M&lt;/math&gt; for &lt;math&gt;|z - a| &lt; r \ \Leftrightarrow z \in U_r(a)&lt;/math&gt;: :&lt;math&gt;|a_n| \leq \frac{M}{r^n}&lt;/math&gt;
The [[w:en:Liouville's Theorem|Liouville Theorem]] (every [[w:en:Entire Function|holomorphic function bounded on the entire complex plane]] is constant) can be easily proven using the integral formula. This can then be used to easily prove the [[w:en:Fundamental Theorem of Algebra|Fundamental Theorem of Algebra]] (every polynomial in &lt;math&gt;\mathbb{C}&lt;/math&gt; factors into linear factors).
Here's the translation with the specified conditions:

=== Proof 1 ===
The Cauchy integral formula is differentiated partially, allowing differentiation and integration to be swapped:
:&lt;math&gt;\begin{align} f^{(n)}|_{U}(z) &amp; =\frac{\partial^{n}f}{\partial z^{n}}|_{U}(z)=\frac{1}{2\pi\mathrm{i}}\frac{\partial^{n}}{\partial z^{n}}\oint_{\partial U}\frac{f(\zeta)}{\zeta-z}\mathrm{d}\zeta\\  &amp; =\frac{1}{2\pi\mathrm{i}}\oint_{\partial U}f(\zeta)\underbrace{\frac{\partial^{n}}{\partial z^{n}}\frac{1}{\zeta-z}}_{n!/(\zeta-z)^{1+n}}\mathrm{d}\zeta=\frac{n!}{2\pi\mathrm{i}}\oint_{\partial U}\frac{f(\zeta)}{(\zeta-z)^{1+n}}\mathrm{d}\zeta\end{align} &lt;/math&gt;
=== Proof 2a: Cauchy Kernel ===
Developing &lt;math&gt;\frac{1}{\zeta - z}&lt;/math&gt; in the Cauchy integral formula using the geometric series gives (Cauchy kernel):
:&lt;math&gt; \frac{1}{1 - \frac{z - z_o}{\zeta - z_o}} = \sum_{n=0}^{\infty} \left( \frac{z - z_o}{\zeta - z_o} \right)^{n} &lt;/math&gt;

=== Proof 2: Cauchy Kernel - Taylor Series ===
:&lt;math&gt;\begin{align} f|_{U}(z) &amp; =\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{r}(z_o)}\frac{f(\zeta)}{\zeta-z}\mathrm{d}\zeta=\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{r}(z_o)}\frac{f(\zeta)}{\zeta-z_o-(z-z_o)}\mathrm{d}\zeta \\ &amp; {=} \frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{r}(z_o)}\frac{f(\zeta)}{\zeta-z_o}\cdot \frac{1}{1-\frac{z-z_o}{\zeta-z_o}}\mathrm{d}\zeta\, \\ &amp;\overset{|\frac{z-z_o}{\zeta-z_o}|&lt;1}{=} \frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{r}(z_o)}\frac{f(\zeta)}{\zeta-z_o}\sum_{n=0}^{\infty}\left(\frac{z-z_o}{\zeta-z_o}\right)^{n}\mathrm{d}\zeta\\ &amp; =\sum_{n=0}^{\infty}\underbrace{\left(\frac{1}{2\pi\mathrm{i}}\oint_{\partial D_{r}(z_o)}\frac{f(\zeta)}{(\zeta-z_o)^{n+1}}\mathrm{d}\zeta\right)}_{a_{n}}(z-z_o)^{n}\end{align}&lt;/math&gt;

=== Proof 2b: Cauchy Kernel ===
Since the geometric series converges uniformly for &lt;math&gt;|z - z_o| &lt; |\zeta - z_o| = r&lt;/math&gt;, one can integrate term by term, i.e., swap the sum and the integral. The development coefficients are:
:&lt;math&gt;\begin{align} a_{n} &amp; =\frac{1}{n!}f^{(n)}|_{U}(z_o)=\frac{1}{2\pi\mathrm{i}}\oint_{\partial U_{r}(a)}\frac{f(\zeta)}{(\zeta-z_o)^{n+1}}\mathrm{d}\zeta\\ &amp; =\frac{1}{2\pi\mathrm{i}}\int_{0}^{2\pi}\frac{f(z_o+re^{\mathrm{i}t})}{(re^{\mathrm{i}t})^{n+1}}\mathrm{i}re^{\mathrm{i}t}\,\mathrm{d}t=\frac{1}{2\pi r^{n}}\int_{0}^{2\pi}f(z_o+re^{\mathrm{i}t})e^{-\mathrm{i}nt}\,\mathrm{d}t\end{align}&lt;/math&gt;

=== Proof 3: Estimation of the Coefficients ===
For the coefficients &lt;math&gt;a_n \in \mathbb{C}&lt;/math&gt;, the following estimate holds. There exists a &lt;math&gt;M &gt; 0&lt;/math&gt; such that &lt;math&gt;|f(z)| \leq M&lt;/math&gt; for &lt;math&gt;|z - z_o| = r&lt;/math&gt;. Then, for &lt;math&gt;n \in \mathbb{N}0&lt;/math&gt;, we have:
:&lt;math&gt;\begin{align} |a_{n}|&amp;=\left|\frac{1}{2\pi r^{n}}\int_{0}^{2\pi}f(z_o+re^{\mathrm{i}t})e^{-\mathrm{i}nt}\,\mathrm{d}t\right|\\ &amp;\leq\frac{1}{2\pi r^n}\int_0^{2\pi}\underbrace{|f(z_o+re^{\mathrm{i} t})|}_{\leq M}\,\mathrm{d}t\leq \frac{M}{r^{n}}\end{align}&lt;/math&gt;

=== Proof 4: Liouville's Theorem ===
If &lt;math&gt;f&lt;/math&gt; is holomorphic on all of &lt;math&gt;\mathbb{C}&lt;/math&gt; and bounded, i.e., &lt;math&gt;|f(z)| = |\sum_{n=0}^{\infty} a_n z^n| \leq M&lt;/math&gt; for all &lt;math&gt;z \in \mathbb{C}&lt;/math&gt;, then, as before, for all &lt;math&gt;r &gt; 0&lt;/math&gt;, we have:
:&lt;math&gt;|a_n| \leq \frac{M}{r^n}&lt;/math&gt;
Since &lt;math&gt;r&lt;/math&gt; was arbitrary, it follows that &lt;math&gt;a_n = 0&lt;/math&gt; for all &lt;math&gt;n \in \mathbb{N}&lt;/math&gt;. Therefore, from the boundedness of &lt;math&gt;f&lt;/math&gt;, we conclude:
: &lt;math&gt;f(z) = a_0&lt;/math&gt;
Thus, every bounded holomorphic function on all of &lt;math&gt;\mathbb{C}&lt;/math&gt; is constant (Liouville's theorem).

=== Example ===
Using the integral formula, integrals can also be computed:
:&lt;math&gt; \oint_{\partial U_2(0)} \frac{e^{2\zeta}}{(\zeta + 1)^4} \mathrm{d}\zeta = \frac{2\pi \mathrm{i}}{3!} \frac{\mathrm{d}^3}{\mathrm{d}z^3} e^{2z} |_{z = -1} = \frac{8 \pi \mathrm{i}}{3 e^2} &lt;/math&gt;

== Cauchy Integral Formula for Cycles ==
A generalization of the integral formula for circular contours is the version for cycles:

Let &lt;math&gt;G \subseteq \mathbb{C}&lt;/math&gt; be a domain, &lt;math&gt;f \colon G \to \mathbb{C}&lt;/math&gt; holomorphic, and &lt;math&gt;\Gamma&lt;/math&gt; a [[w:en:zero homologous|zero homologous]] [[w:en:cycle|cycle]] in &lt;math&gt;D&lt;/math&gt;. Then, for all &lt;math&gt;z \in D&lt;/math&gt; not on &lt;math&gt;\Gamma&lt;/math&gt;, the following integral formula holds:
:&lt;math&gt; n(\Gamma, z) \cdot f(z) = \frac{1}{2\pi \mathrm{i}} \int_\Gamma \frac{f(\zeta)}{\zeta - z} \mathrm{d}\zeta &lt;/math&gt;

Here, &lt;math&gt;n(\Gamma, z)&lt;/math&gt; denotes the [[w:en:winding number|winding number]] or [[w:en:revolution|revolution]] of &lt;math&gt;\Gamma&lt;/math&gt; around &lt;math&gt;z&lt;/math&gt;.

== Cauchy Integral Formula for Polycycles ==
The Cauchy integral formula has been generalized to the multidimensional complex space &lt;math&gt;\mathbb{C}^n&lt;/math&gt;. Let &lt;math&gt;U_1, \ldots, U_n&lt;/math&gt; be disk domains in &lt;math&gt;\mathbb{C}&lt;/math&gt;, then &lt;math&gt; U := \prod_{i=1}^n U_i &lt;/math&gt; is a [[w:en:Polycylinder|Polycylinder]] in &lt;math&gt;\mathbb{C}^n&lt;/math&gt;. Let &lt;math&gt;f \colon U \to \mathbb{C}&lt;/math&gt; be a holomorphic function and &lt;math&gt;\xi \in U&lt;/math&gt;. The Cauchy integral formula is given by
:&lt;math&gt; f(z_1, \ldots, z_n) = \frac{1}{(2\pi \mathrm{i})^n} \oint_{\partial U_n} \cdots \oint_{\partial U_1} \frac{f(\xi_1, \ldots, \xi_n)}{(\xi_1 - z_1) \cdots (\xi_n - z_n)} \mathrm{d} \xi_1 \cdots \mathrm{d} \xi_n &lt;/math&gt;

=== Restrictions in Multidimensional Space ===
Since the Cauchy integral theorem does not hold in higher-dimensional space, this formula cannot be derived analogously to the one-dimensional case. Therefore, this integral formula is derived using [[w:en:Induction (Mathematik)|induction]] from the Cauchy integral formula for disk domains. Using the [[w:en:Multiindex|multi-index]] notation, the formula can be simplified to:
:&lt;math&gt; f(z) = \frac{1}{(2\pi \mathrm{i})^n} \oint_{\partial U} \frac{f(\xi)}{(\xi - z)} , \mathrm{d} \xi &lt;/math&gt;
with &lt;math&gt;\partial U = \partial U_1 \times \cdots \times \partial U_n&lt;/math&gt;.

=== Polycycles ===
Polycycles are defined using a vector of radii, where &lt;math&gt; M := \max_{\xi \in U} |f(\xi)| &lt;/math&gt; and &lt;math&gt; r = (r_1, \ldots, r_n) &lt;/math&gt; is the radius of the polycycle &lt;math&gt; U := \prod_{i=1}^n U_i &lt;/math&gt;.&lt;ref&gt;
for the derivatives of the holomorphic Function &lt;math&gt;f&lt;/math&gt; as well as Cauchy's inequality 
:&lt;math&gt;\left|D^k f(z)\right |\le \frac{M \cdot k!}{r^k},&lt;/math&gt;

== See also ==

*[[cycle]]

*[[Cauchy's Integral Theorem for Cycles]]

*[[null-homologous|zero homologous]]

== References == 
 &lt;references /&gt;

== Literature ==
*Kurt Endl, [[w:de:Wolfgang Luh|Wolfgang Luh]]: ''Analysis.'' Volume 3: ''Function Theory, Differential Equations.'' 6th revised edition. Aula-Verlag, Wiesbaden 1987, ISBN 3-89104-456-9, p. 153, Theorem 4.9.1.
*Wolfgang Fischer, [[w:de:Ingo Lieb|Ingo Lieb]]: ''Function Theory.'' 7th improved edition. Vieweg, Braunschweig, 1994, ISBN 3-528-67247-1, p. 60, Chapter 3, Theorem 2.2 (''Vieweg-Studium. Advanced Mathematics Course'' 47).
[[Category: Function Theory]] [[Category: Theorem (Mathematics)|Cauchy's Integral Formula]]

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* Source: Wikiversity  https://en.wikiversity.org/wiki/Complex%20Analysis/Cauchy's%20Integral%20Theorem%20for%20Disks
* see [[v:en:Wiki2Reveal|Wiki2Reveal]] for the functionality of [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Complex%20Analysis/Cauchy's%20Integral%20Theorem%20for%20Disks&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Cauchy's%20Integral%20Theorem%20for%20Disks&amp;coursetitle=Complex%20Analysis Wiki2Reveal].
&lt;!-- * Next contents of the course are [[]] --&gt;;

=== Translation and Version Control ===

This page was translated based on the following [https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Cauchy-Integralsatz_für_Kreisscheiben source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity:
* Source: [[v:de:Kurs:Funktionentheorie/Cauchy-Integralsatz_für_Kreisscheiben|Cauchy-Integralsatz_für_Kreisscheiben]] - URL: https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Cauchy-Integralsatz_für_Kreisscheiben
* Date: 12/10/2024

&lt;span type=&quot;translate&quot; src=&quot;Kurvenintegral&quot; srclang=&quot;de&quot; date=&quot;12/10/2024&quot; time=&quot;17:04&quot; status=&quot;inprogress&quot;&gt;&lt;/span&gt;

&lt;!-- Translation Reference --&gt;
&lt;noinclude&gt;[[de:Kurs:Funktionentheorie/Cauchy-Integralsatz_für_Kreisscheiben]]&lt;/noinclude&gt;


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|-
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|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/patron|patron]]&lt;/span&gt;&lt;span class=&quot;prop other&quot;&gt;patron index&lt;/span&gt;
|-
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|-
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|class=&quot;props&quot;| &lt;span class=&quot;prop main&quot;&gt;[[Boolf prop/3-ary/dominion|dominion]]&lt;/span&gt;
|-
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|class=&quot;props&quot;| &lt;span class=&quot;prop main nameless&quot;&gt;[[Boolf prop/3-ary/nameless 2|nameless 2]]&lt;/span&gt;
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[[Category:Boolf prop/3-ary| ]]</text>
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  <page>
    <title>User:Atcovi/Health Psychology/Chapter 5 - Diverse Understandings of Stress</title>
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      <comment>/* Introduction */</comment>
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      <text bytes="3049" sha1="dsy1ggbimdjbms5fyubunufncnh1c2d" xml:space="preserve">== 5.1 - What is stress and how do we measure it? ==
'''Introduction'''

* '''Stress''' - An upset of homeostasis (easiest definition). Measured as a stimulus, a response, and as an interaction. Stress can be subjective and different based on a person's personal experiences, but excessive stress leads to poor health outcomes and cause detrimental health consequences, such as heart attacks. 
* Majority of researchers agree the best way to know when a person is stressed is too look at how their ''body responds to a situation.'' Sympathetic nervous system activates? Their stressed! Stress in the early days were mostly biologically defined.
** '''Hans Selye''', in 1956, detected a specific response pattern which animals went through ([[w:General_Adaptation_Syndrome|General Adaptation Syndrome]]). 
** Some saw stress as the &quot;perceived demands on the organism that exceed the resources to meet those demands&quot; (the demands are bigger than the resources). 
* A '''stressor''' is something that disrupts the body's 'homeostatic balance', while the '''stress response''' is the physical and mental response to the stressor. 

'''Measuring Stress'''

* Two broad categories of stress measurement are '''physiological measures''' and '''self-reports'''. A '''self-report questionnaire''' (like the Life Experiences Survey or Social Readjustment Rating Scale [though widely criticized as it bases ''life events'' as 'automatic centers of stress', when that isn't always the case]) is a great way for someone to express if they are stressed or not. Other ways could be measuring the physiological response to stress, such as a blood test or measuring one's heart beat - but even this may not be accurate because other activities, not negative, stressful events, could trigger these physiological response. 
* The '''Hassles Scale''' consists of 117 events and accounts for 'small hassles', like a noisy neighbor, which can add up. Also found to be more accurate than the SRRS for measuring frequency and intensity of headaches/back pains in college students.  
* The '''Urban Hassles Index''' measures stressors that usually affect teenagers in urban environments.
* Parents can benefit from the '''Parenting Daily Hassles Intensity Scale''' and the '''Family Daily Hassles Inventory'''.
* Some other questionnaires shift from one-time events and daily hassles to ''major chronic stressors''. Examples are the '''Gurung 2004 21-item scale''' and the '''Trier Inventory of Chronic Stress'''. 
* The '''Perceived Stress Scale''' hones down on the subjectivity of stress, which differs person by person. One of the most commonly used stress scales in today's world. 

'''Stress Over Time'''

* How are we stressed by certain things? Theorists like Walter Cannon and Robert Sapolsky believed that physiological responses to stress were developed through centuries of evolution. Early stressors were probably '''acute physical stressors''' in response to wild animals and beasts. Humans started to face '''chronic stressors''' as life expectancy increased.</text>
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      <comment>+5.2</comment>
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      <text bytes="3086" sha1="lq8nv1ao0q6pocn1hzgxz3bdja9wmwx" xml:space="preserve">== 5.1 - What is stress and how do we measure it? ==
'''Introduction'''

* '''Stress''' - An upset of homeostasis (easiest definition). Measured as a stimulus, a response, and as an interaction. Stress can be subjective and different based on a person's personal experiences, but excessive stress leads to poor health outcomes and cause detrimental health consequences, such as heart attacks. 
* Majority of researchers agree the best way to know when a person is stressed is too look at how their ''body responds to a situation.'' Sympathetic nervous system activates? Their stressed! Stress in the early days were mostly biologically defined.
** '''Hans Selye''', in 1956, detected a specific response pattern which animals went through ([[w:General_Adaptation_Syndrome|General Adaptation Syndrome]]). 
** Some saw stress as the &quot;perceived demands on the organism that exceed the resources to meet those demands&quot; (the demands are bigger than the resources). 
* A '''stressor''' is something that disrupts the body's 'homeostatic balance', while the '''stress response''' is the physical and mental response to the stressor. 

'''Measuring Stress'''

* Two broad categories of stress measurement are '''physiological measures''' and '''self-reports'''. A '''self-report questionnaire''' (like the Life Experiences Survey or Social Readjustment Rating Scale [though widely criticized as it bases ''life events'' as 'automatic centers of stress', when that isn't always the case]) is a great way for someone to express if they are stressed or not. Other ways could be measuring the physiological response to stress, such as a blood test or measuring one's heart beat - but even this may not be accurate because other activities, not negative, stressful events, could trigger these physiological response. 
* The '''Hassles Scale''' consists of 117 events and accounts for 'small hassles', like a noisy neighbor, which can add up. Also found to be more accurate than the SRRS for measuring frequency and intensity of headaches/back pains in college students.  
* The '''Urban Hassles Index''' measures stressors that usually affect teenagers in urban environments.
* Parents can benefit from the '''Parenting Daily Hassles Intensity Scale''' and the '''Family Daily Hassles Inventory'''.
* Some other questionnaires shift from one-time events and daily hassles to ''major chronic stressors''. Examples are the '''Gurung 2004 21-item scale''' and the '''Trier Inventory of Chronic Stress'''. 
* The '''Perceived Stress Scale''' hones down on the subjectivity of stress, which differs person by person. One of the most commonly used stress scales in today's world. 

'''Stress Over Time'''

* How are we stressed by certain things? Theorists like Walter Cannon and Robert Sapolsky believed that physiological responses to stress were developed through centuries of evolution. Early stressors were probably '''acute physical stressors''' in response to wild animals and beasts. Humans started to face '''chronic stressors''' as life expectancy increased.

== 5.2 - Main Theories of Stress ==</text>
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  <page>
    <title>Complex Analysis/Chain</title>
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      <comment>New resource with &quot;A chain is a formal linear combination of[[Complex Analysis/Trace|Trace of Curve]], we have  == Definition - Chain ==  Let &lt;math&gt;G \subseteq \mathbb C&lt;/math&gt;, let &lt;math&gt;n \in \mathbb N&lt;/math&gt;, and let &lt;math&gt;\gamma_i \colon[a_i, b_i] \to G&lt;/math&gt; be curves in &lt;math&gt;G&lt;/math&gt; and &lt;math&gt;n_i\in \mathbb Z&lt;/math&gt;. Then the formal linear combination &lt;math&gt;\sum_{i=1}^n n_i\gamma_i&lt;/math&gt; is called a chain in &lt;math&gt;\mathbb C&lt;/math&gt;. The set of all chains in &lt;math&gt;G&lt;/math&gt;, which i...&quot;</comment>
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      <text bytes="3642" sha1="eq8qlk3y2w9wy6i7udsn09j87yommre" xml:space="preserve">A chain is a formal linear combination of[[Complex Analysis/Trace|Trace of Curve]], we have

== Definition - Chain == 
Let &lt;math&gt;G \subseteq \mathbb C&lt;/math&gt;, let &lt;math&gt;n \in \mathbb N&lt;/math&gt;, and let &lt;math&gt;\gamma_i \colon[a_i, b_i] \to G&lt;/math&gt; be curves in &lt;math&gt;G&lt;/math&gt; and &lt;math&gt;n_i\in \mathbb Z&lt;/math&gt;. Then the formal linear combination &lt;math&gt;\sum_{i=1}^n n_i\gamma_i&lt;/math&gt; is called a chain in &lt;math&gt;\mathbb C&lt;/math&gt;. The set of all chains in &lt;math&gt;G&lt;/math&gt;, which is naturally an abelian group, is denoted by &lt;math&gt;C(G)&lt;/math&gt;.

== Definition - Trace of a Chain ==
The ''trace'' of a chain &lt;math&gt;\Gamma&lt;/math&gt; is the union of the traces of the individual curves &lt;math&gt;\gamma_i&lt;/math&gt;, i.e.

&lt;center&gt;&lt;math&gt; \mathrm{Trace}(\Gamma) := \bigcup_{i=1}^n \mathrm{Trace}(\gamma_i) &lt;/math&gt;&lt;/center&gt;
==Cycle== 
A chain &lt;math&gt;\Gamma = \sum_{i=1}^n n_i \gamma_i \in C(G)&lt;/math&gt; with &lt;math&gt;\gamma_i \colon[a_i, b_i] \to G&lt;/math&gt; is called a cycle if each point of &lt;math&gt;G&lt;/math&gt; occurs equally often as the starting and ending point of curves in &lt;math&gt;G&lt;/math&gt;, i.e., if

&lt;center&gt;&lt;math&gt; \sum_{i=1}^n n_i |\{i: \gamma_i(a_i) = z \}| = \sum_{i=1}^n n_i|\{i: \gamma_i(b_i) = z\}| &lt;/math&gt;&lt;/center&gt;
holds for every &lt;math&gt;z \in G&lt;/math&gt;.
===Interior and Exterior Region=== 
Let &lt;math&gt;\Gamma&lt;/math&gt; be a cycle in &lt;math&gt;\mathbb C&lt;/math&gt;, with the help of the [[w:en:winding number|winding number]] one can consider a decomposition of &lt;math&gt;\mathbb C&lt;/math&gt; into three parts determined by &lt;math&gt;\Gamma&lt;/math&gt;, namely:

*The image set of &lt;math&gt;\mathrm{Trace}(\Gamma)&lt;/math&gt;

*The ''exterior region'', those points that are not traversed by &lt;math&gt;\Gamma&lt;/math&gt;, i.e. &lt;center&gt;&lt;math&gt; A_\Gamma := {z \in \mathbb C \setminus \mathrm{Trace}(\Gamma) : n(\Gamma, z) = 0}&lt;/math&gt;&lt;/center&gt;

*The ''interior region'' consists of those points that are traversed by &lt;math&gt;\Gamma&lt;/math&gt;, i.e. &lt;center&gt;&lt;math&gt; I_\Gamma := {z \in \mathbb C \setminus \mathrm{Trace}(\Gamma) : n(\Gamma, z) \ne 0}&lt;/math&gt;&lt;/center&gt;
== Page Information ==
This learning resource can be presented as a '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Kurs:Funktionentheorie/Kette&amp;author=Kurs:Funktionentheorie&amp;language=en&amp;audioslide=yes&amp;shorttitle=Chain&amp;coursetitle=Kurs:Funktionentheorie Wiki2Reveal Slide Set]'''.

=== Wiki2Reveal ===

This '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Kurs:Funktionentheorie/Kette&amp;author=Kurs:Funktionentheorie&amp;language=en&amp;audioslide=yes&amp;shorttitle=Chain&amp;coursetitle=Kurs:Funktionentheorie Wiki2Reveal Slide Set]''' was created for the course '''[https://en.wikiversity.org/wiki/Kurs:Funktionentheorie Kurs:Funktionentheorie]'''' using the [https://niebert.github.io/Wiki2Reveal/ Wiki2Reveal Link Generator].
&lt;!--
* The content of the page is based on the following contents:
** [https://en.wikipedia.org/wiki/Kurs:Funktionentheorie/Kette https://en.wikiversity.org/wiki/Kurs:Funktionentheorie/Kette]
--&gt;
* [https://en.wikiversity.org/wiki/Kurs:Funktionentheorie/Kette The page] was created as a document type [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron SLIDE].
* Link to the source in Wikiversity: https://en.wikiversity.org/wiki/Kurs:Funktionentheorie/Kette
* See also further information about [[v:en:Wiki2Reveal|Wiki2Reveal]] and under [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Kurs:Funktionentheorie/Kette&amp;author=Kurs:Funktionentheorie&amp;language=en&amp;audioslide=yes&amp;shorttitle=Chain&amp;coursetitle=Kurs:Funktionentheorie Wiki2Reveal Link Generator].
&lt;!-- * Next content of the course is [[]] --&gt;;

[[Category:Wiki2Reveal]]</text>
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      <text bytes="3642" sha1="g7nna5njuh0ta0oali5jxs8zfnt3ub7" xml:space="preserve">A chain is a formal linear combination of[[Complex Analysis/Trace|Trace of Curve]], we have

== Definition - Chain == 
Let &lt;math&gt;G \subseteq \mathbb C&lt;/math&gt;, let &lt;math&gt;n \in \mathbb N&lt;/math&gt;, and let &lt;math&gt;\gamma_i \colon[a_i, b_i] \to G&lt;/math&gt; be curves in &lt;math&gt;G&lt;/math&gt; and &lt;math&gt;n_i\in \mathbb Z&lt;/math&gt;. Then the formal linear combination &lt;math&gt;\sum_{i=1}^n n_i\gamma_i&lt;/math&gt; is called a chain in &lt;math&gt;\mathbb C&lt;/math&gt;. The set of all chains in &lt;math&gt;G&lt;/math&gt;, which is naturally an abelian group, is denoted by &lt;math&gt;C(G)&lt;/math&gt;.

== Definition - Trace of a Chain ==
The ''trace'' of a chain &lt;math&gt;\Gamma&lt;/math&gt; is the union of the traces of the individual curves &lt;math&gt;\gamma_i&lt;/math&gt;, i.e.

&lt;center&gt;&lt;math&gt; \mathrm{Trace}(\Gamma) := \bigcup_{i=1}^n \mathrm{Trace}(\gamma_i) &lt;/math&gt;&lt;/center&gt;
==Cycle== 
A chain &lt;math&gt;\Gamma = \sum_{i=1}^n n_i \gamma_i \in C(G)&lt;/math&gt; with &lt;math&gt;\gamma_i \colon[a_i, b_i] \to G&lt;/math&gt; is called a cycle if each point of &lt;math&gt;G&lt;/math&gt; occurs equally often as the starting and ending point of curves in &lt;math&gt;G&lt;/math&gt;, i.e., if

&lt;center&gt;&lt;math&gt; \sum_{i=1}^n n_i |\{i: \gamma_i(a_i) = z \}| = \sum_{i=1}^n n_i|\{i: \gamma_i(b_i) = z\}| &lt;/math&gt;&lt;/center&gt;
holds for every &lt;math&gt;z \in G&lt;/math&gt;.
===Interior and Exterior Region=== 
Let &lt;math&gt;\Gamma&lt;/math&gt; be a cycle in &lt;math&gt;\mathbb C&lt;/math&gt;, with the help of the [[w:en:winding number|winding number]] one can consider a decomposition of &lt;math&gt;\mathbb C&lt;/math&gt; into three parts determined by &lt;math&gt;\Gamma&lt;/math&gt;, namely:

*The image set of &lt;math&gt;\mathrm{Trace}(\Gamma)&lt;/math&gt;

*The ''exterior region'', those points that are not traversed by &lt;math&gt;\Gamma&lt;/math&gt;, i.e. &lt;center&gt;&lt;math&gt; A_\Gamma := {z \in \mathbb C \setminus \mathrm{Trace}(\Gamma) : n(\Gamma, z) = 0}&lt;/math&gt;&lt;/center&gt;

*The ''interior region'' consists of those points that are traversed by &lt;math&gt;\Gamma&lt;/math&gt;, i.e. &lt;center&gt;&lt;math&gt; I_\Gamma := {z \in \mathbb C \setminus \mathrm{Trace}(\Gamma) : n(\Gamma, z) \ne 0}&lt;/math&gt;&lt;/center&gt;
== Page Information ==
This learning resource can be presented as a '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Kurs:Funktionentheorie/Kette&amp;author=Kurs:Funktionentheorie&amp;language=en&amp;audioslide=yes&amp;shorttitle=Chain&amp;coursetitle=Kurs:Funktionentheorie Wiki2Reveal Slide Set]'''.

=== Wiki2Reveal ===

This '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Kurs:Funktionentheorie/Chain&amp;author=Kurs:Funktionentheorie&amp;language=en&amp;audioslide=yes&amp;shorttitle=Chain&amp;coursetitle=Kurs:Funktionentheorie Wiki2Reveal Slide Set]''' was created for the course '''[https://en.wikiversity.org/wiki/Kurs:Funktionentheorie Kurs:Funktionentheorie]'''' using the [https://niebert.github.io/Wiki2Reveal/ Wiki2Reveal Link Generator].
&lt;!--
* The content of the page is based on the following contents:
** [https://en.wikipedia.org/wiki/Kurs:Funktionentheorie/Kette https://en.wikiversity.org/wiki/Kurs:Funktionentheorie/Kette]
--&gt;
* [https://en.wikiversity.org/wiki/Kurs:Funktionentheorie/Kette The page] was created as a document type [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron SLIDE].
* Link to the source in Wikiversity: https://en.wikiversity.org/wiki/Kurs:Funktionentheorie/Kette
* See also further information about [[v:en:Wiki2Reveal|Wiki2Reveal]] and under [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Kurs:Funktionentheorie/Kette&amp;author=Kurs:Funktionentheorie&amp;language=en&amp;audioslide=yes&amp;shorttitle=Chain&amp;coursetitle=Kurs:Funktionentheorie Wiki2Reveal Link Generator].
&lt;!-- * Next content of the course is [[]] --&gt;;

[[Category:Wiki2Reveal]]</text>
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      <text bytes="3642" sha1="gook78beu545wz40natdt90m6wbjnbu" xml:space="preserve">A chain is a formal linear combination of[[Complex Analysis/Trace|Trace of Curve]], we have

== Definition - Chain == 
Let &lt;math&gt;G \subseteq \mathbb C&lt;/math&gt;, let &lt;math&gt;n \in \mathbb N&lt;/math&gt;, and let &lt;math&gt;\gamma_i \colon[a_i, b_i] \to G&lt;/math&gt; be curves in &lt;math&gt;G&lt;/math&gt; and &lt;math&gt;n_i\in \mathbb Z&lt;/math&gt;. Then the formal linear combination &lt;math&gt;\sum_{i=1}^n n_i\gamma_i&lt;/math&gt; is called a chain in &lt;math&gt;\mathbb C&lt;/math&gt;. The set of all chains in &lt;math&gt;G&lt;/math&gt;, which is naturally an abelian group, is denoted by &lt;math&gt;C(G)&lt;/math&gt;.

== Definition - Trace of a Chain ==
The ''trace'' of a chain &lt;math&gt;\Gamma&lt;/math&gt; is the union of the traces of the individual curves &lt;math&gt;\gamma_i&lt;/math&gt;, i.e.

&lt;center&gt;&lt;math&gt; \mathrm{Trace}(\Gamma) := \bigcup_{i=1}^n \mathrm{Trace}(\gamma_i) &lt;/math&gt;&lt;/center&gt;
==Cycle== 
A chain &lt;math&gt;\Gamma = \sum_{i=1}^n n_i \gamma_i \in C(G)&lt;/math&gt; with &lt;math&gt;\gamma_i \colon[a_i, b_i] \to G&lt;/math&gt; is called a cycle if each point of &lt;math&gt;G&lt;/math&gt; occurs equally often as the starting and ending point of curves in &lt;math&gt;G&lt;/math&gt;, i.e., if

&lt;center&gt;&lt;math&gt; \sum_{i=1}^n n_i |\{i: \gamma_i(a_i) = z \}| = \sum_{i=1}^n n_i|\{i: \gamma_i(b_i) = z\}| &lt;/math&gt;&lt;/center&gt;
holds for every &lt;math&gt;z \in G&lt;/math&gt;.
===Interior and Exterior Region=== 
Let &lt;math&gt;\Gamma&lt;/math&gt; be a cycle in &lt;math&gt;\mathbb C&lt;/math&gt;, with the help of the [[w:en:winding number|winding number]] one can consider a decomposition of &lt;math&gt;\mathbb C&lt;/math&gt; into three parts determined by &lt;math&gt;\Gamma&lt;/math&gt;, namely:

*The image set of &lt;math&gt;\mathrm{Trace}(\Gamma)&lt;/math&gt;

*The ''exterior region'', those points that are not traversed by &lt;math&gt;\Gamma&lt;/math&gt;, i.e. &lt;center&gt;&lt;math&gt; A_\Gamma := {z \in \mathbb C \setminus \mathrm{Trace}(\Gamma) : n(\Gamma, z) = 0}&lt;/math&gt;&lt;/center&gt;

*The ''interior region'' consists of those points that are traversed by &lt;math&gt;\Gamma&lt;/math&gt;, i.e. &lt;center&gt;&lt;math&gt; I_\Gamma := {z \in \mathbb C \setminus \mathrm{Trace}(\Gamma) : n(\Gamma, z) \ne 0}&lt;/math&gt;&lt;/center&gt;
== Page Information ==
This learning resource can be presented as a '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Kurs:Funktionentheorie/Chain&amp;author=Kurs:Funktionentheorie&amp;language=en&amp;audioslide=yes&amp;shorttitle=Chain&amp;coursetitle=Kurs:Funktionentheorie Wiki2Reveal Slide Set]'''.

=== Wiki2Reveal ===

This '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Kurs:Funktionentheorie/Chain&amp;author=Kurs:Funktionentheorie&amp;language=en&amp;audioslide=yes&amp;shorttitle=Chain&amp;coursetitle=Kurs:Funktionentheorie Wiki2Reveal Slide Set]''' was created for the course '''[https://en.wikiversity.org/wiki/Kurs:Funktionentheorie Kurs:Funktionentheorie]'''' using the [https://niebert.github.io/Wiki2Reveal/ Wiki2Reveal Link Generator].
&lt;!--
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&lt;!-- * Next content of the course is [[]] --&gt;;

[[Category:Wiki2Reveal]]</text>
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      <text bytes="169" sha1="cpkszp4e4fnoeyum3tf00pqjlnmvoyl" xml:space="preserve">{{Welcome}} [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 13:17, 16 December 2024 (UTC)</text>
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      <text bytes="169" sha1="1geuyn4y55v3y2o6vtgvl7v5wr0r96n" xml:space="preserve">{{Welcome}} [[User:RockTransport|RockTransport]] ([[User talk:RockTransport|discuss]] • [[Special:Contributions/RockTransport|contribs]]) 13:18, 16 December 2024 (UTC)</text>
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  <page>
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      <comment>New resource with &quot;== Introduction == '''Chain''' and '''cycle''' are mathematical objects studied in [[w:de:Complex analysis|complex analysis]] but also appear as special cases in [[w:de:Algebraic topology|algebraic topology]]. A chain generalizes a [[w:de:Path (mathematics)|curve]], and a cycle generalizes a closed curve. They are primarily used in integration in complex analysis.  == Definitions == === Chain === A chain on a set &lt;math&gt;G \subset \mathbb{C}&lt;/math&gt; is defined as a finite i...&quot;</comment>
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      <text bytes="8955" sha1="brz3deoxv0zlrcbafbka0lxsn9h86hy" xml:space="preserve">== Introduction ==
'''Chain''' and '''cycle''' are mathematical objects studied in [[w:de:Complex analysis|complex analysis]] but also appear as special cases in [[w:de:Algebraic topology|algebraic topology]]. A chain generalizes a [[w:de:Path (mathematics)|curve]], and a cycle generalizes a closed curve. They are primarily used in integration in complex analysis.

== Definitions ==
=== Chain ===
A chain on a set &lt;math&gt;G \subset \mathbb{C}&lt;/math&gt; is defined as a finite integer linear combination of paths &lt;math&gt;\gamma_1,\ldots, \gamma_k&lt;/math&gt;:
&lt;math&gt;\Gamma := \sum_{i=1}^k n_i\gamma_i \quad n_i \in \mathbb{Z}&lt;/math&gt;.
&lt;math&gt;\gamma_1,\ldots, \gamma_k&lt;/math&gt; are generally continuous [[w:de:Curve (mathematics)|curves]] in &lt;math&gt;G&lt;/math&gt;.

=== Integration over a chain ===
Let &lt;math&gt;f:G \to \mathbb{C}&lt;/math&gt; be integrable, and let &lt;math&gt;\Gamma&lt;/math&gt; be a chain of piecewise continuously differentiable paths (paths of integration) &lt;math&gt;\gamma_1,\ldots, \gamma_k&lt;/math&gt; in &lt;math&gt;G \subset \mathbb{C}&lt;/math&gt;. The integral over the chain &lt;math&gt;\Gamma&lt;/math&gt; is defined by:
:&lt;math&gt;\int_\Gamma f(z) \, dz := \sum_{i = 1}^k n_i \int_{\gamma_i} f(z) \, dz&lt;/math&gt;

=== Definition: Cycle ===
'''Version 1:''' A cycle is a chain &lt;math&gt;\Gamma := \sum_{i=1}^k n_i\gamma_i&lt;/math&gt;, where every point &lt;math&gt;a \in \mathbb{C}&lt;/math&gt; appears as the starting point as many times as it appears as the endpoint of the curves &lt;math&gt;\gamma_i&lt;/math&gt;, taking multiplicities &lt;math&gt;n_i&lt;/math&gt; into account.

'''Version 2:''' A cycle is a chain &lt;math&gt;\Gamma := \sum_{i=1}^k n_i\gamma_i&lt;/math&gt; consisting of closed paths &lt;math&gt;\gamma_1, \ldots, \gamma_k&lt;/math&gt;.

=== Connection Between Version 1 and Version 2 ===
Version 2 is essential for complex analysis. Based on the properties of Version 1, any cycle &lt;math&gt;\Gamma := \sum_{i=1}^k n_i\gamma_i&lt;/math&gt; can be transformed into a chain &lt;math&gt;\hat{\Gamma} := \sum_{i=1}^m \hat{n}_i \hat{\gamma}_i&lt;/math&gt; of closed paths &lt;math&gt;\hat{\gamma}_1, \ldots, \hat{\gamma}_m&lt;/math&gt;.
If the paths &lt;math&gt;\gamma_1, \ldots, \gamma_k&lt;/math&gt; are piecewise continuously differentiable, then the closed paths &lt;math&gt;\hat{\gamma}1, \ldots, \hat{\gamma}m&lt;/math&gt; are also continuously differentiable. For all holomorphic functions &lt;math&gt;f:G \to \mathbb{C}&lt;/math&gt;, it holds that:
&lt;math&gt;\int\Gamma f(z) , dz = \int{\hat{\Gamma}} f(z) , dz&lt;/math&gt;.

=== Trace of a path ===
The '''trace''' of a path &lt;math&gt;\gamma : [a,b] \to G&lt;/math&gt; is defined as:
&lt;math&gt;\operatorname{Trace}(\gamma_i) := \operatorname{Image}(\gamma) := { \gamma(t) ,| , t \in [a,b] }&lt;/math&gt;.

=== Trace of a cycle/chain ===
The trace of a chain &lt;math&gt;\Gamma&lt;/math&gt; is the union of the [[w:de:Image (mathematics)|images]] of its individual curves, i.e.:
&lt;math&gt;\operatorname{Trace}(\Gamma) := \bigcup_{i=1}^N\operatorname{Image}(\gamma_i)&lt;/math&gt;.
If &lt;math&gt;\operatorname{Trace}(\Gamma) \subset \mathbb{C}&lt;/math&gt; is a subset of &lt;math&gt;G \subset \mathbb{C}&lt;/math&gt;, then &lt;math&gt;\Gamma&lt;/math&gt; is called a cycle '''in''' &lt;math&gt;G&lt;/math&gt; if and only if the trace &lt;math&gt;\operatorname{Trace}(\Gamma) \subseteq G&lt;/math&gt; lies in &lt;math&gt;G&lt;/math&gt;.

=== Winding number ===
The '''[[w:de:Winding number (mathematics)|winding number]]''' is defined analogously to that of a closed curve but uses the integral defined above. For &lt;math&gt;z \not\in \operatorname{Trace}(\Gamma)&lt;/math&gt;, it is given by:
&lt;math&gt;n(\Gamma , z) := \frac{1}{2\pi \mathrm{i}} \int_\Gamma \frac{\mathrm{d}\zeta}{\zeta - z} \in \mathbb{Z}&lt;/math&gt;.

=== Interior points of a cycle ===
The '''interior''' of a cycle consists of all points for which the winding number is non-zero:
&lt;math&gt;\operatorname{Int}(\Gamma):={z\in\mathbb{C}\setminus\operatorname{Trace}(\Gamma) : n(\Gamma , z) \neq 0}&lt;/math&gt;.

=== Exterior points of a cycle ===
Analogously, the '''exterior''' is the set of points for which the winding number is zero:
&lt;math&gt;\operatorname{Ext}(\Gamma):={z\in\mathbb{C}\setminus\operatorname{Trace}(\Gamma) : n(\Gamma , z) = 0}&lt;/math&gt;.

=== zero-homologous cycle ===
A cycle is called '''null-homologous''' for a set &lt;math&gt;G\subseteq\mathbb{C}&lt;/math&gt; if and only if the interior &lt;math&gt;\operatorname{Int}(\Gamma)&lt;/math&gt; lies entirely within &lt;math&gt;G&lt;/math&gt;. This is equivalent to the winding number vanishing for all points in &lt;math&gt;\mathbb{C} \setminus G&lt;/math&gt;.

=== Homologous cycles ===
Two cycles &lt;math&gt;\Gamma_1&lt;/math&gt;, &lt;math&gt;\Gamma_2&lt;/math&gt; are called '''homologous''' in &lt;math&gt;G\subseteq\mathbb{C}&lt;/math&gt; if and only if their formal difference &lt;math&gt;\Gamma_1-\Gamma_2&lt;/math&gt; is null-homologous in &lt;math&gt;G&lt;/math&gt;.

== Integral Theorems ==
Chains and cycles are important in complex analysis because, as mentioned, they generalize curve integrals. In particular, the integral over a cycle generalizes the closed curve integral. The [[w:de:Cauchy integral theorem|Cauchy integral theorem]], the [[w:de:Cauchy integral formula|Cauchy integral formula]], and the [[w:de:Residue theorem|residue theorem]] can be proven for cycles.

== Relation to Homology Theory ==
To indicate that chains and cycles are special cases of objects in [[w:de:Homology theory|homology theory]] of algebraic topology, they are sometimes referred to as 1-chains and 1-cycles.&lt;ref&gt;[[w:de:Otto Forster|Otto Forster]]: ''Riemann surfaces'', Springer 1977; English edition: ''Lectures on Riemann surfaces'', Graduate Texts in Mathematics, Springer-Verlag, 1991, ISBN 3-540-90617-7, Chapter 20&lt;/ref&gt;. In algebraic topology, the term 1-cycle or p-cycle is commonly used instead of cycle.&lt;ref&gt;{{Literature| Author=Wolfgang Lück| Title=Algebraic Topology: Homology and Manifolds| Publisher=Vieweg| Year=2005}}&lt;/ref&gt;. Additionally, note that the plural of cycle is &quot;cycles,&quot; while the plural of Zykel is &quot;Zykel&quot; in German.

=== Embedding in Homology Theory ===
The terms chain and cycle are special cases of [[w:de:Mathematical object|objects]] in [[w:de:Topology (mathematics)|topology]]. In [[w:de:Algebraic topology|algebraic topology]], one considers [[w:de:Chain complex|complexes of p-chains]] and constructs [[w:de:Homology group|homology groups]] from them. These groups are [[w:de:Invariant (mathematics)|invariants]] in topology. A very important [[w:de:Homology theory|homology theory]] is that of [[w:de:Singular homology|singular homology groups]].

=== 1-Chain of the Singular Complex ===
A chain, as defined here, is a 1-chain of the [[w:de:Singular complex|singular complex]], which is a specific chain complex. The operator defined in the section on cycles, &lt;math&gt;\partial \colon C_1(X) \to \operatorname{Div}(X)&lt;/math&gt;, is the first [[w:de:Boundary operator|boundary operator]] of the singular complex. The group of divisors is therefore identical to the group of 0-chains. The group of cycles, defined as the kernel of the boundary operator &lt;math&gt;\partial&lt;/math&gt;, is a 1-[[w:de:Chain complex|cycle]] in the sense of the singular complex.

=== Algebraic Topology ===
In algebraic topology, one considers both the kernel of the boundary operator and the image of this operator, constructing a corresponding homology group from these two sets. In the case of the singular complex, one obtains [[w:de:Singular homology|singular homology]]. In this context, the previously defined terms homologous chain and null-homologous chain take on a more abstract meaning.

== See also ==

[[Global Cauchy Integral Theorem]]

[[w:de:Stokes' theorem|Stokes' theorem]]

[[w:de:Smooth function|smooth function]]


== References ==

{{Literature
| Author=Wolfgang Fischer, Ingo Lieb
| Title=Complex Analysis
| Edition=8th
| Publisher=Vieweg
| Location=Braunschweig
| Year=2003
| ISBN=3-528-77247-6
}}

[[w:de:Otto Forster|Otto Forster]]: ''Riemann surfaces'', Springer 1977; English edition: ''Lectures on Riemann surfaces'', Graduate Texts in Mathematics, Springer-Verlag, 1991, ISBN 3-540-90617-7, Chapter 20


== Notes ==
&lt;references /&gt;

[[Category:Complex analysis]]

== Page Information ==
This '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Kurs:Funktionentheorie/Zyklus&amp;author=Kurs:Funktionentheorie&amp;language=de&amp;audioslide=yes Wiki2Reveal slide deck]''' was created for the learning unit '''[https://de.wikiversity.org/wiki/_Kurs:Funktionentheorie Kurs:Funktionentheorie]''''. The link for the [[v:en:Wiki2Reveal|Wiki2Reveal slides]] was generated using the [https://niebert.github.io/Wiki2Reveal/ Wiki2Reveal link generator].

=== Wiki2Reveal ===

[https://de.wikiversity.org/wiki/Kurs:Funktionentheorie/Zyklus The page] was created as a [https://de.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type.

Link to the source on Wikiversity: https://de.wikiversity.org/wiki/Cauchy-Riemann-Differentialgleichungen

See also further information about [[v:en:Wiki2Reveal|Wiki2Reveal]] and under [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Cauchy-Riemann-Differentialgleichungen&amp;author=Kurs:Funktionentheorie&amp;language=de&amp;audioslide=yes Wiki2Reveal link generator].</text>
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      <text bytes="991" sha1="pqfe6x2o78gq5h55ss0ocm97h2kxxz0" xml:space="preserve">== Blocked ==

&lt;div class=&quot;user-block&quot; style=&quot;background:#ffe0e0; border:1px solid pink; padding:0.5em; margin:0.5em auto; min-height: 40px&quot;&gt;
[[File:Modern clock chris kemps 01 with Octagon-warning.svg|40px|left|alt=Stop icon with clock]] You have been '''[[Wikiversity:Blocking policy|blocked]]''' temporarily from editing for abuse of editing privileges. Once the block has expired, you are welcome to make constructive contributions to Wikiversity. If you think you were blocked in error and/or want to contest this block, consider adding the following tag below this notice: &lt;code&gt;&lt;nowiki&gt;{{unblock|your reason here ~~~~}}&lt;/nowiki&gt;&lt;/code&gt;
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If using the above tag does not help, either an administrator may have declined the request after the unblock request was reviewed by an administrator or you may have been blocked from editing your talk page.&lt;/div&gt;

—[[User:Atcovi|Atcovi]] [[User talk:Atcovi|(Talk]] - [[Special:Contributions/Atcovi|Contribs)]] 19:26, 16 December 2024 (UTC)</text>
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      <comment>{{Information
|Description=LIB.2A: Shared Libraries (20241217 - 20241216)
|Source={{own|Young1lim}}
|Date=2024-12-17
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
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{{Information
|Description=LIB.2A: Shared Libraries (20241217 - 20241216)
|Source={{own|Young1lim}}
|Date=2024-12-17
|Author=Young W. Lim
|Permission={{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}
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== Licensing ==
{{self|GFDL|cc-by-sa-4.0,3.0,2.5,2.0,1.0}}</text>
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      <comment>New resource with &quot; == Introduction == This page about ''Laurent Series'' can be displayed as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Laurent%20Series&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Laurent%20Series&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]'''. Single sections are regarded as slides and modifications on the slides will immediately affect the content of the slides. The following aspects of ''Laurent Series...&quot;</comment>
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== Introduction ==
This page about ''Laurent Series'' can be displayed as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Laurent%20Series&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Laurent%20Series&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]'''.
Single sections are regarded as slides and modifications on the slides will immediately affect the content of the slides.
The following aspects of ''Laurent Series'' are considered in detail:
* (1)
* (2)
* (3)

== Objective ==
This learning resource about ''Laurent Series'' in Wikiversity has the objective to ...

== Target Group ==
The target group for ''Laurent Series'' of the learning resource is

The target groups for ''Laurent Series'' of the learning resource are:
* Bachelor/Master students with the subsect
*

== Learning Prerequisites ==
This [[Open Educational Resources|learning resource]] about ''Laurent Series''

== Learning Tasks / Activities ==
Learning activities focus on
* '''()'''
* '''()'''

== References ==
&lt;references/&gt;

=== See also ===
* [[Wiki2Reveal]]
* [[w:en:Laurent Series|Laurent Series]]
* [[Open Educational Resources]]
== Page Information ==
You can display this page as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Laurent%20Series&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Laurent%20Series&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]'''

=== Wiki2Reveal ===
The '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Laurent%20Series&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Laurent%20Series&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator].
&lt;!--
* Contents of the page are based on:
** [https://en.wikipedia.org/wiki/Laurent%20Series https://en.wikiversity.org/wiki/Laurent%20Series]
--&gt;
* [https://en.wikiversity.org/wiki/Laurent%20Series This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type.
* Source: Wikiversity  https://en.wikiversity.org/wiki/Laurent%20Series
* see [[v:en:Wiki2Reveal|Wiki2Reveal]] for the functionality of [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Laurent%20Series&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Laurent%20Series&amp;coursetitle=Complex%20Analysis Wiki2Reveal].
&lt;!-- * Next contents of the course are [[]] --&gt;;

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== Introduction ==
This page about ''Laurent Series'' can be displayed as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Laurent%20Series&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Laurent%20Series&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]'''.
Single sections are regarded as slides and modifications on the slides will immediately affect the content of the slides.
The following aspects of ''Laurent Series'' are considered in detail:
* (1)
* (2)
* (3)

== Objective ==
This learning resource about ''Laurent Series'' in Wikiversity has the objective to ...

== Target Group ==
The target group for ''Laurent Series'' of the learning resource is

The target groups for ''Laurent Series'' of the learning resource are:
* Bachelor/Master students with the subsect
*

== Learning Prerequisites ==
This [[Open Educational Resources|learning resource]] about ''Laurent Series''

== Learning Tasks / Activities ==
Learning activities focus on
* '''()'''
* '''()'''

== References ==
&lt;references/&gt;

=== See also ===
* [[Wiki2Reveal]]
* [[w:en:Laurent Series|Laurent Series]]
* [[Open Educational Resources]]
== Page Information ==
You can display this page as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Laurent%20Series&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Laurent%20Series&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]'''

=== Wiki2Reveal ===
The '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Laurent%20Series&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Laurent%20Series&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator].
&lt;!--
* Contents of the page are based on:
** [https://en.wikipedia.org/wiki/Laurent%20Series https://en.wikiversity.org/wiki/Laurent%20Series]
--&gt;
* [https://en.wikiversity.org/wiki/Laurent%20Series This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type.
* Source: Wikiversity  https://en.wikiversity.org/wiki/Laurent%20Series
* see [[v:en:Wiki2Reveal|Wiki2Reveal]] for the functionality of [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Laurent%20Series&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Laurent%20Series&amp;coursetitle=Complex%20Analysis Wiki2Reveal].
&lt;!-- * Next contents of the course are [[]] --&gt;;

=== Translation and Version Control ===

This page was translated based on the following [https://de.wikiversity.org/wiki/Laurent-Reihe Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity:
* Source: [[v:de:Laurent-Reihe|Laurent-Reihe]] - URL: https://en.wikiversity.org/wiki/Laurent-Reihe
* Date: 12/17/2024

&lt;span type=&quot;translate&quot; src=&quot;Kurvenintegral&quot; srclang=&quot;de&quot; date=&quot;12/17/2024&quot; time=&quot;17:04&quot; status=&quot;inprogress&quot;&gt;&lt;/span&gt;

&lt;noinclude&gt;[[de:Laurent-Reihe]]&lt;/noinclude&gt;

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        <id>2387134</id>
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== Introduction ==
This page about ''Laurent Series'' can be displayed as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Laurent%20Series&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Laurent%20Series&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]'''.
Single sections are regarded as slides and modifications on the slides will immediately affect the content of the slides.
The following aspects of ''Laurent Series'' are considered in detail:
* (1)
* (2)
* (3)

== Objective ==
This learning resource about ''Laurent Series'' in Wikiversity has the objective to ...

== Target Group ==
The target group for ''Laurent Series'' of the learning resource is

The target groups for ''Laurent Series'' of the learning resource are:
* Bachelor/Master students with the subsect
*

== Learning Prerequisites ==
This [[Open Educational Resources|learning resource]] about ''Laurent Series''

== Learning Tasks / Activities ==
Learning activities focus on
* '''()'''
* '''()'''

== References ==
&lt;references/&gt;

=== See also ===
* [[Wiki2Reveal]]
* [[w:en:Laurent Series|Laurent Series]]
* [[Open Educational Resources]]
== Page Information ==
You can display this page as '''[https://niebert.github.io/Wiki2Reveal/wiki2reveal.html?domain=wikiversity&amp;title=Laurent%20Series&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Laurent%20Series&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]'''

=== Wiki2Reveal ===
The '''[https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Laurent%20Series&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Laurent%20Series&amp;coursetitle=Complex%20Analysis Wiki2Reveal slides]''' were created for the '''[https://en.wikiversity.org/wiki/Complex%20Analysis Complex Analysis]'''' and the Link for the [[v:en:Wiki2Reveal|Wiki2Reveal Slides]] was created with the [https://niebert.github.io/Wiki2Reveal/ link generator].
&lt;!--
* Contents of the page are based on:
** [https://en.wikipedia.org/wiki/Laurent%20Series https://en.wikiversity.org/wiki/Laurent%20Series]
--&gt;
* [https://en.wikiversity.org/wiki/Laurent%20Series This page] is designed as a [https://en.wikiversity.org/wiki/PanDocElectron-Presentation PanDocElectron-SLIDE] document type.
* Source: Wikiversity  https://en.wikiversity.org/wiki/Laurent%20Series
* see [[v:en:Wiki2Reveal|Wiki2Reveal]] for the functionality of [https://niebert.github.io/Wiki2Reveal/index.html?domain=wikiversity&amp;title=Laurent%20Series&amp;author=Complex%20Analysis&amp;language=en&amp;audioslide=yes&amp;shorttitle=Laurent%20Series&amp;coursetitle=Complex%20Analysis Wiki2Reveal].
&lt;!-- * Next contents of the course are [[]] --&gt;;

=== Translation and Version Control ===

This page was translated based on the following [https://de.wikiversity.org/wiki/Laurent-Reihe Wikiversity source page] and uses the concept of [[Translation and Version Control]] for a transparent language fork in a Wikiversity:
* Source: [[v:de:Laurent-Reihe|Laurent-Reihe]] - URL: https://de.wikiversity.org/wiki/Laurent-Reihe
* Date: 12/17/2024

&lt;span type=&quot;translate&quot; src=&quot;Kurvenintegral&quot; srclang=&quot;de&quot; date=&quot;12/17/2024&quot; time=&quot;17:04&quot; status=&quot;inprogress&quot;&gt;&lt;/span&gt;

&lt;noinclude&gt;[[de:Laurent-Reihe]]&lt;/noinclude&gt;

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    <title>Category:Theorem (Mathematics)</title>
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        <id>2387134</id>
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      <comment>New resource with &quot;This category contains theorems on Mathematics.&quot;</comment>
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      <contributor>
        <username>Bert Niehaus</username>
        <id>2387134</id>
      </contributor>
      <comment>New resource with &quot;This category contains all learning resources about [[Complex Analysis]].&quot;</comment>
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      <text bytes="73" sha1="3xod92nmx07smrb9ovdxnxyhw29xii9" xml:space="preserve">This category contains all learning resources about [[Complex Analysis]].</text>
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      <contributor>
        <username>Bert Niehaus</username>
        <id>2387134</id>
      </contributor>
      <minor />
      <comment>added [[Category:Mathematics]] using [[Help:Gadget-HotCat|HotCat]]</comment>
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      <text bytes="99" sha1="c4e9zqhnluinw884ipas7fydx5yxjhj" xml:space="preserve">This category contains all learning resources about [[Complex Analysis]].

[[Category:Mathematics]]</text>
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